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1. 0 333333 15 751313 The result is conflicting one pair agrees and the other does not a matter sometimes resolved by saying that division distributes to the left but not to the right It is simpler and clearer to note that the monadic function a amp does not distribute over addition but that the function amp a does We will hereafter speak only of the distributivity of monadic functions For example double and halve both distribute over addition Exercises 1 Does 8a distribute over subtraction Test your assertion 2 Repeat the experiments of this section using conformable that is equal in number of items lists a b and c 3 Repeat the experiments of this section using conformable tables a B and c 4 Experiment with the dyadic cases of the functions g and gaf for various values of the proverbs and g such as f and g and state clearly the effects of the conjunctions and amp b f g cis equivalent to f b g c and b f amp g c is equivalent to g b f g c 5 Comment on the assertion that the equivalence of g and gef is a test of the distributivity of over g 6 Experiment with the conjunction dtest 12 x y y amp x in testing for distributivity Include dtest andb amp 3 dtest candb 38 dtest c B Linearity A function that distributes over addition is said to be inear Linear functions prove to be important in almost every
2. 112 15 11 112161 11361 2361 We first verify that 2 3 6 1 represents the correct sum d 2 3 61 10 d 10 s10 365 1996 2361 2361 2361 The reason that the representation d is preferred is that its items can be simply written side by side to give the normal decimal form whereas the items of s10 would give the quite different result 1121511 Similar remarks apply to bases other than 10 Exercises 1 Perform the carries on the base 8 sum d8 that is 3 12 6 9 2 Enter x 4 1000 to obtain four random integers less than 1000 Then obtain their base 10 represetations sum them and perform the carries 40 Exploring Math necessary to obtain a normalized representation Verify the correctness of the final results 3 Repeat Exercise 2 for bases other than 10 4 The method for adding multi digit decimal numbers commonly taught requires a sequence of carries interleaved with the additions whereas the method used here first performs all additions and then performs the carries Which is the least error prone Which is the easier to re check by repeating all or part ofthe process 5 Give a clear statement in English of the carrying or normalization process commonly taught Include the case of bases other than 10 as well as the case where a carry occurs from the leading position thus increasing the number of items in the list As suggested in the last exercise the hand procedure for normalization can
3. g 15 b 4 s 3 n 7 h b s i n An AP beginning at b with steps of size s h 4 7 10 13 16 19 22 h h 2 13 13 13 13 13 13 13 b s n 1 2 13 n b s n 1 2 91 h 91 Exercises Write formal proofs for each of the foregoing results 7 Define a function such that b s n gives the mean of the arithmetic progression beginning at b and continuing with increments s for a total of n items B Inductive Proof An inductive proof of the equivalence of two functions proceeds by first assuming that they are equal for some unstated value of the integer argument n and using that assumption called the induction hypothesis to prove that they are therefore equal for the next argument n 1 It is then shown that they are indeed equal for some specific argument n k It therefore follows that they are equal for all values k k 1 k 2 and so on without limit For example Chapter 9 Proofs 57 ssq i gt 0 ssq 5 Sum of squares of first 6 non negative integers 55 ssq i 6 015 14 30 55 Using rational constants such as 2r6 for 2 6 we then define a putative equivalent function g adopt the induction hypothesis that nis equal to g n and use it to prove that n 1 equals g n 1 g 1r6 amp 3r6 amp amp 2 2r6 amp amp 3 ssq n 1 i gt n tl Definition of ssq i gt n n 1 Sum of first terms plus last term ssq n n 1 Definition of ssq g n n 1
4. to the values x With a left argument less than x the Vandermonde function v produces non square power tables as illustrated below x 1 2 3 4 1 amp V 2 amp V 3 amp V 4 amp V CV x Br 1 1 1 1 1 11 1 1i 1 1 2 1 2 4 1 2 4 8 12 4 8 I1l1 311 3 911 3 9 27 1 3 9 27 1l1 4 1 4 16 1 4 16 64 1 4 16 64 Although these matrices are not square they may be used with the generalized inverse function denoted by 3 as illustrated below 3Vx 2 25 0 75 1 25 0 75 _1 55 1 15 1 35 _0 95 0 25 0 25 0 25 0 25 2 VX 1 0 5 0 0 5 0 3 0 1 0 1 0 3 f c 1 3 3 1 amp p vc3 3 V x mp f x 11 5 _13 7 10 5 vc3 p x 8 3 26 1 64 9 124 7 cp x 8 27 64 125 The matrix product 3 V x mp f x used above can be written more simply as a matrix divide by using the dyadic case of the function Thus x 3 V x 11 5 13 7 10 5 Finally we define a conjunction FIT such thatn FIT x gives an n element list of coefficients that fits the function at the points x Thus 94 Exploring Math FIT 2 y x amp V 3 FIT 38V c 3 FIT y 0 1 1 7 cp y 1 00133 1 10388 1 22004 1 3498 1 49317 1 65015 1 82073 y 1 1 10517 1 2214 1 34986 1 49182 1 64872 1 82212 1 3 3 1sp d 3 FIT fx dp x 8 3 26 1 64 9 124 7 fx 8 27 64 125 Exercises 30 Experiment with the conjunction FIT for various values of its par
5. 70 Exploring Math plot sin sin cos x 06 0 4 SET gt SSS 8 RR Ih N AIS Aw SH A 4 AS N uy ALA OG e ww voy IN N EHEN A Wa 05 N NER BOX WML RR N Fk Chapter 11 Coordinates and Visualization 71 surface plot sin grid 0 3 30 73 Chapter 12 Linear Functions That wholly consisted of lines like these C S Calverley A Distributivity The properties of commutativity and associativity introduced in Chapters 3 and 9 concerned a single function the important property of distributivity concerns a pair of functions It is commonly treated as a relation between two dyadic functions as illustrated below abe 345 Assign the names aandbandc a b c 345 d a b c 27 le a b a c 27 The general equivalence of the results d and e is expressed by saying that times distributes over addition However this distributivity might equally be expressed with the sum as the left argument of times as follows btc a 27 g b a c a 27 Times also distributes over subtraction a fact that may be illustrated as follows a b c a b a c b c a b a c a 1_31_31_31_31 Does division distribute over addition It can be tested as follows 74 Exploring Math a b c a b axXc b c a b a c a
6. w SPOT sort w table i w A w table sort table 5 A table with more rows than columns may be displayed more compactly by transposing it Try the following transpose transpose table The function A applies to lists of numbers as well as to lists of letters words and when applied to lists such as i 3 and i 4 produces tables that show its behaviour more clearly The following experiment uses the link function to box tables and link them together for more convenient comparison i i 24 i A SPOT i A ABCD i A 0123 I I I I I l I l l I I l l l l SPOT ABCD 0 SPTO ABDC 0 SOPT ACBD 0 SOTP ACDB 0 STPO ADBC 0 STOP ADCB 0 PSOT BACD 1 PSTO BADC 1 POST BCAD 1 POTS BCDA 1 PTSO BDAC 1 PTOS BDCA 1 OSPT CABD 2 OSTP CADB 2 OPST CBAD 2 OPTS CBDA 2 OTSP CDAB 2 OTPS CDBA 2 TSPO DABC 3 TSOP DACB 3 TPSO DBAC 3 NNFPRPOOWWRFRFPOOWWNHNOOWWNHNF EH RFP ONONFRFPOWOWHFNOWOWNNFWEWD le TPOS DBCA 3 ol TOSP DCAB 3 1 TOPS DCBA 3 ol B Proofs Although proofs are an important and many would say the essential part of mathematics we will spend little time on them in this book In introducing his book Proofs and Refutations The Logic of Mathematical Discovery 4 Imre Lakatos makes the following point Its modest aim is to elaborate the
7. 105210 042 054 056 105310 053 056 0359 44 22 oOOoooooo fo fo 100 Exploring Math T 1 2 3 O51 0j2 053 Roots Sa sy ea a na Ess HF SF FE SF en I 11 2 3 051 0j2 053 Sa asa a la a EHE BESSER SEHE m SEES SEIFE FIIR oe SE HE 1 2 3 051 052 03531 1 211 1 41421 1 73205 0 70710750 707107 1j1 1 2247451 22474 I 311 1 25992 1 44225 0 86602550 5 1 09112j0 629961 1 2490250 721125 1053111 0 7692395_0 638961 0 4548325j_0 890577 4 81048 3 700415j_3 07371 2 18796j_4 2841 1053211 0 940542j_0 339677 0 852887j 0 522096 2 19328 2 062875j_0 745007 1 870625 _1 1451 105311 0 973427j_0 228999 0 933693j_0 358074 1 68809 1 64323j_0 386571 1 57616j_0 604461 b antaa ee eae ae SO re ei ee Do a at ee re SS ee er ee a eee 1 Exercises 1 Comment on the foregoing tables including the two part representation that appears to be used for each complex number 2 Enter e T c and comment on the results 3 Study the tables for other functions such as and and perhaps even and and and Two part representations for individual numbers are not uncommon e The result of 36 4 is represented as 9 25 using an integer part anda fractional part joined by a dot e The result of 23 10 5 can also be represented as 23e5 using a factor and an exponent joined by the letter e e The rational 2 3 can be represented as 2r3 using a numerator and denominator joined by the letter r e Two pi cubed 2 0 1 3 can be represented as 2p3
8. 14 63 exponent 102 exponential 24 89 90 91 112 113 114 115 Exponential Family 114 Extended Topics 122 factor 80 102 factorial 12 19 24 49 89 Fahrenheit 117 falling factorials 122 false 20 Fibonacci series 91 Finance 122 Fractals 7 125 fractional part 102 function 2 3 5 6 10 12 13 19 24 25 27 30 31 34 37 38 40 41 42 43 44 45 46 47 49 50 51 52 53 58 59 60 61 66 68 75 76 77 78 79 80 81 83 84 85 87 88 89 90 91 92 93 94 95 96 97 98 99 102 104 105 106 109 110 111 112 113 114 115 Function Tables 17 generalized inverse 95 geometric figures 65 80 geometric progression 58 gerund 50 51 gopen 67 70 gpolygon 67 grammar 23 24 27 63 Grammar 23 24 graph 65 66 67 68 89 91 Greatest Common Divisor 20 guesses 14 Help 122 help menu 63 84 heron 66 Heron s formula 66 68 hierarchical rules 24 hyperbolic 92 113 114 115 identities 103 104 identity 11 49 56 79 90 Identity 5 98 imaginary 92 100 101 102 104 107 114 115 Index 127 improve 14 28 95 indexing 52 83 Indo European 35 125 induction hypothesis 59 110 induction hypothesis 59 INDUCTIVE PROOF 59 Infinite rank 50 Inner Product 78 integer part 102 integers 3 9 12 40 49 59 65 98 99 interpreted 24 51 52 intervals 120 inverse 10 38 51 78 93 95 97 98 99 106 Inverse 117 Inv
9. Hint Begin with the coefficients c 1 1 1 1 1 1 and apply the function de to it Pursuing the idea suggested in the exercise we have c 1 1 1 1 1 1 dce c 12345 Since the second element of the derivative de c is twice the value of the corresponding element of c we replace the third element by one half its value to compensate c 1 1 1r2 1 1 1 dce c 11345 Since the third element of de c is now six times its required value of one half we replace the fourth element of c by 1x6 and so on de c 1 1 1r2 1r611 110 545 dc c 1 1 1r2 1r6 1r24 1 110 5 0 1666667 5 dc c 1 1 1r2 1r6 1r24 1r120 1 1 0 5 0 1666667 0 04166667 dc dce c 1 1 0 5 0 1666667 It should now be clear that the coefficients are the reciprocal factorials c i 6 1 1 0 5 0 1666667 0 04166667 0 008333333 Chapter 17 Calculus 111 dce c 1 1 0 5 0 1666667 0 04166667 ce i Coefficients for exponential ce 6 1 1 0 5 0 1666667 0 04166667 0 008333333 ce 10 p x i 4 Ten term approximation to exponential 1 2 71828 7 38871 20 0634 NX 1 2 71828 7 38906 20 0855 We have in effect defined the exponential as that function which satisfies i e is the solution of an equation that requires it to equal its own derivative We may write such equations more clearly in terms of the following derivative adverb D 0 D 1 The scalar first derivative adverb 183 D The derivative of the cube 3 amp amp 2 0 183 D x i 6 Applied to an argu
10. I i 4 1000 4 3 4 5 4 2 10 4 2 10 010012 5 2 5 3 5 1113 5 11 10 0 1 0 1 71 0 0 4 2 0 3 4 2 03 1600211013 115 5 0145 5 01 L I amp mp x 5 _32 18 _37 Lx 5 _32 18 _37 Exercises 8 Using the result of14 x from Exercise 7 try to determine by hand the value of the matrix m such that msmp x gives the same result Compare your result with l L4 I i 4 9 Compare the results of the function m amp mp derived in Exercise 8 with the result of 24 when applied to the argument z 2 7 1 8 2 8 10 Repeat Exercise 8 for the function L6 11 Repeat Exercise 8 for the function L3 1 The error produced in Exercise 9 illustrates the fact that the domain of the matrix product representation of a linear function is restricted to arguments of a specific number of items even though the linear function from which it is derived has a wider domain E Why The Name Linear Why is a function that distributes over addition called linear We will attempt to answer this by applying an arbitrary linear vector function to geometric figures beginning with the right angled and isosceles triangles of Chapter 11 is 3 4 9 4 6 8 rt 3 4 9 4 9 7 Jm 2 2 10 17 45 78 Exploring Math mp L m mp 1 rt L rt We may plot these resulting triangles by hand or by the methods of Chapter 11 to try to assess the effec
11. In typing the examples on your computer enter only the part in Courier followed by pressing the Enter key but do not enter anything that appears in Roman Thus 3 2 Addition 5 three 3 Assign the name three to 3 three 2 Use the assigned name in a sentence 5 b 2 b b 4 In experiments on a sequence of numbers it will be easier to make the entries and to compare the results if we treat them as a list This may be illustrated as follows 2 0 0 2 1 2 2 2 4 2 0 1 2 3 4 5 0246810 a 0 1 2 3 4 5 2 a 02 46 8 10 ata 02 46 8 10 Comparisons can be shown more clearly by using the equals function as follows 2 a ata Chapter 1 Exploration 3 111111 a 2 The list a to the power 2 that is the square 0149 16 25 a a 0 1 4 9 16 25 a 3 a a a The cube equals a product of three factors 111111 Lists of integers whole numbers are so useful that a special function is provided for making them Enter the following expressions and comment on the results i 6 The first six non negative integers whole numbers 012345 a i 6 Read aloud as a is the list i 6 b 6 The integers in repeatable random order b 512430 atb 524775 a b O 1 4 12 120 2 a The even numbers divisible by 2 02 46 8 10 14 2 a The odd numbers 1357911 a b 011000 As shown by the last result the lists a and b are not equal but they are similar in the sense that one can be obtained from the other by shuffling
12. Induction hypothesis 1r6 n 3r6 n 2 2r6 n 3 n 1 Definition of g 1r6 n 3r6 n 2 2r6 n 3 1 2 n n 2 1r6 n 1 3r6 n 1 2 2r6 n 1 3 g n 1 Definition of g The lines of the foregoing proof that are not annotated concern the use of manipulations from elementary algebra including the expansion of the square and the cube of the sum n 1 The inductive proof may now be completed by showing that the functions are equal for the argument 0 Exercises 8 Enter n 6 and then enter the lines of the foregoing proof to verify that they each give the same result It is advisable to enter such a sequence in a text or script file then execute it observe the result and return to the script file to correct any errors and re try To open the script file hold down the control key and press n to execute it hold down both the control and shift and press w to see the result switch to the execute window by holding down control and pressing the tab key return to the script window by the same action 9 Define the function s i gt and an equivalent function t that does not use summation Give an inductive proof that they are equivalent A recursive definition of a function provides a clear statement of the value of f n 1 in terms of the value of n this fact is obviously valuable in the construction of an inductive proof But how does one find a function such as g This matter will be treated in Chapter 1
13. On the other hand if grammar is important why was it not treated first In learning our native language we spend years at it and become quite proficient before we even hear of grammar However grammar becomes important for more advanced use of the language in clear writing and speaking Moreover the teaching of grammar relies on many examples of the use of the language that would not be familiar to a beginner Similarly more advanced and independent writing in J will require knowledge of its grammar Moreover we will find it helpful to refer to sentences from earlier chapters to illustrate and motivate discussions of the grammar In learning a second language a student has the advantage of already appreciating the purposes and value of language as well as some knowledge of grammar from her native tongue On the other hand one may be seriously misled by such knowledge and the student is sometimes best advised to forget her native language as much as possible one may know too many things that are not true The beginner in J will already know much of two relevant languages English and Mathematical Notation to be referred to as MN The knowledge of English grammar is very helpful especially when we recognize certain analogies between e English verbs action words and functions such as and and e Nouns on which verbs act and the arguments such as 3 and 4 and sToP to which functions apply e Pronouns such as a and b and
14. br bv10i b t ar br t 3 27 27 18 6 54 54 36 5 45 45 30 This table of products may now be summed to collect those corresponding to the same powers of ten that is diagonally as follows s 3 27 6 27 54 5 18 54 45 36 45 30 s 3 33 86 117 81 30 10 s a b 728540 728540 This may also be expressed by using the oblique adverb which applies its function argument to each of the diagonals Thus Js t 3 33 86 117 81 30 Exercises 10 Carry out by hand the process defined by ar br for various values of ar and br and test the correctness of the resulting products 11 Experiment with the expression lt ar br to get a clear view of the behaviour of the oblique adverb 12 Define and test a function TIMES such that ar TIMES br gives the standard decimal representation of the product of numbers whose decimal representations are ar and br Chapter 7 Decimal and Other Number Systems 45 A clearer view of the justification for the diagonal sums used in the expression t can be obtained by producing a table of powers of ten which multiplied by t gives products weighted by the appropriate powers of ten a 365 b 1996 ar bvl0i a br bv10i b t ar br ea i ar eb i br exp ea eb wp 10 exp wpt t wp wpt 300000 270000 27000 1800 60000 54000 5400 360 5000 4500 450 30 wpt 728540 a b 728540 TIMES N ar TIMES br 728540 10 ar TIMES br a b
15. dc 1 i For example c 6 5 4 3 li c 0123 c i c 0589 1 c i c 589 d dc c 589 c amp p d amp p x i 7 6 18 56 138 282 506 828 5 22 57 110 181 270 377 Exercises 4 Use the fact that the polynomial 0 0 0 1 amp p is equivalent to the cube function to compare the use of the function de with the derivatives of powers obtained in the preceding section Compare de 0 0 0 1 amp p x i 7 with3 D x 5 Comment on the polynomial de de c amp p This is the second derivative of c amp p that is the rate of change of the rate of change For example if c amp p gives the position of a vehicle then de c sp gives its speed and de de c amp p gives its acceleration 110 Exploring Math D Differential Equations Many important functions are simply related to their rates of change their first or second derivatives For example capital invested at compound interest increases at a rate proportional to its value and the exponential or growth function denoted by increases at a rate equal to itself In other words the exponential is equal to its derivative Is there a polynomial with this property Clearly not since the derivative of a polynomial is of lower degree possessing one less term However it is possible to define a power series having the desired property Exercises 6 Try to develop a rule or function to generate the coefficients of a power series that equals its derivative
16. to multiplication in the sense illustrated below 2 8 _16 Chapter 15 Arithmetic 97 _2 8 38 2 More precisely if i is an integer then the functions si and amp i are inverse i 8 amp i 2 _16 amp i amp i 2 Again amp i is not a proper inverse because it may lead out of the class of integers producing the class of rationals For example amp i 2 0 25 Exercises 6 Illustrate the fact that the rationals are closed under multiplication and division D Irrational Numbers The square function is closed on the rationals and the square root provides an inverse For example r 3 5 0 6 r 0 36 S r 0 6 Again is not a proper inverse because there is no rational whose square is 2 and the result is to introduce a further class of irrationals Because there is at least one rational between any pair of distinct rationals their average it might seem impossible that there could be any numbers that are not rational However the school of Pythagoras produced a rather straightforward argument to show that the square root of 2 the length of the hypotenuse of a right triangle with sides of unit length is not a rational E Complex Numbers Because there is no rational whose square is negative the square root applied to a negative argument leads to the further class of complex numbers Thus 98 Exploring Math 1 051 Ja i 6 012345 Sia 0 1 1 41421 1 73205 2 2 23607 Si a
17. 11 1210 00 0 0000111 1310 0 0 O 0000011 1410 0 0 O 0000001 1510 0 0 O 000000 0 lt table a table a gt table a Relations Il012345 I 101234 5 101234 5 IOIO 1 1 1 1 1 01 0000 010000000 11lo0 1 1 1 1 1 0 1 0 O O 0111100000 1210 0011211210 01000I211 10 0 0 O 1310 0 0 O 1 1 3i0 0 O 1 0 0 3 1 1 1 000 1410 0 0 O O 11 4 10 0 0 O 1 0 4 1 111700 1510 0 0 0 O 0I5 0 O O O O 1 5 111110 table a table a Power and outof 101 2 3 4 5 101234 5 10110 0 0 o0 0jol 11111 1 j1 12 1 1 1 1 111101234 5 1211 2 4 8 16 32 2 0 013610 1311 3 9 27 81 2431310001410 jal 4 16 64 256 10241410 0001 5B 1511 5 25 125 625 312515100000 1 table a Roots 20 Exploring Math 1010 1 je l j1 0 1 2 3 4 5 1210 1 1 41421 1 73205 2 2 23607 1310 1 1 25992 1 44225 1 5874 1 70998 4 0 1 1 18921 1 31607 1 41421 1 49535 1510 1 1 1487 1 24573 1 31951 1 37973 Exercises Produce and examine bordered tables for the following functions lt a 2 lt gt Produce and examine bordered tables for the following commuted functions Produce and examine bordered tables
18. 8 0 1 0 _0 1666667 0 0 008333333 0 _0 0001984127 ec cos t i 8 10 _0 5 0 0 04166667 O _0 001388889 0 The power series for an ordinary polynomial that is one with a finite list of coefficients ends with non significant zeros but the series for a transcendental function continues with non zero terms However the coefficients for the exponential sine and cosine diminish rapidly in magnitude This rapid decline accounts for the utility of a small number of terms in approximating functions Exercises 14 Predict and confirm the results of cos cos sin sin t i 8 15 Repeat Ex 14 for cc times cc plus sc times sc t i 8 16 Repeat Ex 14 for t i 8 times t i 8 17 The function h 1 2 3 amp p 4 3 amp p is a polynomial Determine its coefficients by hand and confirm the result by enteringh t i 8 18 Read Section 9D Expansion of Book 2 Chapter 14 Polynomials 89 If and g are polynomials then g g is equivalent to On the other hand division for an arbitrary pair such as g may be not a polynomial but a power series For example f 1 amp p g 1 _1 _1sp f g t i 8 11235 813 21 The foregoing Taylor series may be surprising it is the Fibonacci series in which each item is the sum of the two preceding it This matter is discussed in Concrete Mathematics 3 and in Concrete Math Companion 2 D Parity A function E is said to be even if E is equivalent to E that is E
19. 8 Recursion 51 08 h 108 gt n n f g h Q li 53 Chapter 9 Proofs Drug thy memories lest thou learn it lest thy heart be put to proof Tennyson A Introduction It is probably advisable to begin by reviewing the brief discussion of proofs at the end of Chapter 2 The final experiment of Chapter 1 showed a relation between the sum of the first n odd numbers and the square of n We will first reproduce it here n 20 odds 1 2 a i n 20 odds 135 7911 13 15 17 19 21 23 25 27 29 31 33 35 37 39 odds n n 400 400 odds 1 4 9 16 25 36 49 64 81 100 121 144 169 196 225 256 289 324 361 400 1 a 1 a 149 16 25 36 49 64 81 100 121 144 169 196 225 256 289 324 361 400 But is the indicated relation true for any positive integer n If you are already convinced that it is any proof may seem pointless However you might still ask why it is true The following should be helpful in answering this q 1 2 i n 6 First six odd numbers r q Odds in reverse order q r 1357911 119753 1 54 Exploring Math q r qtr a 2 qtr 136136 12 12 12 12 12 Die 6 6 6 6 6 36 The foregoing shows the rather obvious fact that sums over a list over the reversed list and over one half of the sum of the lists all agree But the half sum of the lists has a pattern whose sum is easily expre
20. 98 99 100 Readings 122 reciprocal 24 89 105 107 112 Recursion 49 recursive proof 110 Refutations 14 125 reg 70 105 106 relation 1 9 12 34 35 42 55 75 relations 9 10 12 14 34 Relations 9 repeatable 3 repeated approximations 120 report 29 30 31 32 34 Reports 29 representation 28 37 38 39 40 43 45 46 68 79 83 84 87 88 102 104 105 106 107 Representations Of Functions 83 Research 6 reversal 38 Reverse 31 rfd 68 105 130 Exploring Math right parenthesis 41 right to left 26 rising factorials 122 Roots Representation 88 Roots table 102 Rotate 31 66 SAMPLE TOPICS 122 Save As 64 scan 25 script 59 64 68 Script Windows 64 secant line 109 Secant Slope 109 second derivative 113 selection 52 Self Classify 123 sets 7 Sets 122 Shape 31 signum 46 51 similar 3 9 10 35 98 113 sine 89 90 91 92 113 114 115 slope 89 109 110 solving 120 sort 3 4 9 10 13 specific arguments 34 Spelling 23 28 square 3 118 squares 6 59 95 102 stitch 27 stopping condition 49 stopping value 49 subtotals 6 78 Subtraction 46 successor 97 98 sum 6 24 25 29 39 40 43 51 55 56 57 58 59 60 61 75 77 78 87 89 91 102 103 104 110 111 sum function 6 25 symmetric 4 18 56 57 65 table 13 17 18 19 24 34 43 44 45 52 68 69 78 79 81 84 88 92 93 101 102 104 110 114 T
21. A q iS 1 11 9 5 7 3 Write a sequence of sentences such as 1 3 5 7 11 9 that uses only associativity and commutativity to move from the first expression to the last and enter them all to test their equivalence 4 Use the words Comm and Assoc to annotate your solution to Exercise 3 to provide a formal proof of the equivalence of q and 117 A q 5 The proof that q is equivalent to n n is completely formal except for one omission Complete it Following Lakatos s point that a formal or informal proof may suggest further lines of inquiry we note that the list sum q q gave items with a common value This is of course a proposition that is not true for every list q but depends upon some property of q What is that property The point is that q is an arithmetic progression successive items increase by the addition of a fixed constant in this case 2 The sum of the first and last items therefore equals the sum of the item just following the first and just preceding the last and so on for further pairs This is more easily stated and seen by reversing the list to bring corresponding pairs together Thus q q 1357911 119753 1 56 Exploring Math q q 12 12 12 12 12 12 The method of proof can therefore be applied to find expressions equal to the sum of any geometric progression For example g i n 6 g DW HR HAQ ou 01 54 WM g g 5555 5 u n n 1 2 15
22. J as illustrated below 90 Exploring Math 5 amp 0 6 amp 0 i 8 O 1 1752 3 62686 10 0179 27 2899 74 2032 201 713 548 316 1 1 54308 3 7622 10 0677 27 3082 74 2099 201 716 548 317 The adverbs o andE produce the odd and even parts of functions to which they are applied For example o is equivalent to opex and E is equivalent to epex Exercises 23 Compare the coefficients t i 8 and opex t i 8 and epex t i 8 24 Comment on the functions cos E and cos Oand sin E andsin O 25 The function j multiplies its argument by 031 the imaginary square root of negative _1 Comment on the even function j E ej E is the cosine Try entering j t i 8and j E t i 8 E Linearity Since c d p x equals c p x d p x it appears that a polynomial is in some sense a linear function of its coefficients We will now consider a series of examples to clarify this vague statement producing the matrix that represents the linear function and a simple expression for it as a power table mp c 1 3 3 1 d 2 1 0 4 x 123 4 c p x c amp p x p amp x c 18 27 64 125 8 27 64 125 8 27 64 125 oook oOoOrO oroo HOoo m p amp x 1 I The matrix that represents the linear function eerer BWNHE Mm oo hr BOF 2 6 m mp c 8 27 64 125 m amp mp c 8 27 64 125 The matrix m that represents the desired linear function of the coefficients looks like a pow
23. O 0j1 051 41421 051 73205 052 0j2 23607 Taken together with the rationals these imaginary square roots of negative numbers form the class of complex numbers closed under square root as well as under addition subtraction multiplication and division Exercises 7 Read Section 9F Real and Complex Numbers of Book 2 8 Read Chapter 7 Permutations of Book 2 9 Read Chapter 8 Classification and Sets of Book 2 99 Chapter 16 Complex Numbers A Introduction The following tables illustrate some of the consequences of adding the imaginary square root of minus one to the number system T 1 bylover x Bordered table adverb by amp adapted from Ch 3 over li _1 051 c 1 4 i i 4 012 3 0 051 0j2 0353 T c Addition table I O 1 2 3 0 091 052 053 O O 1 2 3 0 091 052 053 1 1 2 3 4 21131132 153 21 2 3 4 5 2 231 252 2531 31 3 4 5 6 3 341 352 353 O O 1 2 3 0 091 052 053 l0j1 0j1 151 251 351 051 052 053 054 105210j2 152 252 352 052 053 054 055 053 033 153 2453 353 053 034 055 056 4 T c Multiplication table 4 0 1 2 30 0391 052 053 4 I 010 0 0 0 0 0 ol 110 1 2 30 031 0j2 0j3 210 2 4 60052 0j4 0j6l 310 3 6 90053 0j6 059 ojo 0 0 0 105110 031 052 053
24. SS 1 4 2 amp 3 SS 1 13 r 10 i 6 x 0 Secant slope of cube with run of 2 at 1 1 0 1 0 01 0 001 0 0001 le 5 x i 7 r 183 SS x application 1 7 0 01 3 31 0 0001 3 0301 le 6 3 003 le 8 3 0003 le_10 3 00003 Slopes of cube for various runs and points of 19 37 12 61 27 91 12 0601 27 0901 12 006 27 009 12 0006 27 0009 12 0001 27 0001 61 49 21 48 1201 48 012 48 0012 48 0001 91 76 51 75 1501 75 015 75 0015 75 0001 127 109 81 108 18 108 018 108 002 108 108 Exploring Math B Derivative As the run decreases in size the slope appears to be approaching a limit which we may interpret as the derivative the slope of the tangent at the point x However a zero value for the run gives only the meaningless ratio of 0 divided by 0 0 amp 3 SS x 0000000 For the case of the cube this derivative may be obtained exactly because the cube of x r is x 3 3 x 2 r 3 x r 2 r 3 and the rise found by subtracting x 3 is 3 x 2 r 3 x xr 2 r 3 Dividing this by the run gives 3 x 2 3 x r r 2 Setting r to zero in this expression gives 3 x 2 the derivative of the cube at the point x The function for the derivative of the cube may therefore be expressed and used as follows d3 3 82 d3 x 0 3 12 27 48 75 108 This result may be compared with the final row of the table of secant slopes Similar analysis for other powers yields d4 4 amp 3 for the derivative
25. Use the arguments x and y to test the assertions that each of the following functions is linear x 2 7 1 8 y 3 14 2 L4 L5 L4 L4 L6 L4 1 L5 is not linear 16 illustrates the fact that the inverse of a linear function is linear L4 gives subtotals and L gives differences try L4 L6 x and L6 L4 x to test the assertion that they are inverse functions D Inner Product Applied to the sum and product the dot conjunction produces the matrix product function that is for the arguments used in the preceding section equivalent to the function defined there mp The space before the dot is essential wimp d Using w and d and t from the preceding section 11 t amp mp d 11 24 32 For any square matrix m that is m the function mp m is a linear vector function For example m 5 4 4 10 m mp 1 2 718 x D IA Chapter 12 Linear Functions 77 y 3 14 2 m L x L y L xty L x L y I4 2 10 I3 _5 11 5 _32 18 37 14 8 16 18 9 40 34 55 9 _40 34 55 14 2 03 I5_5 01 Conversely for any linear function a matrix m that represents it in the function L m amp mp 1 can be obtained by simply transposing the table produced by applying the function to the appropriate identity matrix For example 1 L 1 L m _
26. adverbs Finally the real and imaginary parts of the function j are the cosine and sine respectively For example j x cos sin x 1 ol 1 0l 0 540302 0 841471 0 540302 0 841471 0 416147 0 909297 0 416147 0 909297 0 989992 0 14112 0 989992 0 14112 0 653644 _0 756802 0 653644 _0 756802 0 283662 _0 958924 0 283662 _0 958924 Exercises 12 Study the plot of sine versus cosine in Section 9J of Book 2 13 See Chapters 3 Vector Calculus and 4 Difference Calculus of Book 3 115 Chapter 18 Inverses and Equations A Inverse Functions The many scattered references to inverse in the index suggests the ubiquity of the notion in math The general reason for its importance appears in the following example if we use heat amp 4 to compute the output of an electric heater as a function of the voltage applied we will commonly need the inverse volts amp 4 to determine what voltage would be required to produce a desired amount of heat Thus heat amp 4 volts amp 4 heat volts heat i 5 10 12 3 4 0 4 16 36 64 0 1 2 3 4 A method for obtaining the inverse of a composition of two functions may be seen in the following example cff m s Celsius from Fahrenheit m 100r180 amp Multiply by conversion factor s amp 32 Subtract a conversion constant cff temp 40 32 212 Celsius fo
27. by a negative sign or at least one digit As shown in Exercise 4 an r may be used in a number to denote a rational fraction as in 2r3 for two thirds and a list may be represented by a list of numbers as in 2 3 2r3 4 The spelling rules of J determine how words are formed from the string of characters that comprise a sentence They can be clarified by applying the word formation verb to a quoted sentence For example 4 3 2 1 i 6 I 1 14 3 2 11 1 1i 161 It should also be noted that redundant spaces may be inserted in a sentence to improve readability as in a i 6 instead ofa i 6 A Introduction 29 Chapter 5 Reports Cornelius the centurion a man of good report Acts If a is a list of twelve monthly receipts for a year then a quarter by month report ofthe same receipts can be obtained as follows Wo Ur ONN The sum over the quarters is given by A two year report for constant receipts of 10 can be obtained by 10 10 10 10 10 10 10 10 qm 4 3 a 1 74520669358 o O A qm 15 20 21 ten 2 4 3 10 ten 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 A more realistic report can be obtained by applying the repeatable random generator to this array 30 Exploring Math yqm ten ym woauhr ONN 0 oO A ooano o ro o ooW Ul The sums over the years of this report are yqm 179 11 2 3
28. ee Re er ART Ei urn ee 53 B Inductive Proof uuuseeneneeeenennennennnnesnnnnneennnnennennneenennenen 56 MOONS een 61 A TnitrOGuctl Of ae cicscsede ena a e T ana 61 B Editing nungen 62 C Script Windows unueenseessennensnsnsnnsnnnnennnnnennennonnensennnennsennen nennen 62 Coordinates and Visualization uu02202200220000 63 A Introduction t e een nannte 63 C Plotting Multiple Figures 2042004 2 nerneennenseenneennennee nenn 67 D Plotting Functions Seieren a ias 68 Linear FUNCTIONS 2 02 ccceeceececceceeceeceececceceeceececceceeeeeeees 73 Ax DASEPIDULLVIEY 22222228 a coe ER E debe ad Oe 73 B Linearity co ceciate dak ans RoR RO RR eh er 74 C Linear Vector Functions cne ches apts Rae 75 Di Inner Product a 42 52 0er o aeai een 76 E Why The Name Linear nnersesnnneennensennsennnennen nennen 77 Representations Of FUNCTIONS ccssseeeeeseeeeeeeeeeees 81 A Introducuon za aio oc RE RRA Mie BE 81 Polynomialszz z 2 0 22242002 2420240 edeien ahera ner detains dsaie esas 85 A Coefficients Representation eeeseeesesserseenseenseensennennennne nennen 85 B Roots Representation ise lt a ena ala 86 C Versatility u ieatinin cia thins ena 87 De Parity 220 Siok forse E athe calle taeda dap coset te re RR 89 E Eineatityaasza in E Ran Re aR eS 90 F Polynomial Approximations ccccesccssecseesseesecseeeseeeseeseeeneee
29. interpreted as follows When the argument n is 0 then the signum on the right returns 0 choosing the leading function in the gerund giving a result of 0 otherwise the result is the nth odd number that is gt lt plus the sum for an argument n 1 that is sod lt Exercises 2 For convenience certain constant functions are provided directly without the need for the rank operator Experiment with the constant functions _9 and _8 and so on through 9 Use 1 and 0 to simplify the definitions of and sod above 3 Because increment gt is the inverse of decrement lt the expression gt lt is of the form gi f g where gi is the inverse of g We say that this is a case of applying f under g and denote it by amp g Use this fact to simplify the definition of sod and check the resulting behaviour Recursive definition essentially specifies a function in terms of the same function applied to a simpler case and its use can enormously simplify many definitions For example the Tower of Hanoi puzzle is stated as follows A set of n drilled discs of different diameters stacked as a pyramid on a peg A is to be moved one at a time to a peg C without ever placing a larger on a smaller A third peg B may be used as intermediary The process for two discs may be expressed by the table 50 Exploring Math AB AC BC which is to be interpreted row by row as follows Move the top disc from A to B M
30. inverse log 9 Experiment with the expressions 6 amp 10 amp 3 and 6 amp 10 amp 3 and comment on the results The first multiplies its arguments by adding their base 10 logarithms and applying the anti log that is ten to the power the second uses subtraction to obtain the quotient The dyadic case of the function f amp g is similar to the monadic but applies the preparation function g to each of the arguments 10 Define the function saf lt suppress after first and experiment with the expressions saf band saf amp b for various values of the Boolean list b such as b 0 0 1 0 1 1 0 Comment on the results saf suppresses all ones after the first in a Boolean list safe suppresses all before the last by first reversing the list and again reversing the resulting list after applying the function saf D Equations A function such as 3 _4 1 amp p may not have a known inverse but we can obtain the inverse of a given argument such as y 6 by solving the equation y x that is by finding a value x that satisfies the indicated relation If we know values a and b such that is monotonic in the interval from a to b and if y lies in the interval from a to b then a suitable solution x can be obtained by simple repeated approximations take the average of a and b consider the intervals bounded by it and each of them and choose as a new interval the one whose function values still embr
31. it A window may be saved as a file under the name shown on the window by pressing Control s and can be re opened at any time by pressing Control o It can also be saved under any chosen name by using Save As or Save Copy As from the file menu Exercises Select the item Session Manager from the User Manual and from it select the item Script Windows Read the discussion of their use 63 Chapter 11 Coordinates and Visualization It was their belief that if they stared long enough at these mystic curves and angles red ink would turn into black Alva Johnson A Introduction Take a sheet of graph or squared paper ruled with equidistant vertical and horizontal lines choose some point of intersection as the origin to be labelled 0 0 and label vertical lines from left to right and horizontal lines from bottom to top with symmetric integers as follows 9 8 7 6 5 4 3 210123456789 Any point of intersection may then be labelled by two coordinates the first or x coordinate specifying the vertical line through it and the second or y coordinate the horizontal Such a coordinate system makes it possible to describe geometric figures and leads to analytic or coordinate geometry For example p 3 4 A single point q 9 4 r 6 8 s 9 7 t 8 6 is p q r Isosceles triangle rt p q s Right angled triangle qd p q s r Quadrilateral pg p q s r t Pentagon Properties of the geometric figures can be obtained fro
32. of 184 d5 5 X84 for the derivative of amp 5 and so on We define a corresponding adverb for the derivative of any power D 1 x amp x _ 1 2Dx 0246 8 10 12 3 Dx 0 3 12 27 48 75 108 4Dx 0 A 32 108 256 500 864 None of this constitutes a proof that the derivatives of all powers follow this pattern but it does suggest an induction hypothesis for a recursive proof This matter is treated in Book 3 Exercises 1 If amp 3 is the cube and g 5 is five times the cube what is the derivative of g Five times the derivative of that is 5 3 amp 2 Since any secant slope of g is five times the slope of the same is true of the limiting value that is the derivative 2 Ifh 2 amp 4 what is the derivative of the sum s gth Chapter 17 Calculus 109 The sum of the derivatives of g and h that is 5 3 amp 2 2 4 amp 3 3 Ifc 3 1 4 2 _andE 0 1 2 3 _ are constant functions then t C E is a weighted sum of powers What is its derivative der C E E 1 0 Try der 1 2 3 4 5 C Polynomials The preceding Exercises developed the fact that the derivative of a weighted sum of powers is itself such a sum with the exponents decreased by 1 Since a polynomial is a weighted sum of powers its derivative is also a polynomial of degree one less The derivative of c amp p is d amp p where the coefficients d are obtained from c by applying the following function
33. of the principal domains of _1 amp 0 and _2 amp 0 Apply them to the argument _1 1 C Under SS UL idr 10 amp Inverse of decimal representation i e decimal value dr idr I Decimal representation dr x 213 213 idr dr x 213 dr idr dr x I213 2 1 3 213 az amp 0 Append zero az dr x 2130 idr az dr x Decimal value with appended zero 2130 x 10 2130 The foregoing elaborates the familiar idea that a number can be multiplied by ten by appending a zero to its decimal representation The full expression may be paraphrased in English as Obtain the decimal representation append a zero then evaluate the resulting list in decimal that is apply the function inverse to the decimal representation It illustrates the form _1 g that occurs so often that it is also provided by the conjunction amp as follows idr az dr x 2130 az amp dr x 2130 118 Exploring Math The general idea is that amp g applies under g in the sense that g prepares the argument for the function and the preparation is finally undone For example amp y 4 Double under natural logarithm 16 y Is equivalent to squaring 16 amp 10 amp y 4 16 amp 10 amp y 4 2 y 2 Exercises 8 Paraphrase the foregoing expressions in detail amp takes the natural logarithm of its argument doubles it and applies the exponential
34. the figures observed Include the following sequence red lt 255 0 0 gdopen a red POLY SCALE lt reg 6 gdshow gdopen b red POLY SCALE lt 1A reg 6 gdshow gdopen c red POLY SCALE lt reg 6 gdshow lt reg 0 i 6 D Plotting Functions This section illustrates the use of various facilities for plotting functions load plot plot x 2 35711 In to argument x this and the following plot the horizontal axis is labeled with the default values from 0 4 The next plot after that uses the form x x to label this axis according to the Chapter 11 Coordinates and Visualization 69 The alternative function PLOT stick line amp plot draws vertical sticks to each point as well as the lines between the points Similarly BAR stick amp plot produces barcharts Enter the definitions of these functions and experiment with them plot x NB Plot square function 140 plot x x NB square Versus argument 140 120 100 80 60 40 20 Entering load graph also makes available a function called steps that produces a grid from one value to another in a specified number of steps For example steps 2 4 10 2 2 2 2 4 2 6 2 8 3 3 2 3 4 3 6 3 8 4 We will give it an alternative name as follows grid steps grid 2 4 10 NB 2 to 4 in 10 steps 2 2 2 2 4 2 6 2 8 3 3 2 3 4 3 6 3 8 4 sin 1 amp 0 cos 2 amp 0 plot sin x grid 0 6 100
35. this might be rather difficult to do in a long list of words it is easy to overlook repetitions and you may not even know how many anagrams to expect all together We will now use the anagram function a for this purpose Its left argument chooses one of many permutations to apply to the list right argument Thus w SPOT 8 A w POST 12 A 8A w The permutation 12 A is the inverse of 8 A SPOT 012345678A wW SPOT SPTO SOPT SOTP STPO STOP PSOT PSTO POST 30 A w lindex error 30 A w The last result shows that there is a limit to the valid left argument properly so since there is a limit to the number of different permutations of a list But how many are there In the case of a two item list aB there are clearly only two Chapter 2 Whatis Math 11 possibilities the identity permutation that leaves the list unchanged and the one that gives BA Thus O 1 A AB AB BA Write down all permutations of the list ABc to convince yourself that there are six possible permutations Thus i 6 A ABC Exercises 1 Produce all anagrams of various three letter English words to find those words that have the largest number of proper English words among their anagrams 2 Did you find any word more prolific than apt 3 Find all English words among the anagrams of spor In solving the last exercise above it was necessary to find the largest left argument of A permitted This could be done by experiment Thu
36. to any permutation of a list This may be tested as follows q 1 2 i n 6 117 A q 11195 73 q 117 A q _1 A q 36 36 36 But why is summation symmetric We may for example ask whether the notion applies to other functions as in product over maximum over gt and Chapter 9 Proofs 55 subtraction over beginning with the following tests r q q q gt q q 36 10395 11 6 r xr gt xr r 36 10395 11 6 What is it about the functions and gt that make and gt symmetric The answer is that they are both associative and commutative These matters are examined further in Exercises but the main point is that any conjecture may lead to further sub conjectures that can be identified and pursued until the reader reaches assertions that are satisfying to him As Lakatos shows assertions satisfactory for one reader or purpose may not be satisfactory for another Exercises 1 Addition is said to be associative because a sequence of additions can be associated by parenthesizing them in any way without changing the result For example 1 2 3 4 and 1 2 3 4 and 1 2 3 4 and 1 2 3 4 are all equal Test the associativity of addition by entering a variety of equivalent expressions 2 Repeat Exercise 1 for product and maximum 3 The completely parenthesized form of q is 1 3 5 7 9 11 and the corresponding form of 117
37. to drop the Studio menu in J then click on Labs and then on Graph Utilities Exercises 4 Enter the foregoing sequence of graphics sentences in a script window and use the Selection option from the run menu to execute it 5 Display each of the polygons defined in this section in various colors in particular display rt in red and without clearing the window is in green 6 Permute the coordinates of the polygons as in1 A pg and discuss the appearance of the resulting figures 7 Enter rot j rfd and rfd amp 180p_1 and experiment with rot by plotting the results of the following forms 45 rot rt 45 amp rot amp gt rt is rts 8 Experiment with and comment on the function rotate introduced by the graphics file B Visualization The examples of Section A illustrate the fact that the coordinate representation and the graphic representation of figures are complementary each provides certain advantages For example the graph of Exercise 6 shows how easy it is to distinguish an improper polygon in which sides cross a matter that would not be easy to spot in a table of coordinates On the other hand for the computation of properties such as areas coordinates are far superior For the particular triangles rt and is and even for rts and iss plotted by hand in Exercise 3 the computation of area appears simple but this simplicity is deceptive as illustrated by the rotated figure of rts in Exe
38. toggle switches or door locks But do not forget your own safety danger lurks in electrical devices as well as in wilderness parks Finally in choosing a device for exploration favour the older models modern typewriters and digital clocks may be totally inscrutable At least one author Ivan Illich has claimed to see a sinister motive in this claiming that modern design is deliberately inscrutable in order to keep ordinary people like us in ignorance But can exploration be applied to abstract non physical notions such as math Yes it can With an ordinary hand calculator you can explore the relation between multiplication and addition by using it to multiply two by three then to add two plus two plus two and then comparing the results If the calculator has a button for power you can even explore that less familiar notion by doing two to the power three and comparing the result with two times two times two 2 Exploring Math But the abilities of a calculator are limited and for a general exploration of math we will use a computer equipped with suitable software called J It is available from Website http www jsoftware com We will assume that you have J at hand on a computer and will simply show examples of exploring math with it 3 2 5 3 2 6 3 2 1 These examples are in a uniformly spaced font Courier that differs from the Roman font used elsewhere We will use this difference to append comments to some of the examples
39. using a factor and an exponent joined by the letter p e The complex number 3 4 _1 is represented as 334 using a real part and an imaginary part joined by the letter 3 e Further cases may be found in the discussion of constants in the J dictionary The monadic function used in the table T a is called magnitude it yields the square root of the sum of the squares of the real and imaginary parts of an argument When applied to a real non complex number it is sometimes called the absolute value Functions defined on real numbers are extended to complex numbers without change except that they apply to the new element _1 according to the normal rules The extended functions can therefore be examined in terms of elementary algebra B Addition The sum of complex numbers can be analyzed in terms of their real and imaginary components as follows i _1 ar 5 ai 2 br 3 bi 4 a art i ai b br i bi 5j2 354 Chapter 16 Complex Numbers 101 The following sequence of identities shows that the components of a sum are the sums of the components a b arti ai br i bi Definitions of a and b ar br i ai i bi Addition is associative and commutative artbr i ai bi Multiplication by i that is is distributes over Exercises 4 Enter the foregoing sequence and check that each of the sentences yield the same result 5 Write and enter a corresponding sequence for multiplication C Multiplicat
40. x is a list B Roots Representation The product x r is said to be a polynomial expressed in terms of the list of roots x It is called a polynomial because any such function can also be expressed in a coefficients representation Thus x 4 r 2 3 5 x r 2 1 1 x r _30 31 10 1p x 2 The monadic case ofp applied to the boxed roots yields the coefficients of the other representation p lt r _30 31 101 Exercises 7 Define a polynomial in terms of roots function pir such that r pir x evaluates a polynomial with roots r for the argument x 8 Why are the elements of the list r in the function r pir called roots Each of the elements is a zero or root of the function in the sense that it yields a zero result For example enter pir 1 Oandrspir r 9 Every function of the form rapir can be represented in the form csp Is the converse true Try to define a list s such that sspir is equivalent to dep where d 2 p lt r Then look at the result ofp dand of p d p x i 8 10 Discuss the result of p d The dyadic function p covers both the coefficients and roots representations If the left argument is open not boxed it is treated as a list of coefficients If it is boxed and contains two items the last item is the boxed list of roots and the first is the boxed multiplier If it contains a single item b it is equivalent to 1 b that is a multiplier of 1 Chapter 14 Polyno
41. 1 N 10 11 12 15 551112 2 Exercises 6 Although the formal definition of the process carried out by N is rather involved the hand calculation of it is quick and trivial Confirm this by performing it on various lists checking the accuracy of your work by applying the function 10 amp to each list and its normalized form 7 The copula used in the definition of NORM differs from the used elsewhere Its use localizes the assigned name so that it bears no relation to the same name used outside the definition Experiment with the distinction by defining a function GNORM that is identical to NORM except for the use of global assignment and compare the behaviour of the two functions A name can be erased by using 4 55 asin 4 55 lt c B Addition In the example d10 bv10i 365 1966 we have already seen how the decimal representations of two numbers may be added to obtain a representation of the sum we may now obtain a standard representation by applying the function n Thus Chapter 7 Decimal and Other Number Systems 43 d10 bv10i 365 1966 d10 365 996 d10 112 15 11 N d10 2361 Exercises 8 Use bv10i to compute the table of decimal representations of the list of numbers a b c where a 365 and b 1996 and c 29 From this table compute the standard representation of the sum at b c 9 Use ar bv10i aandbr bv10i b and cr bv10i c to obtain the decimal representa
42. 1 0 0 5 0 8660254 _0 5 0 8660254 1 1 2246le 16 5 _0 8660254 5 _0 8660254 0 0 reg _1 i 0 Function for regular polygons 104 Exploring Math lt reg 3456 Boxed polygons of 3 6 sides 1 o 1 ol 1 O 1 O 0 309017 0 951057 0 5 0 866025 I_0 5 0 866025 O 1 0 809017 0 587785 _0 5 0 866025 _0 5 _0 866025 _1 0 0 809017 0 587785 1 0 0 _1 0 309017 _0 951057 _0 5 _0 866025 l l l 0 5 _0 866025 Compare the function reg with that used in Chapter 11 and use the plotting functions of that chapter in the following Exercises Exercises 8 Plot the figures reg 4 2 reg 4 in contrasting colors 9 Use the function rot of Chapter 11 to plot rotated figures E Division Since amp b division by b is the inverse of amp b multiplication by b division is easily expressed in a polar representation the magnitude is the quotient of the magnitudes and the angle is the difference of the angles For example asb 0 92j_0 56 a b a b 5 38516 0 3805064 5 0 9272952 1 07703 _0 5467888 la b 1 07703 A complex number may be normalized by dividing it by its magnitude yielding a complex number with magnitude 1 For example b 5 0 6j0 8 Ib 5 1 norm Jnb norm b 0 6j0 8 Since a normalized number can be restored by multiplying its norm by a real number
43. 112345 6 1 0 1 2 34 02 4 6 81023456 7 2 1 0 1273 103 6 91215134567 8 3 2 1 0 1 2 104 8121620145678 9a 32 1 0 1i IO 5 10 15 20 25 5 6 7891015 4 3 2 1 0 18 Exploring Math Much can be learned from such tables For example the multiplication table is symmetric that is each row is the same as the corresponding column and its transpose a a is the same as the table a a itself This implies that the arguments of multiplication may be exchanged without changing the product or as we say multiplication is commutative The same may be said of addition but not of subtraction which is non commutative as is obvious from its table Tables for both negative and positive arguments are even more interesting For example try each of the three tables with the following symmetric argument 6 5 4 3 210123456 Note how the multiplication table is broken into quadrants of exclusively positive or exclusively negative numbers by the row and column of zeros and try to explain why this occurs The symbol in the sentence a a denotes an adverb because it applies to the verb to produce a related verb that is in turn used to produce a table It is much easier to interpret a table if it is bordered by its arguments We will use a second adverb called table for this purpose For example b 2 3 5 7 11 a table b Bordered multiplication table I12 3 5 711 l0 0 0 O 0 O 111 2 3 5 711 I2 4 6 10 14
44. 22 I3l 6 9 15 21 33 l4 8 12 20 28 44 15110 15 25 35 55 table a Bordered addition table 101234 5 10101234 5 11112345 6 12123456 7 13134567 8 14145678 9 1515 6 7 8 9 10 table i 6 I I6 5 4 3 2 10 1 2 3 4 5 6 I_6 36 30 24 18 12 60 6 12 _18 24 30 _36 I_5I 30 25 20 15 10 50 5 10 _15 _20 _25 _30 I4 24 20 16 12 8 40 4 8 12 16 20 24 I_31 18 15 12 9 6 30 3 6 9 12 15 18 1321212 10 8 6 4 20 2 4 6 8 10 12 I1 6 5 4 3 2 101 2 3 4 5 6l O 0 0 0 0 0 00 0 0 0 0 0 Ol Chapter 3 Function Tables 19 111 6 5 4 3 210 1 2 3 4 5 6 2 12 10 8 6 4 20 2 4 6 8 10 12 3118 15 2 9 6 30 3 6 9 12 15 18 al 24 20 16 12 8 40 4 8 12 16 20 24 5 _30 25 20 15 10 50 5 10 15 20 25 30 6 _36 30 24 18 12 60 6 12 18 24 30 36 Tables also provide an interesting and effective way to explore unfamiliar functions Often the display of a bordered function table provides a precise and easily remembered definition of the function For example lt table i 5 Relation I5 4 3 21012345 3 0 1 1 1 1111111 1 4 0 0 1 1 11111112 1 3 0 0 O 1 111111121 1 2 0 0 0 O 1111111 I1 0 O O O 0111111 1010 0 0 O 00111 11 11 0 0O 0 O 00011
45. 333333 0 cp i 8 015 14 30 55 91 140 gi 8 015 14 30 55 91 140 Exercises Chapter 9 Proofs 59 10 Study the discussion of proofs in Section D of Chapter 5 of Book 2 11 Find a function equivalent to the sum of cubes and construct an inductive proof of the equivalence c 8 ZERO 5 FIT scubes x i 6 12 For many functions the coefficients for an equivalent or approximate polynomial may be conveniently obtained by using the Taylor adverb t as in f t i 6 Experiment with this for the functions amp 4 amp 4 amp 2 gt 4 lt 4 A 61 Chapter 10 Tools Without tools he is nothing with tools he is all Carlyle A Introduction This chapter concerns tools for exploration They are fully treated in Burke s J User Manual available on line under the help menu in the J system but should themselves be explored in the manner used for math in preceding chapters For example an overall view of the tools available may be obtained by dropping the menus This can be done by clicking the mouse on each ofthem but they can also be dropped by first pressing the alt key then the down arrow then the left or right arrow to move over the menus The alt key will roll up a menu With a menu dropped use the up and down arrows to select an item and press enter to execute it Alternatively an underscored letter in an item can be entered to execute it Some menu items can be invoked directly withou
46. 4 But for present use in further experiments with inductive proofs we provide the following methods The function g is an example of a polynomial a sum of weighted powers of the argument the weights being 0 1r6 3r6 2r6 They may be obtained as follows w ssq a a a i 5 58 Exploring Math _2 99066e 14 0 1666667 0 5 0 3333333 _6 50591e_14 6 w _1 7944e 13 1 3 2 _3 90354e_13 Because matrix divide produces its results by approximation the extreme items of 6 w are not quite zero They can be zeroed by the following function in which the first argument specifies the tolerance in number of decimal digits ZERO gt 10 amp 8 ZERO 6 w 01320 14 ZERO 6 w _1 7944e 13 1 3 2 _3 91687e 13 For convenience in experimenting with a variety of functions we will adopt from Section F of Chapter 14 the conjunction FIT so defined thatn FIT x gives the n item list of coefficients of a polynomial that best fits the function at the points x For example V i FIT 2 y x amp V 3 FIT 2 38V c 3 FIT b 0 2 i 5 1 00238 0 9203119 0 7569838 cp b 1 00238 1 21672 1 49162 1 82708 2 2231 b 1 1 2214 1 49182 1 82212 2 22554 As remarked g is an example of a polynomial and the coefficients produced by FIT can preferably after being zeroed be used with the polynomial function p to produce an equivalent function Thus c 8 ZERO 4 FIT ssq a i 5 0 0 1666667 0 5 0 3
47. 6 10 15 8 14 16 Because yqm has three categories or axes we call it a rank 3 report or array Its rank 2 cells are the two quarter by month tables seen in its display and its rank 1 cells are the eight rows arranged in effect in a 2 by 4 array The sums over the quarters in each year are the sums over the two rank 2 cells yielding a 2 by 3 array for the two years and three months in each quarter Thus 2 yqm 15 20 21 11 13 22 Similarly the sums over the three months in each quarter are a 2 by 4 array given by 1 yqm 12 7 21 16 5 9 10 22 Exercises 1 Enter the foregoing expressions and verify that they reproduce the foregoing results 2 The function reproduced the same result because it is a repeatable random number generator Try the expression ten several times to show that the results do not repeat 3 Predict and verify the results of 3 yqmand 0 yam 4 Experiment with the box function as in lt 3 4 5 and lt yqm and lt 2 yqm and lt 1 yqm 5 The sentence yqm gives the shape of the array yqm Apply to other results such as yqm and 2 yqm and 1 yqm Chapter 5 Reports 31 6 The function gives the number of items or major cells in its argument Apply it to various arguments The expression k can be used to apply any function to the rank k cells of its argument For example the mean or average function can be used as follows mean mean 3 4 5 6 4 5 mean m
48. 728540 728540 Exercises 13 Perform hand calculations of products using the process defined by the function TIMES and compare its use with the commonly taught process Which requires the most writing Which is the more error prone Which is the easier to re check by re calculation of parts of the process D Subtraction Subtraction leads to the question of representing negative arguments We will use lists of negative numbers with the standard form limited as it is for positive arguments to numbers whose magnitudes are less than the base For example 10 3 6 5 _365 10 3 4 25 _365 The function bv10i 108 amp _1 can be used to obtain the representation of a negative number by applying it to the magnitude and then multiplying the resulting list by _1 Thus c _ 365 ar 1 bvl0i a 46 Exploring Math A corresponding function for either positive or negative arguments can be obtained by multiplying not by _1 but by the signum of the argument 365 0 365 10 1 REP10 10 amp _1 l REP10 _365 3 6 5 REP10 365 365 With this representation of negative numbers expressions for addition apply equally for subtraction For example a 365 b 1996 t REP10 a b t 1 6 3 1 10 t a b _1631 _1631 The normalization function must be modified in the same manner NOR NORM amp N 10 amp NOR NOR N 3 4 25 N 3 4 25 Exercises 14 Read Chapter 4 of Book 2 Arithmetic and t
49. A sum of monomials is called a polynomial and is written in MN in the form 2x 4x 3x7 x having the value 30 if x is 2 A direct translation to J would read as 2 x 0 4 x 1 3 x 2 x 3 The numerous parentheses required suggest a reason for the precedence rules adopted in MN power before multiplication before addition they are precisely the rules that permit the polynomial to be expressed without parentheses Exercises 1 Write a parenthesis free J expression for the foregoing polynomial then assign the value 2 to x and enter the expression to test its validity 2 Use the results of Exercise 1 to define a function py so that2 4 3 1 py x yields the value of the polynomial for any single argument x py i 3 Use py to define a function poly so that it applies to each element of a list x and test it by using it with the arguments 2 4 3 1 andi 8 poly py 1 0 4 Comment on the function 2 4 3 1 amp poly The dyadic function poly represents a family of polynomials 2 4 3 1 amp poly is a specific member of this family The elements of the list2 4 3 1 are called coefficients and poly is said to be a coefficients representation of polynomials 86 Exploring Math 5 The dyadic case of the primitive function p is a coefficients representation of polynomials Experiment with the expression c p x for various values of c and x 6 Experiment with te p x where tc is a table of coefficients and
50. ABLE 18 19 20 23 tables 5 13 17 18 20 30 52 76 95 101 102 tangent 89 110 tangible representations 14 Taylor adverb 61 Taylor series 90 91 94 95 113 114 Taylor series adverb 90 Terminology 33 tetrahedron 81 the reciprocal factorials 112 ties 50 Tools 63 Tower of Hanoi 51 transcendental functions 89 transformations 14 translating 7 transpose 13 18 TRANSPOSITION 31 tree 34 35 true 20 23 34 35 55 57 88 110 Two part representations 102 Under 119 under open 69 Vandermonde matrices 94 Vandermonde matrix 93 94 variable 34 vector 34 66 77 78 79 80 81 83 104 Vector Calculus 115 verbs 5 23 25 27 28 Versatility 89 Index 131 Visualization 7 65 68 125 Vocabulary 123 volume 7 81 weighted sum 77 111 Weighted Taylor coefficients 114 whole numbers 3 7 97 Width 25 26 window 59 64 67 68 word formation 28
51. Exploring Math Kenneth E Iverson Copyright 1996 2002 Jsoftware Inc All rights reserved Table Of Contents Chapter 1 EX PLOPAUION BSPSSRAE SE RRRRAE PERPREREEPRERER EPRPREREEERFERRSEFEEREESEEHEEURSEERE 1 A Introduction Sessel ee eisen 1 B Ramble Or Research ccccccceeccccessssceceessececesscececssseeceeseeeesseeees 6 What Is Math ccccccsecssssccesscesseeseeeseeeesseseeeeeeeeseseeees 9 rU E eTo a E AENEA EAEE EEEE einen elite 9 B Proofs nina ne Ran 13 C SUMMAIY een EE A o i i a aie 14 Function ST ADIOS u 17 Grammar And Spelling uursrrsrrerennnnnnnnn nennen nenn nennen 23 A Introduction 2 24 32 el amp is unless okies thn 23 B Th amp UseOf Grammir eee uu au0 ie a E RR 24 C Punctuation And Other Rules n nennen 25 Dy Spelling u u ns en ceek ig avacevs R RRE na 27 12227812 u CERPBERPREREREPRECHESTEREFUEEEEHETTEEPPRFEPBETEREFTELEEREFUBELEHEGELTFERTE 29 A Introd cti n u 2 en a rar 29 B Transpositi n u auanensnneeen innen ananmindaken 31 Terminology APPRDRPREHESPELPPIEFDBEEPAETERENPEECHEEFENELTEETERETPELPPHFEFEETERENE 33 Decimal and Other Number Systems 0ee 37 A Introductiohu 2 2 2 2 en eee Heel ese Pee rn 37 BSA GION u he Ha es Re 42 C Multiplication 2 2 23 anche heh rn needa tet 44 D Subtractionn assseiaesn nn Br E E 45 FRO GUTS On s caaeaae aaea aaea aa E aaRS aiat 47 AeIE E E rantegianeatacnaes 53 Ae ntrod ctot in
52. RE 121 D Vocabulary and Definitions cccceeceeseceseeeeecnceeeeeseeeeeenseeeeees 122 Chapter 1 Exploration Something lost behind the ranges Lost and waiting for you Go Kipling A Introduction Exploring a city or wild park on foot is more fun and often more instructive than studying it in books lectures or pictures A map or other guide may be helpful but it is important to be able to experiment choosing your own path approaching points of interest from various directions This can give you a sense of the lay of the land that is more useful and more lasting than any fixed tour of important points laid out by someone else Matters other than landscapes may also be explored effectively and enjoyably For example to learn about clockwork begin not with diagrams and discussions of balance wheels springs and escapements but rather with an actual old style wind up alarm clock Explore it by first finding what can be done with it Can you reset the time make it run faster stop it or reset the hour hand independently of the minute hand Having learned what it can do explore the matter of how it does it by removing its cover studying the works and finally taking it apart and re assembling it You may of course not be skillful enough to get it working again Exploration can also be applied to other devices that may be more interesting or more easily available to you toasters typewriters electrical
53. ace the argument y Chapter 18 Inverses and Equations 119 See Section C of Chapter 7 of Book 3 for an executable definition of the foregoing bisection method and Sections D and E for the faster Newton and Kerner methods that employ derivatives The many uses of equations and their solutions in math can mostly be seen as limited means of obtaining inverse functions 120 Exploring Math 121 Chapter 19 Readings A Introduction Reading any math text can serve as a stimulus to further exploration whatever notation it may be expressed in Those such as Book 2 and Book 3 that are expressed in J are particularly accessible to users of this book We will here discuss other books of this type that are easily available because they can be conveniently displayed on the screen by using the Help menu and because selections from them can be printed using the resulting Print menu for study We will here present a few examples from two such books J Phrases and J Dictionary B Phrases After printing the Table of Contents and displaying and reading the first page of the book of J Phrases to learn the conventions used you may choose any chapter for further exploration Some such as Chapter 2 Primitive Notions and Chapter 8 Numbers will provide further elaboration of matters already treated in earlier chapters here Others such as Chapters 12 and 13 Finance and Data enter new territory Chapter 16 Extended Topics provid
54. ameters Include the example used at the beginning of this section and compare the fit provided by the coefficients te with that provided by the five element list tc5 5 FIT sin x 95 Chapter 15 Arithmetic A Introduction As remarked in Chapter 1 arithmetic is that branch of mathematics that deals with whole numbers As treated in Book 2 it includes topics such as permutations polynomials and logic These are usually considered to be advanced topics to be treated only after the introduction of fractions irrational numbers and even complex numbers What are the potential advantages of extending the treatment of arithmetic in this manner e It may serve to defer the treatment of fractions until the student has matured through experience gained in many interesting uses of whole numbers How many cooks fear the use of fractions involved in dividing a recipe Is 2 3 really a number since it cannot be written in decimal although 3 4 can And how many question the point of complex numbers whose mechanics are often elaborated long before any of their interesting uses can be shown e Although polynomials may be of little practical use when limited to integer arguments notions such as the product of coefficients c d remain meaningful and interesting Indeed they provide useful insights into the products of multi digit numbers as shown in Chapters 7 and 14 B Insidious Inverses The familiar counting numbers may be defined
55. as follows there is a first denoted by 1 and a successor function denoted by gt Thus gt 1 2 gt 2 3 gt gt gt 1 96 Exploring Math An inverse predecessor function denoted by lt undoes the work of the successor Thus However lt is not a proper inverse because its application to the first counting number cannot yield a counting number Thus In other words the introduction of a seemingly innocent inverse has broadened the class of counting numbers to define the class of integers which includes zero and negative numbers The introduction of the further classes of rationals irrationals and complex numbers can be viewed in a similar light Exercises 1 Illustrate the fact that the successor and predecessor are proper inverses on the domain of integers Include examples of the powers lt n and gt n for both positive and negative values of n 2 Same and illustrate the use of a function that has a proper inverse on some domain On the domain of permutation vectors permutations of the integers i n the grade is its own proper inverse 3 Experiment with some of the inverse pairs listed in the definition of the power conjunction in the J dictionary 5 4 Read the discussion in the first three pages of Book 2 5 Study Section 2 I Identity Elements and Infinities of Book 2 C Rational Numbers The multiplication of two integers yields an integer Moreover division is inverse
56. be precisely prescribed in English Can it also be defined as a computer executable function in J We begin with a process on a specific argument y 3 4 25 r i 0 Initialize the result as an empty list ci 1 y Current item is last item of argument r 10 ci r Prefix remainder to the result list e lt ci 10 Compute the carry to the next position y y Truncate by dropping the treated item ci ct_1 y Add carry to last item r 10 ci x c lt ci 10 y y ci c _1 y r 10 ci r c lt ci 10 y y 365 10 r 10 3 4 25 365 365 The last two groups of four steps are identical a uniformity that was achieved by truncating the argument each time Complete uniformity would allow the entire process to be stated more compactly and more generally as a repetition or iteration of a fixed procedure defined by the four steps It remains to make the first block uniform initialize the carry to zero and replace the first line of the block as follows r i c 0 ci c _1 y The foregoing process may now be defined as an iteration as follows Chapter 7 Decimal and Other Number Systems 41 NORM 3 0 r i c 0 label_loop if 0 lt y do ci c _l y r 10 ci r c lt ci 10 y y goto_loop end r NORM 3 4 25 365 In the foregoing definition The right argument is denoted by y The block to be iterated is delimited by do and end Repetition of the block is determined b
57. branch of applied math The functions L1 amp 2 and L2 amp 2 and L3 1 are each linear Thus a 3 4 9 4 9 7 b 3 4 9 4 6 8 L1 a b 12 16 36 16 Chapter 12 Linear Functions 75 30 30 L1 a L1 b 12 16 36 16 30 30 Such matters may be expressed more clearly and compactly as follows a L1 amp L1 b 12 16 12 16 136 16136 16 130 30130 30 a L1 amp L1 b C Linear Vector Functions A function of rank 1 applies to each vector in its argument and may be called a vector function We will use the term in a more restrictive sense the result must be the same shape as the argument Thus L3 1 defined in the preceding section is a linear vector function d 42 1 e 235 L3 dte 656 L3 d L3 e 656 If 1 then the function wef is a weighted sum with weights specified by the vector w Moreover it is linear For example w 2 0 3 w amp f d 11 w d 8 0 3 w d 11 w amp f dte w amp f d w amp f e 30 30 Although waf is linear it is not a linear vector function according to our strict definition Such a linear vector function may be defined as follows x 5 12 76 Exploring Math y 7 2 0 g w amp f x amp f y amp f gd 11 24 32 t w X y aI or bd NF ON W h t amp f h d 11 24 32 In general if t is an n by n table then taf is a linear vector function on any vector of n elements Exercises 7
58. e 11 4 78062e 12 _4 9381e 10 _2 14316e _ 911 69003e_ 11 3 0418le 11 _7 82095e 9 2 48923e 8 6 7725e 11 T 05946e 8 sin tc p ar 5 2 x 4 0 a _2 18587e 13 _1 07342e 8 5 27356e 16 4 43534e 14 9 83991e 8 _5 0716e 7 11 04966e 12 1 22129e 11 1 92828e_ 6 6 01583e _6 19 06789e 11 4 9381e 10 _1 62857e 5 _3 95772e 5 2 14316e 9 7 820958 9 _8 83031e 5 0 000183771 The first panel above shows that ve provides the better approximation at the very points on which it was determined the second panel shows that this better performance persists for other points in the range spanned by them and the third shows that the Taylor series generally performs better at points that is x outside the range Chapter 14 Polynomials 93 Exercises 29 Will the use of a larger number of terms in a polynomial approximation improve its fidelity Experiment to test the matter Not necessarily Although the higher order elements of the coefficients t i n may decrease rapidly the power of the argument by which each is multiplied in the polynomial evaluation may rapidly increase The resulting product produced to limited precision may introduce large round off errors We will now develop a polynomial of lower degree that provides a least squares best fit
59. e arguments of the form rt is pg and scale the whole according to the requirements of the entire collection It suffices to modify the functions slide size and scale so as to apply to each box that is under amp open gt and to find the maxima and minima after razing the argument by applying Thus SLIDE 1 amp gt lt lt SIZE 1 amp gt lt gt SCALE amp gt lt amp gt SIZE SLIDE We may then proceed with experiments such as the following which plots the isosceles triangle together with the right triangle displaced two places up and to the right POLY gdpolygon amp gt color 0 255 0 255 0 0 gdopen color POLY SCALE is 2 rt gdshow Exercises 14 Experiment with the plotting of multiple figures using expressions of the form 255 0 0 0 0 255 POLY SCALE rt pg 15 Enter SCALE lt rt and SCALE rt to see that only the former gives the desired result Define a corresponding function M that works in either case 68 16 Exploring Math M SCALE bifo lt gt Observe the results of bifo box if open applied to rt and to lt rt Enter the definition reg j o 2 i and verify that reg 4 and reg 6 give the coordinates of regular polygons inscribed in a unit circle This definitionemploys complex numbers so do not spend time on the definition itself at this point Instead plot the figure reg 6 and various permutations of it and interpret
60. e coordinates of a figure e sliding them to bring the lowest point to 0 0 e sizing them to no more than in magnitude e doubling and subtracting 1 to bring them between _1 and 1 e ravelling them to form a list for use by the plotting function slide 1 lt size 1 gt scale lt size slide slide is Chapter 11 Coordinates and Visualization 65 00 60 34 size slide is 00 10 0 51 lt size slide is 1 1 1 1 0o 1 scale is At 101 The following steps introduced the necessary graphing functions and use them to display the isosceles triangle load graph gdopen a Opens graph window labeled a Use mouse to return focus to J gdpolygon scale is gdshow We then superpose a red right triangle and finally clear the window 255 0 0 gdpolygon scale rt Colors red green blue intensity 0 255 gdshow gclear a A graphics window may be closed by clicking the upper right corner with the mouse The functions provided by the graphics file may be displayed by entering names_z_ However they should for the moment be treated as tools whose internal workings may be ignored provided that their effects are sufficiently understood It will be found most convenient to enter a sequence of graphics commands in a script window opened by entering Control n and to execute them by using the drop down run menu 66 Exploring Math To learn more about the use of graphics use the mouse
61. ea may be computed as follows Roll 32 Sides Roll Width Width Extent available for other two sides Length Sides 2 Length Width Area for given roll and width 60 The whole may be re expressed as a single sentence punctuated as follows Area Roll Width Width 2 Width Although long names such as width and Roll can be helpful in understanding the point of a sentence they can also obscure its structure Briefer but still mnemonic names may be substituted W Width P Roll An abbreviation for the perimeter of the field A P W W 32 W Other grammatical rules make it possible to omit some parentheses The next rule after the rule for parentheses is e A sentence is executed from right to left Consequently the phrase P W W may be re written as P w w Hence A P W W 32 W This can be further simplified by using the fact that multiplication is commutative A W P W W 32 A W P W W 2 Since division is not commutative this trick cannot be repeated but because division by two is equal to multiplication by one half we have A W P W W 0 5 A W 0 5 P W W A W 0 5 P W W Although an unparenthesized sentence or phrase is executed from right to left it is easily read from left to right To illustrate this we will use the right to left execution rules to fully parenthesize the last sentence above A W 0 5 P W W This can now be read from left to right as f
62. ean 2 mean 1 yqm 10 5 3 5 4 5 15 5 1 1 5 3 75 5 5 25 A 2 33333 7 5 33333 3 5 7 5 2 75 3 25 5 5 1 66667 3 3 33333 7 33333 4 7 8 Exercises 7 Experiment with rank cases of the following functions and state in English the meanings of the various results Reverse 28 Rotate Number of items Shape B Transposition Given a year by quarter by month report yqm we may want to see the receipts displayed as a quarter by month by year report qmy If we refer to the successive axes or categories by the indices 0 1 2 we may say that qmy is to be obtained by the transposition 1 2 0 choosing axis 1 then axis 2 then axis 0 Thus qmy 1 2 0 yqm qmy yqm qmy yqm N fo ONN oO A oo o6 o w 14 3 2 2 4 3 mean mean 2 mean 1 qmy 13 75 2 75 4 1 66667 0 5 3 5 4 5 32 Exploring Math 5 3 25 2 33333 315 5 1 1 5 15 25 5 5 7 3 33333 3 5 7 5 5 33333 7 33333 4 7 8 Transpositions may also be used on higher rank arrays as in the following product by year by quarter by month report pyqm 2 2 4 3 10 ypmq 1 0 3 2 pyqm ypmq pyam Boxi
63. er table and may be so expressed in terms of the argument x and its indices as follows e i x 0123 Chapter 14 Polynomials 91 The table m is called the Complete Vandermonde matrix ofx A Vandermonde function may be defined and used as follows v i The Vandermonde function x V x Vandermonde matrix for x 1 8 PRPR BWNHeR Oowo FH 27 64 ly x V x amp mp c Linear function in terms of Vandermonde 8 27 64 125 f c amp p fx 8 27 64 125 Cv V Complete Vandermonde function Jy CV x amp mp c 8 27 64 125 The complete Vandermonde matrix is square and invertible Its inverse provides the inverse linear function which may be used to determine the coefficients of a polynomial that represents the function as illustrated below CV x 4 6 4 1 _4 33333 9 5 _7 1 83333 1 5 4 3 5 Ei _0 1666667 0 5 _0 5 0 1666667 CV x amp mp y The inverse linear function applied to y 1331 CV x amp mp f x Using the fact that y is x 1331 Show the definition of 1 3 3 1sp Exercises 26 Use the foregoing discussion as a model for experimenting with Vandermonde matrices for various values of the arguments x and c and comment on the results The linear function CV x amp mp applies only to arguments 92 Exploring Math that have the same number of items as does x 27 Use x 10 i 10 and y sin 1 amp 0 x to obtain coefficients c such that c amp p agrees with sin for the arg
64. ereees 92 PN aid u OU Go SUERRBELIIERLER EIER TEE OBERE FEEBEETDERDSETUFEDRELTERFLSELSEER 95 As Introduchion senden interne 95 B Insidi us Inyerses u rss anne 95 Ce Rational Numbets ass tesa nee 96 D Irrational NUMbETS cccccescesssccessceesseceeceeseeceeeeesseceeeeesaeensees 97 Ex Complex Numbers uses 97 Complex Numbers ccccccceeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeees 99 A Introduction eueeeeessseessseessneesnsennsneennnennnnennnnennsnensnnennsn een ensn nennen 99 IB Additi n sweet iin 100 C Multiphieation 4 r2 22 2 Ra h eo Han eee 101 Dy Powers and Roots essen nn iai 103 E DINA O a A 104 LO fot a ae eee ee me ee een 107 Aw Secant Slope eraciren echelon chest ke teed ee 107 B Derivative annin eenn ei ean a eNA 108 Gy Aa Daae aa TE K AEE A EE T A 109 D Differential Equations u uneenseesseenenenenennnnnennennn nennen 110 E The Exponential Familys ienr ann a E E R E 112 Inverses and Equations nus2n00nnnnnnnnnnnnnnnnnnnnn nn 115 A Inverse Functions esessecssneessseessneesnseennnensnnnensnenennennn nennen nennen 115 B Monotonic Functions cccccccccecssccsssceesecesecesseeceeeeesseceseeeesaeeesees 116 G Undern neeaeneaeen nenne innen zeaieen cegse vou tiaasTaeeeeen 117 DYEQUAHONS ee A re lenken 118 Readings 2 2 nee ame daaa iaaa ENE 121 A Introduction sense gemeinen nn ine 121 j E A iE I A Hs Se oe eae 121 Cy Sample Topics spee teen anna BR AH
65. erses 117 INVERSES 97 Irrational Numbers 99 irrationals 98 99 iteration 40 41 J 2 7 23 24 25 27 28 33 34 40 63 64 67 87 92 98 102 125 J Introduction and Dictionary 122 J Phrases 122 Kerner 121 Kerner s 122 Lakatos 14 56 57 125 laminate 27 90 languages 14 Least Common Multiple 20 left to right 26 27 65 length 26 66 99 104 105 128 Exploring Math Lewis Thomas 34 Linear functions 76 Linear Functions 75 Linear Vector Functions 77 Linearity 76 92 link 13 27 list 2 3 5 6 10 11 12 17 18 27 28 29 34 38 40 42 43 44 46 56 57 60 87 88 90 96 logic 7 14 97 125 magnitude 46 90 102 104 106 107 major cells 31 math 1 2 4 5 6 7 9 24 33 34 63 76 89 Math 9 mathematical 12 125 matrix 34 60 78 79 81 85 92 93 94 96 matrix product 78 mean 31 32 33 34 39 58 Mixed Bases 47 MN 23 24 33 34 87 monomial 87 Monotonic 118 multiplication 1 5 9 17 18 24 26 44 56 87 90 98 99 100 103 106 107 Multiplication 44 103 native language 23 natural logarithm 120 negative numbers with the standard form limited as it is for positive arguments to numbers 46 Newton 121 Newton s 122 normalization 40 44 47 normalized number 107 noun 9 25 34 50 nouns 24 25 28 Nouns 23 number of items 12 31 40 76 79 90 94 Number of items 31 Numbers 122 numerat
66. es an entree to a wide variety of topics addressed by three authors C Burke D B McIntyre and C Reiter C Sample Topics This section of J Dictionary and Introduction provides brief treatments of a variety of topics You might begin with the discussion of Classification and Sets Sections 8 11 and continue with Directed Graphs and Closure Sections 20 21 The discussion of polynomials Sections 23 28 covers some material already treated here in Chapter 14 but also includes matters such as explicit functions for Newton s and Kerner s methods for finding roots as well as stopes that generalize the notions of falling factorials and rising factorials 122 Exploring Math D Vocabulary and Definitions Begin by printing out the Vocabulary Then with the vocabulary displayed click the mouse on any definition such as Self Classify Equal in the upper left corner A study of the defintion will probably provide all the information you need concerning the conventions used If not display the page of the dictionary headed by III Definitions for details of them 123 References Reiter Clifford A Fractals Visualization and J Second Edition Jsoftware 2000 Iverson Kenneth E Concrete Math Companion Jsoftware 1995 Graham Ronald L Donald E Knuth Oren Patashnik Concrete Mathematics Addison Wesley 1989 Lakatos Imre Proofs and Refutations the logic of mathematical discovery Cambridge University P
67. f its argument much as AREA does State the condition for a non negative volume Try voL 1 A tet The volume is non negative if the vertices of the base triangle are in counter clockwise order when viewed from the leading vertex Use expressions analogous to those used for the area of a triangle to investigate the volume transformation effected by a linear function on a tetrahedron Define a degenerate tetrahedron in which the four points are co planar to illustrate the fact that a linear function on it yields a co planar result 81 Chapter 13 Representations of Functions No computation without representation Adin Falkoff A Introduction A family of monadic functions is commonly represented by a single dyadic function a particular member of the family being obtained by bonding a parameter As an example consider the permutation or anagram function introduced in Chapter 2 a ABCDE 2A a ABDCE f 28A fa ABDCE A family may also be represented in several ways using different dyadic functions For example 014 3 a The indexing or from function ABED p 0 1324 A permutation vector a permutation of i 5 p a ABDCE p amp a A monadic permutation function ABDCE b 0 1 3 2 4 101113 214 bC a The cycle function c ABDCE 82 Exploring Math b amp C a A monadic permutation function ABDCE Since different representations have different uses it is important to
68. f the workings of the definition of p Display the definition of p lt p 4 24 r4 r4321 0001 Exercises 1 Compare the results of 0 4 3 2 1 0 and 4 3 2 1 0 and redefine so that it agrees with for the argument 0 The problem of Exercise 1 could be solved by redefining q and r as follows q gt r 0 amp 0 However it would seem more straightforward to define q as the constant 1 as follows q 1 f 0 domain error f 0 A problem arises because 1 is a noun not a function and the arguments in the gerund p q must both be functions We therefore need a function that returns the constant value 1 when applied to any argument Such constant functions are commonly needed and are produced by the rank conjunction used in Chapter 5 to modify a function as in lt 2 Thus 1 0 x i 4 Rank 0 produces a result for each atom of x 1111 LM 2 Infinite rank gives a single result for any argument x 1 Now is the time Chapter 8 Recursion 49 0123 The function q may therefore be redefined as follows q 1 _ 0 4 3 2 1 0 246211 Finally of rank 0 may be redefined compactly as follows f lt gt 1 _ 0 amp 0 43210 246211 As a second example of recursive definition we will define the sum of the first n odd numbers first met in Chapter 1 sod 0 _ gt lt sod lt sod 4 16 sod 0 1 6 0149 16 25 The definition of sod may be
69. for the following Greatest Common Divisor and Least Common Multiple functions In particular apply them to the argument 0 1 as in table 0 1 and note that with the interpretation of true for 1 and false for 0 as was done by Boole they then represent the logical functions or and and Explain the equality denoted by the following sentence e gt e e gt e e e s 6 First enter at e mt e st e dt e trans Then comment on the results of the following at trans at mt trans mt st trans st dt trans dt The following exercises suggest a sequence of experiments that should be tried only after reviewing the tips on explorations given in Chapter 1 6 Exercises a i 6 t a a a a Double minus half a Chapter 3 Function Tables 21 Dmh Dmh a Contrast the result of the following sentence with those of Exercise 6 a a Average Average a Average 3 141 6 Re enter the sentence a a at a a a from the beginning of this chapter and compare the result with the following a a afa f a 23 Chapter 4 Grammar And Spelling The level is low I can spell all the words that I use but it has not fallen and my grammar s as good as my neighbour s Jacques Barzun W S Gilbert A Introduction We have already made significant use of J why trouble us now with its grammar
70. ion In discussing multiplication we will use further functions illustrated as follows a 5j2 b 3j4 Jca a 5 2 Jcb b 34 Jab a b 52 34 j 4 0j4 354 j cb 354 j b 3j4 102 Exploring Math Multiplication is analyzed in the following sequence of identities a b j ca j cb art j ai br j bi ar br j bi j ai br j bi ar br ar j bi j ai br j ai j bi ar br ar j bi j ai br ai bi ar br ai bi ar j bi j ai br ar br ai bi j ar bi ai br Exercises 6 Express the result of the foregoing sequence in English The real part of a product is the difference of the product of the component lists the imaginary part is the sum of the real part of each multiplied by the imaginary part of the other 7 Re express the final sentence of the sequence in terms of the table ab a b ab j 0 1 0 1 ab The function produces a two element vector representation of a complex argument in terms of its real and imaginary components If we plot the point whose coordinates are given by and draw a line to it from the origin we see the possibility of another two element representation in terms of the length of the line and its angle This is called a polar representation and is given by the function Thus b Angle in radian units rather than degrees 5 0 9272952 b Magnitude also called absolute value 5 for real arguments Mul
71. is is the reversal of the list i y 3210 base i y 1000 100 10 1 y base i y 1996 z 3714 z 8 i z 1996 BV i Equivalent to bv 10 BV 1 996 1996 8 BV 3714 1996 We may also define and explore specific cases of the base value function by combining it with various left arguments bv10 10 amp bv8 86 bv2 2 amp bv8 z 1996 bv2 101 5 What function will yield the representation of a given argument In other words what are the functions inverse to the functions b10 b8 and b2 The adverb _1 gives the inverse of a function to which it is applied Thus inv 1 sqrt sqr sqrt inv sqrt i 6 0 1 1 41421 1 73205 2 2 23607 sqr sqrt i 6 012345 bv8i bv8 inv bv8i 365 1996 0555 3714 bv2 inv 365 1996 00101101101 11111001100 Chapter 7 Decimal and Other Number Systems 39 2 bv2 inv 365 1996 365 1996 We learn to add decimal numbers by adding the items of their representations and performing carries as required What would the result mean if we did not perform the carries For example bv10i bv10 inv d10 bv10i 365 1996 65 0 1 96 3 9 s10 d10 s10 1 12 15 11 10 s10 2361 365 1996 2361 d8 bv8i 365 1996 d8 0555 3714 8 d8 2361 It appears that the sum d10 does indeed represent the correct sum in base 10 Why then do we normally perform the carries We could perform successive carries on the sum s10 as follows
72. it is often convenient to work in terms of normalized numbers and then multiply results by appropriate real scale factors Chapter 16 Complex Numbers 105 The reciprocal of a normalized number is simply related to the number itself For example snb 0 6j_0 8 nb The monad is called the conjugate it reverses 0 6j_0 8 the sign of the imaginary part b b The product with the conjugate is a real number the 25 magnitude is its square root b b We have yet to examine division in terms of the real imaginary representation This may be approached by noting that a b is equivalent to a b that is multiplication by the reciprocal Since we already have expressions for the product and the reciprocal the overall result can be obtained by simple but perhaps tedious algebra A Secant Slope 107 Chapter 17 Calculus If a function is plotted over a range of arguments that includes x and y then the straight line through the pointsx x andy f y is called a secant line and the quotient of the differences y x and y x is called its slope This slope gives the approximate rate of change of the function in the vicinity ofx and y For example f x y 1 3 rise f y f x run y x slope rise Srun 4 The secant slope may be expressed in a function that uses the run as the left argument and in an adverb that may be applied to any function ss 3 2 ss 1 4 ss 1 x 2 f
73. l axis at x Then y is x and conversely x Is _ 1 y A similar treatment of a non monotonic function can illuminate the matter raised in Exercise 4 the square function f graphed on a domain that includes both negative and positive arguments is seen to be an even function and a horizontal line through a point such as y 4 intersects the graph in two points giving two possible values for the inverse Only a strictly monotonic function can have a proper inverse but a non monotonic function may have a useful inverse when restricted to a principal domain in which it is monotonic In the case of the square the non negative real numbers provide such a principal domain and the inverse _1 provides the inverse on it An inverse for arguments not in a principal domain is often easily obtained from the inverse on the principal domain In the case of the square it is simply _ 1 Any periodic function such as the sine or cosine cannot be monotonic but may be when restricted to a suitable domain Chapter 18 Inverses and Equations 117 Exercises 5 Define a function pn that gives both positive and negative inverses of the square function and test it on the argument x 0 1 4 9 16 25 pn 1 Or use or instead of 6 Experiment with the functions n o and their inverses n amp 0 for integer values of n from 0 to 8 Which of the inverses have restricted principal domains 7 What are the limits
74. lish occurs in the next As in English an adverb applies to a verb to produce a related verb Examples occurring in Chapter 1 are The adverb which inserts its argument function between items of the noun to which it applies For example 1 2 3 4 is equivalent to 1 2 3 4 and the function may therefore be called the swm function The adverb which uses its argument function to scan all prefixes of its noun argument 1 2 3 is equivalent to 1 1 2 1 2 3 In English the phrase run and hide uses the copulative conjunction and to produce a new verb that is a composition of the actions described by the verbs run and hide In J is a conjunction that applies its first argument verb to the result of its second argument verb For example a 012345 b 512430 a b 0 a b 0 sumdif 123 45 sumdif 235711 13 Exercises 1 Search earlier chapters for further examples of the various parts of speech 2 State the effect of the adverb in the sentences a b and a b C Punctuation and Other Rules In J a sentence can be completely punctuated so that the only grammatical rule needed to parse it concerns the use of parentheses For example the area of a rectangular field can be computed as follows Length 8 Width 6 Area Length Width Area 48 26 Exploring Math If instead the width and the length of the roll of wire available to enclose the field are given the ar
75. m their coordinate representations For example disp 1 amp Rotate by 1 and subtract 64 Exploring Math disp is WW OY Aa AO length amp 1 Displacements from vertex to vertex Length according to Pythagoras length p Length or distance from origin 5 length disp is Lengths of sides of isosceles triangle 655 heron semip semip Heron s formula for area semip 2 Semi perimeter heron length disp rt Area of the right triangle 9 area heron length disp Area function using Heron area rt 9 area is Area of the isosceles triangle 12 Exercises 1 Plot the points p through t on graph paper and join the appropriate points by straight lines to show the figures is through pg Then use the base and altitude of each triangle to compute their areas and compare with the results of Heron s formula 2 Use the AHD 6 to examine the etymology of the several terms used for figures that differ only in the number of their sides or angles or vertices and suggest a compact common terminology 3 gon 4 gon and n gon from polygon 3 A vertex may be shifted to the left by subtracting a vector with a zero final element Plot the following triangles and use both base times altitude and Heron s formula to compute their areas rts p q r 8 0 is p q s 8 0 Although plotting polygons by hand may be instructive it is also convenient to use the computer to plot them We begin by normalizing th
76. ment 0 3 12 27 48 75 AD x Derivative ofthe exponential applied to argument 1 2 71828 7 38906 20 0855 54 5982 148 413 D x Test of the differential equation satisfied by I1 11 11 The hyperbolic sine 5 amp 0 and the hyperbolic cosine 6 amp 0 introduced in Chapter 14 both satisfy a similar equation but one that involves the second derivative 5 amp 0 5 amp 0 DD x Sinh equals its second derivative 11131311 6 amp 0 6 amp 0 DD x Cosh equals its second derivative I1 1l1 11 1 amp 0 1 amp 0 D D x Sinis minus its second derivative 111111 2 amp 0 2 amp 0 D D x Cos is minus its second derivative 111111 Exercises 7 Use the differential equation satisfied by the hyperbolic cosine together with the approach suggested in Exercise 6 to develop a power series for it coshc ce 0 2 i Use the Taylor series 112 Exploring Math 6 amp 0 t i 6 to confirm this solution 8 Use Taylor series as guides in defining functions to generate power series for the hyperbolic sine cosine and sine 9 Experiment with the weighted Taylor coefficients adverb t for each of the functions treated in Exercises 6 8 study the patterns produced and state its definition 10 Predict and confirm the result of t i 10 11 Study and experiment with the table of derivatives given in Sec B Chapter 2 of Book 3 E The Exponential Family In Chapter 13 we introduced odd and even adverbs that prod
77. ment of such rules in MN lie in the expressions used for polynomials and will be discussed further in the corresponding chapter B The Use of Grammar The rules of grammar determine how a sentence is to be parsed that is the order in which its parts are to be interpreted or executed In particular these rules cover the use of punctuation which can make an enormous difference as illustrated by the following sentences The teacher said George was stupid The teacher said George was stupid The punctuation in J is provided by parentheses as illustrated by the following sentences from Chapter 2 a i 6 b 6 3 a atata 111111 3 a atata Removal of the punctuation yields a quite different result 300000 The parsing of a sentence does not depend on the particular word used but only on the class to which it belongs Thus the English examples used above would be parsed without change if the nouns farmer and Mary were substituted for the nouns teacher and George Similarly the sentence 3 b b b b would parse the same as 3 a atata The classes concerned are called the parts of speech J has only six parts of speech including the punctuation provided by parentheses all of which have Chapter 4 Grammar and Spelling 25 been used in earlier chapters For example the nouns 3 and 2 and the verbs and and occur in the first three sentences in Chapter 1 and the copula analogous to the copulas is and are in Eng
78. mials 87 C Versatility The polynomial is a most important function in math This importance stems from its versatility which in turn stems from a few simple properties The discussion of these properties leads to a number of topics not yet discussed such as complex numbers derivatives power series and transcendental functions including the exponential sine 1 amp 0 and cosine 2 amp 0 Even if you are unfamiliar with such matters you will probably find it fruitful and interesting to use this section as an introduction to them always remembering the injunction of Chapter 1 do not spend too much time on matters that may be at the moment beyond your powers In presenting the properties of polynomials we will use the following in examples c 1 331 d 2 10 4 s ctd p c d c d s p 11 33 112 10 4 3 4 3 5 2 7 9 9 13 12 4 e The sum or difference of two polynomials is itself a polynomial For example the polynomial csp dsp is equivalent to the polynomial g ctd amp p e The product of polynomials is a polynomial csp dsp equals psp e Polynomials can be used to approximate a wide variety of important functions A power series is a polynomial whose coefficients are each expressible as a function of its index For example the reciprocal factorial function expc specifies the power series approximatio
79. mt used in the preceding chapter and pronouns such as it and she used in English e Adverbs such as table in the preceding chapter that apply to verbs functions to produce different but related verbs 24 Exploring Math Knowledge of MN can be very helpful particularly in providing familiarity with numbers and symbols for common functions and with some of the purposes of math On the other hand MN can be very misleading because it shows little concern for simple and consistent grammar For example e The simple forms a b and a b used for some functions of two arguments is abandoned in others as in x for the x n used in J and in for m n the number of ways of choosing m things from n e The rule that a function of one argument precedes its argument as in b and sqrt b is abandoned in the case of the factorial n In J this is written as In e The ambivalent use of the minus sign to denote two different functions as determined by the number of arguments provided subtraction in a b and negation in b is not extended to all functions as it is in J For example a b and b denote divided by and reciprocal a b and b denote power and exponential and a b and b denote the addition table and sum over e The imposition of hierarchical rules of execution for certain functions power is performed before multiplication and division which are performed before addition and subtraction The reasons for the develop
80. n to the exponential funtion Thus expc e8 expc i 8 1 1 0 5 0 1666667 0 04166667 0 008333333 e8 amp p i 4 1 2 71667 7 26667 18 4 1 4 1 2 71828 7 38906 20 0855 e The derivative that is the rate of change or slope of the tangent to the graph of a polynomial is itself a polynomial For example the derivative of csp is 1 c i c amp p e The integral or anti derivative of a polynomial is itself a polynomial For example the integral of c p is 0 c 1 i c amp p e The composition csp d amp p is a polynomial Exercises 88 Exploring Math 11 Experiment with the foregoing examples 12 Define and use plus and times and der 1 i and int 0 1 i Comment on their behaviour der int is an identity function The function plus fails for arguments that do not have the same number of items Try the function plus and examine how the laminate function pads a shorter argument with non significant trailing zeros 13 Explain the reason for the diagonal sums produced by used in the function times See the multiplication of decimal numbers in Section C of Chapter 7 The Taylor series adverb t produces a function that gives the coefficients of a power series For example c amp p t 1 8 13310000 c amp p d amp p t 1 8 2799131240 ABs oT 1 1 0 5 0 1666667 0 04166667 0 008333333 0 001388889 sin 1 amp 0 cos 2 amp 0 sce sin t i
81. ng of various ranks can also be used to clarify displays 33 Chapter 6 Terminology If this young man expresses himself in terms too deep for me Oh what a singularly deep young man this deep young man must be W S Gilbert Special terminology used in various branches of knowledge often imposes a serious burden on a beginner It may sometimes be safely dismissed as pretentious and no better than familiar terms but serious treatment of a topic may well require finer distinctions than those provided by familiar language For example the familiar average may sometimes be substituted for mean as defined in math and statistics However mean refers not only to average the arithmetic mean but also to various ways of characterizing a collection by a single number including the geometric mean harmonic mean and common mean Similarly the grammatical terms adopted in J from English may seem pretentious to anyone familiar with corresponding terms in math but they make possible significant distinctions that are not easily made in MN We illustrate this by a few sentences and the classification of items from them in both J and MN with amp cube with 3 commute into commute pi 7 into 22 2 into cube a i 6 J MN Noun 22 Constant Pronoun pi Variable Verb or Function Function or Operator Proverb cube Adverb or Operator Operator Pro adverb commute Conjunction or Operator amp Operator Pro conjunction with List
82. oduce the first example of a list The comma denotes a catenate verb that appends one list or a single item to another Also experiment with other forms of catenate as in b i 6 a b a b Called stitch a b Called laminate a b Called link Why is it possible to enter a list of numbers asina 0 1 2 3 4 5as well as by using the catenate function as in Exercise 3 Certain results that can be produced by functions can also be entered more simply as constants For example This sentence is equivalent to this constant 3 5 2 3 8 10 3 8 3 5 3r5 3 j 4 3j4 2 3 5 7 2357 Read the first five pages of Part II Grammar of J Dictionary 5 also available in Help as described in Chapter 10 D Spelling The many words in English are each represented by one or more letters from a rather small alphabet The words nouns verbs etc of J are each represented by 28 Exploring Math one or more characters from an alphabet of letters and other symbols For example amp i A Every word of more than one character ends with a dot or a colon Any other sequence beginning with a letter and continuing with letters or digits but not ending with a dot or colon is a name that may be used with a copula as in the following examples a 1 6 Pronoun plus Proverb g Pro adverb p3 amp 2 Proverb The representation of numbers is illustrated by 2 and 2 4 and 0 4 _2 and _2 4 and _0 4 A decimal point must be preceded
83. ok them up in AHD 6 and consult their common Indo European root in the appendix Read the entries in the Indo European sub dictionary of AHD for the roots ag ak ar and gene and look up some of the words derived from them 37 Chapter 7 Decimal and Other Number Systems Sixty four I hear you cry Ask a silly question and get a silly answer Tom Lehrer A Introduction To most people the decimal representation is so familiar and so closely identified with the number itself that it may be difficult to grasp the notion of representation For example what is one to make of the assertion The decimal representation of 365 is 365 We will use lists to clarify the discussion The decimal representation of 365 is 3 6 5 The octal base 8 representation of 365 is 5 5 5 bv The base value function 10 bv 3 6 5 365 8 bv 55 5 365 The main idea of a base or radix representation is embodied in the function which we will now re express in terms of more familiar functions Familiarity with decimals should make it clear that the representation 3 6 5 is to be evaluated by multiplying the first item by 100 the second by 10 and the third by 1 and summing the products Thus r 3 6 5 w 100 10 1 r w 300 60 5 r w 365 The weights w would not be appropriate for a list of other than three items and the following suggests a more general expression 38 Exploring Math y 1996 base 10 y i y Th
84. ollows a is w times the value of the entire phrase that follows it which in turn is 0 5 times the phrase that follows it and so on The foregoing example made no use of adverbs and conjunctions and for a sentence that does include them we need a further rule e Adverbs and conjunctions are applied before verbs For example Chapter 4 Grammar and Spelling 27 a b is equivalent to a b 183 a b is equivalent to 4 amp 3 atb A complete formal statement of the grammar of J may be found in J Dictionary 5 which is also available on the computer by using the Help menu This statement of the grammar should perhaps be studied at some point but it is probably better to begin by reviewing familiar sentences and trying to apply the grammatical rules to them You might review the sentences of earlier chapters as follows Modify and simplify them using the methods suggested in the foregoing examples as well as any others that occur to you Try to read the resulting sentences from left to right using English to paraphrase them Assign values to any names used in the sentences so that they may be entered for execution If you modify a sentence in any way that changes its meaning you will probably be alerted to the fact by seeing a different result upon entering it The following Exercises highlight points that you might well miss in your review 3 Exercises Comment on the sentence a 0 1 2 3 4 5 used in Chapter 1 to intr
85. ons 2 complex numbers 4 70 89 97 98 99 100 102 Complex Numbers 99 101 composition 25 89 117 Concrete Math Companion 7 91 126 Exploring Math conjunction 7 25 33 34 50 60 76 78 96 98 conjunctions 27 76 constant 27 29 34 50 51 57 105 111 constant function 118 constants 27 59 102 conventional notation 7 conventions 123 coordinate geometry 65 coordinate system 65 Coordinates 65 copula 25 28 42 copulative conjunction 25 cosine 89 90 91 92 113 114 115 counting numbers 97 98 cube 3 33 59 109 110 111 113 cursor 5 64 Data 122 Decimal 37 decimal point 28 Decimal representation 119 deductive 14 Definitions 123 degenerate triangle 80 degrees 104 105 delete 5 denominator 102 derivative 89 110 111 112 113 Derivative 110 dervatives 89 determinant 68 80 determinants 69 dfr 105 diagonally 44 Difference Calculus 115 Differential Equations 112 Directed Graphs 122 Displacements 66 distributes 56 75 76 80 103 Distributivity 75 divided by 24 110 Division 106 dropping the menus 63 Editing 64 English 5 10 11 12 23 24 25 27 28 31 33 34 40 104 125 Equal 123 equals 2 3 4 56 57 59 89 91 92 112 113 Equations 120 erase 5 etymology 34 66 even 1 3 9 10 14 18 23 68 79 91 92 97 102 114 115 executable 121 execution 24 26 27 50 64 exploration 1 2 4 7
86. or 102 oblique 44 45 octal 37 odd numbers 3 5 6 51 55 operator 34 51 or 20 origin 65 66 91 104 over 101 Padding 44 Parity 91 parse 24 25 parsed 24 pattern 6 56 110 patterns 4 5 14 114 pentagon 69 perform 34 39 40 52 perimeter 26 66 periodic function 118 permutation 9 10 11 52 56 81 83 84 85 98 permutations 7 10 11 12 52 70 84 97 98 permuted 10 12 permuting 3 Phrases 122 pi 33 34 102 105 Plotting 69 PLOTTING 70 polar representation 104 106 polygons 66 68 70 105 106 polynomial 60 61 87 88 89 90 91 92 93 94 95 111 112 Polynomial Approximations 94 polynomials 7 24 87 88 89 91 97 122 Polynomials 87 POLYNOMIALS 111 power 1 3 24 87 89 90 91 92 93 95 98 110 112 113 114 Power 19 power series 89 112 POWERS AND ROOTS 105 predecessor 97 98 Primitive Notions 122 principal domain 118 Pro adverb 28 33 Pronoun 28 33 Pronouns 23 proof 14 55 56 57 58 59 60 61 110 proofs 7 9 13 14 55 58 60 61 Proofs 13 55 proper inverse 98 99 proposition 57 Proverb 28 33 punctuation 24 Punctuation 25 Pythagoras 66 99 quadrant 66 quadrants 18 quotient 106 109 Index 129 radian units 104 radians 105 Ramble 6 random 3 random generator 29 rank conjunction 50 rank k 31 ranks 32 rate of change 89 109 111 Rational Numbers 98 rationals
87. or Vector a Vector Table or Matrix ar a Matrix Report or Array at a a Array 34 Exploring Math In the foregoing MN makes the same distinction made by noun and pronoun in J but uses the terms constant and variable The term variable may prove somewhat misleading particularly when used for a pronoun such as pi for the ratio of the circumference to the diameter of a circle which is not expected to vary The following sentences may be used to clarify the choice of the word variable sqr The square function in J sqr 0 0 0 sqr 2 2 2 sqr 0 0 0 sqr 2 2 2 sqr 3 3 3 Each of these sentences express a true relation in the sense that each comparison yields 1 However the first pair are true only for the specific arguments 0 and 2 and for no other The last three suggest correctly that the indicated relation remains true for any argument or as we say the argument is allowed to vary This generality is commonly indicated by using a pronoun argument or as stated in MN a variable sqr x x x In MN the term operator or functional is used for both of the cases distinguished in J by adverb and conjunction Moreover in MN the term operator is also commonly used to refer to functions The terms list table and report are used in J with meanings familiar to anyone whereas the corresponding terms vector matrix and array might be known only to specialists The familiar use of vector is as a carrier a
88. or permuting the items It is rather easy to see that a and b are similar but for longer lists similarity is not so easy to spot For example are the following lists similar p 2 15 91040131318 7 10 16 0110130718 q 7 4 713010112 1313150109 18 10 8 O 16 A good general method for determining similarity is to first sort each list to ascending order and then compare the results sort sort p 0001124778 9 10 10 10 13 13 13 15 16 18 sort q 0001124778 910 10 10 13 13 13 15 16 18 4 Exploring Math sort p sort q k ea pg Dee les tHe Ci Wes Kye ces ee We Bot Feet Kl eas UO Eee Ue R sort p sort q The last sentence above uses to match the two lists giving 1 if they agree in every item and 0 otherwise This makes a comparison possible without reading all the items that result from an equals comparison Exercises are commonly used by a student or teacher to test a student s understanding in order to decide what best to do next We will also use them to suggest further exploration A few tips on carrying out such explorations Before pressing the enter key think through what the result should be experiments will teach much more if this rule is always followed On the other hand do not hesitate to try anything you choose the result may be unintelligible or it may be an error message but no serious harm can occur Use lists in experiments Their results often show interesting patterns Do not hesitate to try things t
89. otally unknown For example Sia O 1 1 41421 1 73205 2 2 23607 This result will probably convince you that you have discovered the symbol for the square root and you might experiment further as follows roots a roots roots 012345 e However do not spend too much time on results that may be at the moment beyond your powers It may be better to defer further exploration until you have learned some further math such as complex numbers For example Si a O 0531 0j1 41421 051 73205 052 0j2 23607 e Explore a complex sentence by experimenting with its parts For example i 4 Function for symmetric lists 4 32101234 3 6 gt 3 7 gt 3 The function f g is f atop applied to the result of g Chapter 1 Exploration 5 i gt 3 0123456 13 Identity function Exercises 1 What are the commonly used names for the functions or verbs denoted here by plus times minus or addition multiplication or product subtraction 2 Enter plus to assign the name plus to the addition function and then experiment with the following expressions 3 plus 4 2 11 zero 0 one 1 two 2 three 3 four 4 times three plus four times two 3 As illustrated by the preceding exercise much math could be expressed in English words without forcing students to learn the difficult special notation of math Would you prefer to stick to English words 4 Experiment with the following editing facilities for co
90. ough test Thus sum odds 1 4 9 16 25 36 49 64 81 100 121 144 169 196 225 256 289 324 361 400 1 a 1 a 149 16 25 36 49 64 81 100 121 144 169 196 225 256 289 324 361 400 Hereafter we will suggest many experiments without showing the results expecting students to use the computer to produce them B Ramble or Research The main point of this book is to introduce a new tool for exploring math and to foster its use by applying it to a variety of topics In other words it provides a ramble through a variety of topics rather than a systematic study of any one of them Rambles through any subject can be much more rewarding and more self directed if one has a systematic knowledge of at least some aspect of it For example amateur shell collecting is more interesting to one with some knowledge of molluscs and their classification walks through parks are more rewarding to one with some systematic knowledge of plant animal or insect life and walks through hills and mountains are made more interesting by a knowledge of elementary geology However any book on rambling would surely fail if stuffed with serious digressions on the systematic study of each interesting point as it is discovered It is better to provide the reader with effective but unobtrusive pointers to other sources Books 2 and 3 provide deeper studies of two branches of math arithmetic and calculus Being that branch of math that deals with whole numbers arithme
91. ove from A to C Move from B to C The case of n discs can be expressed in terms of the case of one fewer as follows Move n 1 discs to the intermediary peg B then move the remaining largest disc to c and finally move the n 1 discs from B to c We will use this fact to make a recursive definition as follows H m b 1 amp m lt H 1 A be lt H 2 A b 0 2 amp p ABC Pegs IHp AC 2Hp AB AC BC 3 Hp Transposed table AACABBA CBBCACC Exercises 4 Use discs and pegs or numbered cards and labelled positions on a table to carry out the instructions in the foregoing tables to verify that they provide proper solutions to the Hanoi puzzle Also enter the expression 3 H pand test it as well 5 Give an expression for the number of moves required for n discs 6 Explain the behaviour of the definition of a using experiments to show the permutation provided by the function A the selection provided by the indexing function and the purpose of the monadic function Also redefine the main function m using indexing to perform the necessary permutations 7 Experiment with the function Hv H 8 Read the definition of agenda in 5 and experiment with the use of for self reference in recursive definitions 9 Compare the following recursively defined function n with the first definition of NORM in the preceding chapter f 0 10 amp lt amp 10 0 Chapter
92. point that informal quasi empirical mathematics does not grow through a monotonous increase of the number of indubitably established theorems but 14 Exploring Math through the incessant improvement of guesses Italics added by speculation and criticism by the logic of proofs and refutations The main point of the present book is to exploit a new tool for the exploration of relations and patterns that can be used by both mathematicians and laymen to find those guesses that are amenable to and worthy of proof We will defer further discussion of proofs to Chapter 9 partly to allow the reader to garner guesses that can be used to illuminate the discussion We will however recommend the reading of Lakatos at any point The book is highly entertaining instructive and readable by any layman with the patience to look up the meanings of a small number of words such as polyhedron polygon and convex The following quotes from Lakatos reflect his view of the importance of guessing Just send me the thereoms then I shall find the proofs Chrysippus I have had my results for a long time but I do not yet know how I am to arrive at them Gauss If only I had the theorems Then I should find the proofs easily enough Riemann I hope that now all of you see that proofs even though they may not prove certainly do help to improve our conjecture Lakatos On the other hand those who because of the usual deductive presentation of mathema
93. provide transformations from one to the other The monadic cases of A and c provide such transformations A p 2 A p A a ABDCE b C p 101113 214 C b 01324 The behaviour of these various representations of permutations can be studied by using random permutations generated by the function For example q 9 713264058 A q 288918 A q A i 9 71326405 8 C q 1113 217 5 4 6 0 8 Exercises 1 Generate a table of all permutations of order 4 i 4 A 1 4 2 Use the example of q 9 and C q to illustrate the scheme used in the cycle representation of permutations The third box of c q signifies that item 5 moves to position 7 item 4 to position 5 item 6 to 4 item 0 to 6 and item 7 to 0 Moreover item 8 moves to 8 and therefore remains fixed Use the help menu for discussion of permutations in the introduction to the dictionary the vocabulary and the phrase book 3 Isa permutation a linear function If it is produce the matrix m that represents it in the expression m amp mp Chapter 13 Representations of Functions 83 m q i q 85 Chapter 14 Polynomials A Coefficients Representation A function that is a multiple of a non negative integral power of its argument is called a monomial In MN it is written in the form 3x yielding the value 12 if the argument x has the value 2
94. r equal freezing boiling points 40 0 100 im m I 1 Inverse ofm is s I Inverse of s m s temp _40 0 100 im m s temp _72 0 180 is im m s temp _40 32 212 ffc is im ffc cff temp _40 32 212 116 Exploring Math cff ffc temp _40 32 212 In general if several functions are applied one after the other the inverse is obtained by applying their inverses in reverse order Exercises 1 Define the adverb FI _1 fix and invert and predict and confirm the results of applying it to each of the following functions cff m s is im cff ffc 2 Repeat Exercise 1 for the following functions perhaps using the simpler I _1 instead of FI A N NARA R The last function gives a domain error because is a constant function giving 1 for any argument and a constant function cannot have an inverse 3 Repeat Exercise 1 for the following functions x amp 2 amp 3 amp 30 4 Although 2 and _2 both yield 4 the inverse function yields only 2 when applied to 4 Comment on this matter B Monotonic Functions A strictly monotonic function is one that tends in the same direction as its argument increases A graph of such a function as for example provides a visualization of its inverse as follows at any point y on the vertical axis draw a horizontal line to intersect the graph of and from the point of intersection draw a vertical line to intersect the horizonta
95. rcise 7 Moreover the determinant function provides an even simpler statement of area than does Heron s formula and yields additional important information Thus det rt 1 0 5 340 5 940 5 970 5 det rt 1 0 5 9 det 1 A rt 1 0 5 _9 AREA det 1 amp 0 5 AREA rt 9 Exercises Chapter 11 Coordinates and Visualization 67 9 Ifyou are familiar with the computation of determinants from high school check the foregoing results by hand 10 The result of AREA is positive if the coordinates are in counter clockwise order when plotted and are negative if clockwise Test this for various triangles 11 What is the significance of a zero result from AREA 12 Enter t 7 2 10 to generate a random table of seven points Referring to these points by the letters a through c determine which of the last five lie on opposite sides of the line determined by the first two Enter L 0 1 t and compare signs of the areas of the triangles c L and D L etc 13 Compute the area of the pentagon pg of Section A Referring to the points by A E compute the three signed areas A B CandA C D andA D E and add them C Plotting Multiple Figures As illustrated by Exercise 4 different figures may be displayed together However as seen from the same exercise they are scaled independently and therefore do not give a satisfactory picture We will now rectify this by developing functions that will handl
96. ress 1976 Hui Roger K W and Kenneth E Iverson J Dictionary Jsoftware 1998 This text is available on line in the J system as discussed in Chapter 10 American Heritage Dictionary of the English Language Houghton Mifflin Any edition that includes the appendix of Indo European roots Thomas Lewis et cetera et cetera Notes of a Word Watcher Little Brown and Company 1990 absolute value 102 104 addition 1 5 9 18 24 44 46 56 57 75 76 80 87 100 Adverbs 23 27 agenda 50 53 alphabet 28 ambivalent 24 anagram 10 12 83 and 20 appendices 7 approximating functions 90 area transformation 80 arguments 6 18 23 24 31 34 38 46 50 69 78 79 87 90 94 97 104 109 arithmetic 7 33 57 58 97 Arithmetic 97 arithmetic progression 57 array 29 30 34 assign 5 87 associative 57 103 associativity 57 75 atom 34 50 atop 5 average 31 33 99 axes 30 31 axioms 14 axis 31 91 backspace 5 base 8 37 40 125 Index base value 37 38 bisection 121 block 40 41 Boole 20 Boolean 120 Bordered 18 101 box 13 30 64 69 70 84 boxed roots 88 Boxing 32 by 24 101 calculus 7 Calculus 109 carries 34 39 40 44 catenate 27 Celsius 117 Classification 122 Closure 122 Coefficients Representation 87 Colors 67 commutative 18 26 57 103 commutativity 57 75 commuted 20 companion volume 7 Comparis
97. rrecting errors e Correct a line being entered by using the cursor keys marked with arrows to move the cursor to any point and then type or erase using the delete or backspace keys The cursor need not be returned to the end of the line before entering the line e Revise any line by moving the cursor up to it and pressing enter to bring it down to the input area for editing Not only is it important to think through the expected result of an experiment before executing it on the computer but it is also a good practice to look for patterns in any lists or tables you may see Then verify your observations by doing calculations by hand for short lists and then test them more thoroughly on the computer For example the list of odd numbers 1 2 a 135791 may be added by hand to give 36 Now add only the first five of the list the first four and so on down to the first one Do you see a pattern in these results If not compare them with the following list of squares 1 a 1 a 14 9 16 25 36 6 Exploring Math It appears that for any value of n the sum of the first n odd numbers is simply the square of n This may be tested further as follows n 20 a i n odds 1 2 a odds 135 79 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 sum sum odds 400 n n 400 The sum function gives the sum of its arguments but calculation of the subtotals the sum of the first one the first two etc would provide a more thor
98. ry some of its Exercises Note particularly the section on Mixed Bases 47 Chapter 8 Recursion re back currere to run AHDJ 5 The factorial function introduced in Chapter 2 was seen to be a product of the first positive integers Thus In 4 24 4 13 4 3 2 4 3 2 1 4 3 2 1 24 24 24 24 It would therefore appear that n might be defined simply as n n 1 Such a definition is said to be recursive because the function being defined recurs in its own definition But a sequence of the form fn n f n 1 n n 1 f f n 2 would continue forever through n 0 and n _1 etc and it is clear that two further pieces of information are required when to stop the process and the value of the function for the argument at the stopping point For the present case of the factorial the stopping condition could be that the argument be 1 and the stopping value could be given by the identity function The three required functions are p f lt q r 1 amp The complete definition may now be expressed and used as follows f p q r 4 24 0 1 2 3 4 5 12 6 24 120 48 Exploring Math In the definition of the conjunction ties the functions p and q to form a gerund from which the agenda conjunction selects one for execution according to the index 0 or 1 provided by its right argument function r Once is defined as above we can experiment with p and the other functions to see some o
99. s 22 A SPOT TOSP 23 A SPOT TOPS 24 A SPOT index error 24 A SPOT i 24 A SPOT SPOT SPTO SOPT SOTP STOP STOP PSOT PSTO POST POTS PTSO PTOS OSPT OSTP OPST OPTS 12 Exploring Math OTSP OTPS TSPO TSOP TPSO TPOS TOSP TOPS But what is the general relation between the number of permutations and the number of items in the list to be permuted Although we are dealing with English words and anagrams rather than with numbers this is a proper mathematical question because it concerns relations The question can be answered in the following steps In a four letter word the first position in an anagram can be filled in any one of four ways Having filled the first position the next can be filled from the remaining three letters in three different ways The next position can be filled in two ways The last position can be filled in one way The total number of ways is the product of these that is four times three times two times one This product over all integers up to a certain limit 4 in the present example is so useful that it is given its own name factorial and symbol Thus 14 24 A 3 2 1 24 101234567 1 1 2 6 24 120 720 5040 The number of items in a list is a function that is also provided with a symbol w3 APT w3 3 i w3 012345 i w3 A w3 APT ATP PAT PTA TAP TPA Exercises Chapter 2 Whatis Math 13 4 Comment on the following experiments sort
100. s in disease vector It might be thought that a vector carries its items but the actual etymology of the term in math is quite different although not as bizarre as that of matrix New terminology should be approached by using dictionaries to learn the etymology of terms both old and new For example a verb is defined as a word that amongst other things expresses an action the corresponding word function comes from a root meaning to perform Attention to etymology is also rewarding in every day work For example the meaning of atom appears clearly in its derivation a not tem cut as something that could not be cut The American Heritage Dictionary 6 presents etymology in a particularly revealing manner all words derived from a given root are listed together in an appendix This highlights surprising and insightful relations such as that between tree and true As a further example the root tem that occurs in atom also occurs in anatomy microtome and tome Incidentally tome does not necessarily mean a big book but rather one of the volumes cut from a book such as the 24 tomes of the original Oxford English Dictionary Lewis Thomas a noted bio chemist explored the pleasure and profit of etymology in his delightful book et cetera et cetera 7 It is well worth reading l Chapter 6 Terminology 35 Exercises Speculate on the possible relation between the similar sounding words tree and true Then lo
101. ssed as a product 23 qtr eine n n n n 166666 Zi 6 6 6 6 6136136 The last agreement between n n and n n is based on the fact that multiplication is defined as repeated addition The foregoing attempted to show why two results were equal by exhibiting their equivalence to other results where the equivalence was already known or obvious This is perhaps the only way to answer the question why However the equivalences assumed may be made clearer by laying out the steps of the argument as a proof that is as a succession of equivalent statements annotated by the justification of the equivalence of each to the one preceding it Thus q 1 2 i n q Summation is symmetric unaffected by ordering 2 q q Half sum of equals is an identity 2 qt q Summation is symmetric 2 qt q Summation distributes over division n n n n The definition of multiplication Such a list of supposedly equivalent sentences can be tested for careless errors by assigning a suitable value to the argument n entering them on the computer and comparing the results This putative proof has not proved anything but it has as Lakatos would say broken the original conjecture into a collection of sub conjectures each of which may be profitably examined Consider the first assertion that summation is symmetric and gives the same result when applied
102. t dropping the menu by pressing a key usually while holding down the control key as indicated to the right of the item s name For example as shown in the help menu the F1 key may be pressed to display the J vocabulary and any entry in the vocabulary may be chosen for display by double clicking on it with the mouse A definition is then displayed and may also be printed by using Print topic in the file menu Exercises 1 Using items from the help menu display and read various pages from the User Manual including Chapter 1 2 Display and read a few sections from the introduction to the J dictionary 3 Read the section on grammar in the J dictionary 62 Exploring Math B Editing As remarked in Chapter 1 a previously entered line can be brought to the input area for editing and re entry by moving the cursor up to it and pressing enter Moreover a line containing any phrase can be found by pressing Control f to highlight the search entry box entering the phrase in it and pressing enter Repeated searches on the same phrase will find successive occurrences of it Pressing Control d drops a menu of previous entries one may be selected for use by pressing the up arrow C Script Windows Enter Control n to open a script window enter one or more J sentences in it and press Control Shift w to execute the sentences The execution occurs in the execution window and can be viewed by entering Control Tab to switch back to
103. th is about numbers So it is but numbers are not the only nor even the most important concern of math It would be more accurate to say that math is concerned with relations and with proofs of relations Although the first chapter dealt only with numbers it should be clear that the interesting aspects were the relations between results For example a i 6 b 6 b 512430 3 a 0369 12 15 atata 0369 12 15 3 a atata addition 11131311 a b 011000 sort sort b 012345 sorta 012345 sort a sort b 111111 The first six non negative integers The integers in random order The relation between multiplication and The lists a and b are not equal But are similar one is a permutation of the other 10 Exploring Math We will further illustrate this matter of relations by examples that do not concern numbers For example the word Post is said to be an anagram of the word spot because the letters of spot can be permuted to give the word PosT Thus spot and post are similar in the sense already defined for lists The similarity of these words may be tested as follows w SPOT x POST sort w OPST sort x OPST sort w sort x 1111 Sorting w produces oPsT Is it an anagram We will say that it is although it is not an English word You could and should attempt to write down all distinct anagrams of sPoT finding a surprising number of English words among them However
104. tic is the most elementary and accessible of subjects in math but as treated in Book 2 Chapter 1 Exploration 7 it also provides simple introductions to many more advanced topics including proofs permutations polynomials logic and sets These books are easy to consult because they use the same J notation Moreover they incorporate more systematic introductions and discussions of the notation itself Further texts of this character include Reiter s Fractals Visualization and J 1 and Concrete Math Companion 2 On the other hand treatments in conventional notation of a wide variety of topics are more readily available in libraries Use of them in conjunction with the present text will require sometimes difficult translations between J and conventional notation However the effort of translation is often richly repaid as it is in translating from one natural language to another by deeper understanding of the matters under discussion In fact a deep appreciation of the method of exploration proposed here may best be found in an attempt to write a companion volume to some chosen conventional text Some guidance in such an endeavour is provided by Concrete Math Companion 2 published as a companion to Concrete Mathematics 3 A Relations Chapter 2 What Is Math math is the short form of mathematics for which the British use maths preserving the ugly plural form for a singular noun It is commonly thought that ma
105. tics come to believe that the path of discovery is from axioms and or definitions to proofs and theorems may completely forget about the possibility and importance of naive guessing Lakatos Exercises 6 Read the three pages of Section C Chapter 5 of Book 2 C Summary In brief we will interpret math in the following sense it concerns relations and provides languages for expressing them as well as for expressing transformations on tangible representations For example the first four counting numbers can be represented by the list of symbols 123 4 11234 A transformation or function 126 24 Chapter 2 Whatis Math 15 1234 A second transformation 126 24 1 2 3 4 1 2 3 4 Equivalent to the first 1111 17 Chapter 3 Function Tables The pleasures of the table and make it plain upon tables belong to all ages that he may run that readeth it Jean Anthelme Brillat Savarin Habakkuk The effect of multiplication can be shown rather neatly in a succession of products of a list as follows a i 6 Ora 000000 1l a 012345 2 a 0246 8 10 However a more perspicuous table of products with each item of a can be prepared as follows a a 00 00 0 0 012 3 4 5 02 4 6 810 03 6 912 15 04 8 12 16 20 05 10 15 20 25 Similar tables can be prepared for other known functions For example a a at a a a 00 000001234 50 1 2 3 4 5 101 2 3 4 5
106. tions of the numbers of Exercise 1 and use them in expressions to obtain the standard decimal representation of the sum b In the table produced in Ex 8 each of the shorter lists that is 3 6 5 and 2 9 are padded with zeroes on the left a change that does not change the values of the numbers they represent In Ex 9 the representations are not so padded and the lists of differing lengths cannot be added directly They may be added as illustrated below ar br cr I3 6 511 9 9 6 2 9 bvl0 amp gt ar br cr 365 1996 29 bv10i bv10 amp gt ar br cr 5 oro O o Ww 6 9 2 N 239 bv10i bv10 amp gt ar br cr oONGO OH atb c 2390 Padding can also be provided more directly using the fact that the simple opening ofa boxed list pads it albeit on the wrong side gt ar br cr NEW O OV O o u ono Oo pad 1 amp gt pad ar br cr 0365 44 Exploring Math 1996 0029 C Multiplication The commonly taught methods for addition and multiplication both interleave carries with other computations in multiplication each item of the multiplier is applied to the multiplicand and the carries are propagated to give a list of results which are then added to lists for the other items of the multiplier producing a further sequence of carries However as in addition the carries can all be segregated in a final normalization For example a 365 b 1996 ar bvl0i a
107. tiplication is easily expressed in terms of the polar representation the magnitude is the product of the magnitudes and the angle is the sum of the angles For example x a b a b 5 38516 0 3805064 5 0 9272952 26 9258 1 3078 a b 26 9258 a b a b Both representations 5 2 3 4 7 26 5 38516 0 3805064 5 0 9272952 26 9258 1 3078 Chapter 16 Complex Numbers 103 The measure of an angle in radians is the length of arc measured on a circle of radius one unit consequently one half pi radians is a right angle and therefore equivalent to 90 degrees and pi radians is a straight angle of 180 degrees Since the constant 180p_1 is 180 multiplied by the reciprocal of pi the conversions between radians and degrees may be expressed as follows rfd 1r180p16 Radians from degrees dfr 180p 1 amp Degrees from radians rfd 0 45 90 180 0 0 7853982 1 5708 3 14159 dfr rfd 0 45 90 180 0 45 90 180 pid dfr 1 Polar representation in degrees pid a b 0j1 151 _ 150 5 38516 21 8014 5 53 1301 1 90 1 41421 45 1 180 D Powers and Roots We will illustrate the use of powers and roots by developing a function to give the coordinates of regular polygons 2 1 Second square root of _1 031 2 _1 i 4 First four powers of second root of _1 1 0j1 1051 2 _1 i 4 Coordinates of 4 sided polygon square oOorRrOrF FOr O 3 1 Cube root of _1 0 530 8660254 3 _1 i 6 Coordinates of hexagon
108. ts of the linear function Is the right angle of rt retained Do the two equal sides of is remain equal Is the order of the vertices reversed We may also apply the function AREA of Chapter 11 to compare the areas AREA det 1 amp 0 5 det AREA L rt AREA rt 23 AREA L is AREA is 23 The areas of the two triangles appear to be multiplied by the same factor In fact the area transformation produced by a function m amp mp is the determinant of m det det m _23 We now consider three points on a line that is a degenerate triangle having zero area a 3 4 b 5 13 deg a b a 4 3 b 4 3 4 5 13 4 5 10 75 AREA deg AREA L deg Chapter 12 Linear Functions 79 This result suggests correctly that a linear function transforms a line into a line a fact that suggests the use of the term linear for it A point in three dimensional space can be represented by a three element vector such as p 3 1 5 A linear function on such a point must of course be represented by a 3 by 3 matrix m Moreover a tetrahedron may be represented by a 4 by 3 table and the function AREA may be modified to give its volume as follows 12 13 14 15 VOL det 1 amp 1r6 Exercises Use a tetrahedron such as tet 0 0 0 0 0 1 0 1 0 1 0 0 whose volume is easily computed to test the behaviour of the function vou Use a permutation of the vertices of tet to show that voL gives the signed volume o
109. uced the odd and even parts of functions to which they were applied Moreover we saw that the odd part of the exponential was equivalent to the hyperbolic sine and that the even part was equivalent to the hyperbolic cosine Thus O E O E O E x i 6 0 1 1752 3 62686 10 0179 27 2899 74 2032 1 1 54308 3 7622 10 0677 27 3082 74 2099 1 2 71828 7 38906 20 0855 54 5982 148 413 1 2 71828 7 38906 20 0855 54 5982 148 413 O t E t x Coefficients of odd and even parts of 01 0 0 1666667 0 0 008333333 100 5 0 0 04166667 0 5 amp 0 t 6 amp 0 t x Coefficients of hyperbolic sine and cosine O 1 0 0 1666667 0 0 008333333 100 5 0 0 04166667 0 O t E t x Weighted Taylor coefficients 010101 101010 5 amp 0 t 6 amp o t x 010101 101010 Ifj is applied to the argument of the hyperbolic sine to make it imaginary the odd positions of the coefficients of the resulting function 6 amp 0 j are unaffected because they are all zero Moreover those in each fourth place are multiplied by _1 that is the fourth power of 5 1 The function 6 amp 0 3 is therefore equivalent to the cosine Thus 680 8j t x 1 0 _0 5 0 0 04166667 0 Chapter 17 Calculus 113 2 amp 0 t x 10 _0 5 0 0 04166667 0 The sine may also be similarly expressed in terms of the hyperbolic sine Moreover all four of these functions can be expressed directly in terms of the exponential using only the function j and the odd and even
110. uments x Use expressions of the form c p sin 5 2 x to show the comparison clearly 28 Test the use of csp to approximate sin by evaluating c amp p sin z for other arguments such as z 0 65 0 8 and z i 5 2 and comment on the results The approximation is good in the range covered by x 0 0 9 but may be very bad for arguments outside this range F Polynomial Approximations Sections C and E have presented two methods of approximating a function by a polynomial The first used the Taylor series t i n and the second the complete Vandermonde matrix cv x to fit the function exactly at the points x We will first compare their results for the example treated in Exercise 27 sin mp CV 18 amp 0 mp V i x 10 i 10 te sin t i x vc CV x mp sin x sin tc amp p sin vc amp p 5 2 x 0 0 _2 18587e 13 1 5066e 12 _5 27356e 16 _4 43534e 14 4 92267e 12 1 02958e 11 _1 04966e 12 1 22129e 11 1 40716e 11 1 53755e 11 _9 06789e 11 _4 9381le 10 4 78062e 12 1 69003e 11 _2 14316e 9 _7 82095e 9 3 04181e 11 6 7725e 11 sin tc amp p sin vc amp p 5 2 x 0 1 0 _5 27356e_16 1 5066e_12 _4 92267e_12 _4 43534e_ 14 Er 04966e 12 1 02958e _ 11 1 40716e 11 l 1 22129e_ 11 9 06789e _ 1111 53755
111. x equals E x for any argument x Graphically this implies that the graph of an even function is reflected in the vertical axis A function o is odd if o is equivalent to 0 that is o x equals 0 x for any x Consequently the graph of an odd function is reflected in the origin Exercises 19 What is the parity odd or even of each of the functions sine and cosine 20 Enter sin t i 8 and cos t i 8 and comment on the power series of odd and even functions The coefficients of all odd powers of an even function are zero and conversely 21 What are the parities of the products of an even function with an even an even function with an odd an odd with an odd Test your assertions 22 What is the parity of the exponential function The exponential is an example of a function that is neither odd nor even However any function can be expressed as the sum of two functions an odd part and an even part For example opex 2 Q epex 2 opex epex opextepex i 8 1 2 71828 7 38906 20 0855 54 5982 148 413 403 429 1096 63 O 1 1752 3 62686 10 0179 27 2899 74 2032 201 713 548 316 1 1 54308 3 7622 10 0677 27 3082 74 2099 201 716 548 317 1 2 71828 7 38906 20 0855 54 5982 148 413 403 429 1096 63 The odd and even parts of a function may be functions of interest in their own right In the present case opex and epex are the hyperbolic sine and hyperbolic cosine often abbreviated as sinh and cosh denoted in
112. y if followed by a condition The result of the function is the result of the last sentence that is r The entire definition is terminated by a right parenthesis alone on a line A function that works correctly on the argument that guided its definition may not work in general and should be thoroughly tested For example NORM 10 11 12 122 10 NORM 10 11 12 10 10 11 12 122 1122 The discrepancy clearly occurs because the carry computed in the final iteration is not zero and must not be ignored To rectify this we make the condition for repetition depend upon a non zero carry as well as upon a non empty argument NORM 3 0 r i c 0 label_loop if c 0 0 lt y do ci c _l y r 10 ci r c lt ci 10 y y goto_loop end r NORM 10 11 12 1122 42 Exploring Math NORM 1234 5 6 123456 The function may now be generalized to a dyadic definition in which the first argument specifies the base used each occurrence of 10 is replaced by x and the line NORM 3 Oisreplaced by NORM 4 0 NORM 4 0 r i c 0 label _loop if c 0 0 lt y do ci c _l y r x ci r ce lt ci x y y goto_loop end r 8 NORM 5 3 21 10 NORM 10 11 12 15 551112 2 Finally it will be convenient to define a function whose dyadic case is NORM and whose monadic case is 10 amp NORM Thus N 10 amp NORM NORM 8 N 5 3 2

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