Home

HP Solve: Issue 22 (January 2011)

1. 12 O RT 6 23 Q Three resistors in parallel R1 R2 amp R3 have an equivalent resistance of the reciprocal of the sums RI i AD RT 1 1 1 of the reciprocals of each resistance P RI R2 R3 Here is an example Problem See Fig 1 Calculate the R3 30 Ohms equivalent resistance of the following points B A to B This is a 40 ohm resistor R4 with three series resistors R3 R2 R1 in parallel The series equivalent is their sum or 60 Pei ohms 30 20 10 60 ohms in parallel with 40 ohms 1 24 AG Ohms 20 Ohms ohms A to C This is two sets of two series connected resistors R3 amp D R4 R1 amp R2 connected in parallel These equivalent resistors are 70 ohms and 30 ohms in parallel The effective resistance 1s 21 ohms Fig 1 Example series parallel problem A to D This is a 10 ohm resistor connected in parallel with three series connected resistors R2 R3 amp R4 The series equivalent is 20 30 40 90 ohms The equivalent of 90 ohms in parallel with 10 ohms is 9 ohms HP Solve 22 Page 8 Page 1 of 2 B to C This is a 30 ohm resistor R3 connected in parallel to three series connected resistors R2 R1 amp R4 The equivalent of the series resistors is their sum or 80 ohms 80 ohms in parallel with 20 ohms is 21 ohms B to D This 1s two series connected sets R2 amp R3 and R1 amp R4 in parallel This is the equivalent of 50 ohms in parallel with 50 ohms which is 25 ohm
2. 60 The information in the table translates into an efficiency advantage for RPN of ten to thirty per cent depending on which type of algebraic is being used I don t find that to be a persuasive reason to choose RPN over AOS My indifference to keystroke efficiency is related to the way that calculators and computers are used by engineers My perception is that engineers are paid to analyze and solve problems not to enter data or programs into machines Thus keystroke entry tends to be a small bordering on insignificant part of the typical day the typical week and the typical year of an engineering career I look at it this way Ifa typical engineering day involves seven hours of thinking and problem formulation and solution and one hour of data entry would an employer care or even notice if one machine were ten per cent or even twenty five per cent more efficient I admit that there must be some careers where keystroke efficiency 1s important I suspect that those careers are closer to that of a clerk typist than to that of an engineer Additional Algebraic Mechanizations A discussion of algebraic mechanizations in calculators manufactured by HP would not be complete if it did not address user preference 1 e whether to use RPN or algebraic On page 39 of his book A Guide To HP Handheld Calculators and Computers Wlodek Mier Jedrzejowicz wrote Many users and not only those brought up on Friden calculators prefer RPN Others fin
3. how should the manual issue be viewed using today s business outsourced model Many of you reading this have watched HP up close and personal during their active participation at HP Handheld HP Solve 22 Page 39 Page 2 of 3 Conferences HHC s since 2002 when HP GM Fred Valdez broke the mold for HP User Community relationships The record trend reversing attendance at HHC 2007 witnessed the changes that are taking place slowly but consistently A calculator GM Wing first met at HHC 2007 with the support of the post Carly president Hurd demonstrated that HP is changing in ways that brings a smile to the face of the legacy user Product indicators of this very positive change are the HP50g and the HP35s Regarding the latter the HP35s manual 15 available to be down loaded at http h10010 www1 hp com wwpc pscmisc vac us product_pdfs user_guide pdf Printed manuals is still an HP challenge and there have been several solutions proposed to solve this issue After a long period of working on an agreement with at least one third party they could not agree to go forward Business volume 15 the primary issue I am so tempted to get into the HP printed manual business Still having a manual in a useful downloadable form as illustrated at the link above is most helpful If you must have a printed copy you may print it yourself or have it done at a copy store and bind it using the method described in my HHC 2004 paper Personal Low Cost Bind
4. 66 3 To change toggle the radix on the Voyager machines press and hold the lator You may then see the two displays shown in Table 1 key when turning on the calcu HP Solve 22 Page 36 Page 2 of 2 HP s Calculator Manuals HP Solve 22 page 37 Article HP s Calculator Manuals Richard J Nelson Legacy users of HP s Calculators usually remember the manuals that came with their machines in a positive way Popular opinion seems to suggest that newer manuals are not as good as the older manuals and certainly every manual user old or new seems to complain that there are never enough examples Most commonly the manual that describes the machine and how to use it 1s called the owner s manual This 1s a generic term for a manual that comes with your car refrigerator or calculator The owner s manual is assumed to contain the important information related to the product and how to use it HP didn t start out calling their calculator manuals Owner s Manuals and they don t call them by that name today The HP 35A 1972 came with an HP 35 operating manual 36 pp The lower case 1s HP s choice The HP 65A 1974 came with an HP 65 Owner s Handbook 107 pp The HP 67A 1976 came with an HP 67 Owner s Handbook and Programming Guide 353 pp The HP 41C 41CV came with an Operating manual A Guide for the Experienced User 71 pp The HP48G Series machines 1993 came with a User s Guide 592
5. Series5 Series6 Series Series amp Fig 1 A linear plot for the results Solve 22 Page 31 Page 5 of 7 deviation from an expected smooth curve may be due to special schemes used to obtain accurate answers that seem to be most effective for the polynomials of order 10 Figure 2 plots the same data using log log scales The series labeling is the same as Figure 1 Series1 i Series Series3 Seriesd Series5 Series6 Series Series amp Fig 2 A log log plot for the results Applying simple linear and multiple linearized regressions to the data I found the following empirical model Number of observations 88 F statistic 1839 17 Adjusted R Square 0 97688 In NormErr 0 0896701802129991 1 01052899010609 In Root 359 972416836311 N 2 The above model agrees with Figure 2 which shows plots that are somewhat linear on the log log scale What makes the above model interesting 15 that it not only came out as the best model among hundreds of competing models but it also 1s similar to the best model for the PROOT errors on the HP 50G calculator Number of observations 88 F statistic 4229 83 Adjusted R Square 0 98982 In NormErr 0 112444241487657 1 01931043509749 In Root 357 981953542852 N 2 The corresponding coefficients in the above two models have values that are fairly close to one another This is truer for the slopes This
6. Winter 15 here in the northern hemisphere and many HP Solve readers are spending more time indoors because very cold weather is at hand even here in the Sonoran Desert The Consumer Electronics Show is January 6 9 201111 Las Vegas I will be attending and if you will be there let me know Perhaps we can have Saturday night dinner together as a group of HP users like we did last year Here is the content of this issue S01 This is collection of repeating regular columns From the editor This column provides feedback and commentary from the editor RPN Tip 22 Here is an oldie but goodie reprint that explains the basics of RPN One Minute Marvels This OMM reverses the objects on the stack It is short and fast and illustrates some of the criteria used to select OMM s Math problem challenge 3 Here is an equivalent resistance problem that is really very easy to solve IF you use the right approach You must think out of the box on this one S02 HP Algebraic by Palmer Hanson What is an algebraic calculator How do HP algebraic calculators compare with other calculators RPN and Algebraic Palmer is very well known in the algebraic calculator community is one of those few individuals who are able to easily move between the two user interfaces Personally I have to make my brain work extra hard when using an algebraic calculator S03 Better Problem Solving Part I The
7. 1979 About the Author E ZEE Namir Shammas is a native of Baghdad Iraq He resides in Richmond Virginia USA Namir graduated with a degree in Chemical Engineering He received a master degree in Chemical engineering from the University of Michigan Ann Arbor He worked for a few years in the field of water treatment before focusing for 17 years on writing programming books and articles Later he worked in corporate technical documentation He is a big fan of HP calculators and collects many vintage models His hobbies also include A traveling music movies especially French movies chemistry cosmology Jungian psychology mythology statistics and math As a former PPC and CHHU member Namir enjoys attending the HHC conferences Email me at nshammas aol com HP Solve 22 Page 33 Page 7 of 7 Commas in the HP Calculator Display HP Solve 22 page 34 Article Commas in the HP Calculator Display Richard J Nelson The comma delimiter the display of an HP Calculator serves two basic purposes The first and most important purpose is differentiating the two parts of a number the number and its decimal part e g 1618033988 75 US HP Solve readers will look twice at this because such a number looks strange to them The comma used for this example is called the radix mark and the radix mark used in the US and many other English speaking countries 15 a period HP calculators have the feature of setting the radix
8. In its simplest form pressing a calculator key involves two parts First your eye has to search for and find the key Second the key must be pressed If you do not have to do the first part you save of a keystroke and the inverse notation doesn t need to be on the keyboard HP Solve ft 22 Page 24 Page 3 of 4 Do you have an idea for a regular scientific calculator function and how it may be improved to make it easier faster or simpler to use If so send it in to the editor for possible inclusion in this series of articles What do you think Have you tried using the A program in OMM HP Solve Issue 14 Send an email to the editor with your thoughts on how classical problems may be better solved on future calculators Email HP Solve at hpsolve hp com Better Problem Solving Part II will suggest a method of implementing a function that requires a single key press and three inputs The function then decides which of 19 solutions is the correct one and then calculates four outputs If this function is added to all mid range and high end scientific calculators it would save hundreds of hours of involved calculations for just about every technical user The problem is a very common everyday type problem Better Problem Solving Part I Notes 1 You may also calculate the value by Last Year This Year 1 x 100 This method is useable any calculator and it will save keystrokes It still requires that you key the values in the prop
9. RPN the most sensible logic system a pocket calculator can have Have you been reading RPN Tips Are you very familiar with RPN Which of the statements below do not apply to RPN RPN is easy to use because your approach to every problem is always the same B RPN solves problems with the minimum of rules to remember C RPN does not require parentheses to solve problems e g 1 2 3 D RPN is more keystroke efficient than Algebraic 1 e it requires fewer keystrokes to solve problems 1511 paunroqpiad ST uomearmar nur saumsse ameriqas wy Jea aun sr Walqoid 941 3508799 12328 y HP Solve 22 Page 6 Page 2 of 2 HP 48 One Minute Marvel No 9 Stack Reversal One Minute Marvels OMMs are short efficient unusual and fun HP 48 programs that may be entered into your machine a minute or less These programs were developed on the HP 48 but they will usually run on the HP 49 and HP 50 as well Note the HP48 byte count is for the program only Suppose you have 100 objects e g numbers on the stack and you want to take a quick look at the top few If you use the normal stack manipulation commands you could do this but there are other ways that are far more efficient You could assemble the numbers into a list and use the list reverse command to see the objects more easily in the display OR you could just reverse the objects in place using SREV This means the top four a normal HP48 display and more
10. a 10 digit finance calculator of the Voyager Series dealing with currency conversions that involve large values of a foreign currency you really appreciate having the comma delimiters If you use a 48 49 50 graphing calculator you have a 12 digit display but you must use FIX mode to have the number displayed with commas Most users of these machines keep their machine in STD mode and it 1s inconvenient to switch between modes just to read large numbers correctly Have you used more than one model of an HP calculator and noticed how the comma in the various calculators is used Do you frequently use 8 12 digit numbers Do you think that entering and displaying numbers with thousands separators 1s important Do you want HP to bring back the thousands separator in the RPL based machines if it 15 practical This would be best applied to single line displays Send your comments to the HP Solve editor at hpsolve hp com Commas in the HP Calculator Display Notes 1 Ordered pairs complex numbers are shown entered with a comma between them on some machines The comma is used worldwide as a separator between the real and the imaginary parts of a complex number and there is a space after the comma and before the imaginary part e g 3 4 2 The HP Classic series did not provide this feature The HP 71B in 1984 did not Reader question When were the display delimiters first added to HP calculators Was it with the Spice Spike series
11. an HP 67 in that handbook There are 73 steps however 14 additional keystrokes are required because the entry of parentheses are second functions so a total of 87 keystrokes are required Nearly all of the intermediate results which appear in an RPN solution are also available in the HP 33s solution This should not be surprising Examination of the keyboard sequence and the resulting output in the display the table above shows that the HP 33s actually solves the problem from the inside out not because the user asks it to but because that is how it must be done With RPN the user must examine the equation and find the proper inside point at which to begin With ALG the machine essentially does the thinking for the user Other true algebraic mechanizations also do the thinking for the user Solving the Mach Number equation on a calculator in adding machine arithmetic mode requires the appropriate insertion of additional sets of parentheses to circumvent the lack of precedence For the HP 10B the steps in the solution will be 5x 1 2 x 350 661 5 SQ 3 5 1 x 1 6 875 6 x 25500 5 2656 1 1 1 3 2 2 3 286 1 SORT Fig 2 HP 10B keystrokes for the Mach number equation where 79 steps are required However parentheses E x 2 y x and square root of x are second functions with the result that the number of keystrokes required is 106 The numbers below the keystroke listing indicate the extra parentheses
12. comes with a Math Pac and other goodies So you are set to go Demonstrating The PROOT Function Using the PROOT function with the BASIC driven HP 71B is easy Just keep the following simple rules in mind 1 Create an array used to store the polynomial coefficients The array should have n 1 where n 15 the polynomial order elements if you want to take advantage of the MAT commands in the Math Pac 2 Store the polynomial coefficient in a floating point array The first element of that array should contain the coefficient for the term with the highest order and so on This also means that the HP Solve 22 Page 27 Page 1 of 7 constant coefficient 1s stored in the highest index array element Create an array of complex numbers that has n elements Call the PROOT function which stores the calculated real and complex polynomial roots in an array of complex numbers As far as the machine 1s concerned all the roots are complex The real roots are the ones with the imaginary part being zero or a very small number Here 15 a short BASIC program that prompts you to enter the order and the coefficients for a polynomial The program displays the roots using 10 20 40 50 60 70 80 90 OPTION BASE 1 INTEGER N INPUT ENTER POLYNOM ORDER N DIM A N 1 COMPLEX B PROOT FOR I 1 TON DISP PAUSE 100 NEXT I 110 DISP DONE The program performs the following main tas
13. if you are using an HP50g with a small font The utility efficiency of this OMM should be obvious but its primary utility here 15 to illustrate a fast and efficient technique that 15 worth studying Here 15 how the routine works The first command DEPTH returns the number of values on the stack If there are 100 objects on the stack level one would be 100 The number two is then placed on the stack because we want the loop to ROLL the stack from 2 to 100 times SWAP orders the input for the correct input for the FOR loop as 2 through 100 jis a local variable that is used only for the duration of the FOR NEXT loop The first value of j is 2 and itis consumed The second call of j returns 2 to the stack and ROLL 1s executed j 15 incremented to 3 with the execution of NEXT This repeats until the stack 15 rolled down 100 times and the program terminates Is this the only way to accomplish this task Would more efficient methods be faster or use fewer bytes Note that the timing information reverses the digits 1 to 100 about 3 4ths of a second You couldn t press a normal stack sequence of commands faster than this Could the 50g do this with fewer commands What if you used a program that put the objects into a list used the REVLIST command and then exploded the list back onto the stack Would this be a shorter program Would it work as well as SREV As a test include a program as an object on the stack What happens then If you und
14. log o x e Mathematicians generally define log x to be the natural logarithm log x e Computer scientists often choose log x to be the binary logarithm log2 x e n most commonly used computer programming languages the log function returns the natural logarithm log x In an attempt to avoid logarithmic notation confusion the US Department of Commerce National Institute of Standards and Technology NIST recommends to follow the International Organization for Standardization ISO standard titled Mathematical signs and symbols for use in physical sciences and technology IS 31 11 1992 The standard suggests these notations Logarithmic ISO notations The notation Ib x means log x The notation In x means log x 6 The notation Ig x means logio x Calculator keyboard notation usage is not expected to change however and the reader should be aware of other notations being used for logs Perhaps HP could add base 2 log keys and then change the keyboard notations to comply with international standards The constant e is known as Euler s number which has a value of 2 71828182846 This number appears throughout mathematics and its discussion is beyond the scope of this review Euler is also known for Euler s identity named after Leonhard Euler is the equality e 1 0 It is considered the most beautiful theorem in all of mathematics The reader may learn more at http en wikipedia org wiki Euler o27
15. mark to either a comma used in Europe or a period usually the default The second basic purpose of having a comma in the display is as a thousands separator e g 1 618 033 988 75 The thousands separator makes reading large numbers much easier In fact trying to read some numbers without thousands separators 1s what inspired this article See Fig 1 Many HP calculators prior to the introduction of RPL in mid 1986 with the HP28C would show thousands separators in the display With the RPL change in the operating system the display control was changed and the only time you may see large numbers displayed with commas for the easiest reading is in the Fix display mode nanotube 1969530 bestractor 1834980 1821300 jmelton 1818710 cmb7771 1425840 nbruin 1041470 Metalhead 973360 Farshaad 967080 mrenicks 341770 mantrid 847000 a Baal aT TOP T nanotube 1 969 530 bestractor 1 834 980 DADIO 1 821 300 jmelton 1 818 710 cmb7771 1 425 840 nbruin 1 011 470 Metalhead 373 360 Farshaad 967 080 mrenicks 841 770 mantrid 847 000 I asked Dr William Wickes one of the HP architects of RPL about this Here is his reply The HP28C was HP s first multi line RPN calculator so the design team had to rethink the formatting of numbers when two or more numbers would be viewed on the stack at the same time We considered STD mode by definition as meant for unformatted display
16. observation leads me to conclude that the algorithms and its special strategies to handle special cases and strive to obtain very good accuracy used with the HP 71B Mat Pac and with the graphing calculators are basically the same Solve 22 Page 32 Page 6 of 7 Conclusion The Math Pac of the HP 71B offers the first incarnation of the PROOT function This initial implementation lacks the ability to handle complex polynomial coefficients Other than that it does a very good job in finding unique and duplicate roots HP has done well by adding the ability to handle complex polynomial coefficients when it moved the PROOT to the graphing calculators References 1 Hewlett Packard Company Math Pac Owner s Manual for the HP 71 Edition 2 September 1984 2 Hewlett Packard Company HP 71 Owner s Manual Revision E March 1987 3 Hewlett Packard Company HP 71 Reference Manual Edition 4 October 1987 4 Wikipedia The Laguerre s Method March 2010 5 W M Press et al Numerical Recipes The Art of Scientific Computing third edition 2007 Cambridge University Press 6 A Ralston and P Rabinowitz A First Course in Numerical Analysis second edition 1978 Dover Publications 7 Hewlett Packard Company HP 50g 49g 48gll graphing calculator advanced user s reference manual Edition 2 July 2009 8 Kahan Personal Calculator Has Key to Solve Any Equation f X 0 Hewlett Packard Journal December
17. points b 1 for base b because a number raised to the power is itself All the curves approach the y axis but do not reach it because at x 0 the value is a vertical asymptote Computers use base 2 numbers and logs that use two as the base are quite common As figure nine shows the three most common logarithmic bases are 2 e and 10 Logarithms to the base e are called Naperian or natural logarithms for their inventor John Napier 1550 1617 using the notation LN Most scientific calculators have natural log and anti log keys An example is the two HP35s keys as shown in figure 10 These two keys with the six functions shown Fig 10 HP35s LN keys the most mathematically power packed keys on the HP 35s keyboard The three most common logarithmic bases are 2 e and 10 4 Natural logs The natural logarithm of a number x is the power to which e would have to be raised to equal x To calculate the natural log base e of a number use the 5 LN key To calculate the inverse or antilog use the e key See figure ten above Logarithm notations All during our discussions of logarithms we have been using traditional calculator keyboard notations 1 e log x 1s base ten and LN x is base e Every technical field has its own notations and those within a given field may use log x to be LN x as an understanding among themselves For example e Engineers biologists and astronomers often define log x to be the common logarithm
18. the PPC ROM Manual Manuals for computer related products have always and will always fall short from the user s perspective simply because of ignorance resources complexity cost and time The needs of the user don t change it is the manufacturer s continuous attempts to competitively meet those needs that changes If resources were unlimited is a prerequisite that every complainer should use when writing about manuals From the user s perspective I personally would like to see a manual approach similar to the following 1 A photograph of the keyboard with each and every key identified with a description of what the notations and symbols mean 2 A key response description of what happens when any key is pressed Presenting this matrix of key responses will require that the user understand that keys change in their meanings depending on the mode or environment at the moment the key is pressed 3 reference organization that recognizes that the manual is used at least five times more frequently for reference than it is used for explanation initial reading 4 Each page is numbered sequentially Bill Wickes addressed this issue using Section numbering AND page numbering in his famous trilogy of HP48 books If you can t look at the last page in the manual and know the total number of pages in the document you don t understand how important page numbering is Numbering individual sections may be convenient for the document wr
19. the value is entered or by using the key after the value is entered For the HP 19BII the input will be 5x 1 2 x 350 661 5 SQ 3 5 1 x 1 6 875 E 6x 25500 5 2656 1 286 1 SORT HP Solve 21 Page 15 Page 5 of 10 As with the HP 17BII 79 steps involving 85 keystrokes are required The input sequence is very similar to that used with the HP 17BII except that the exponentiation key s labeled not y x The rules for entering the negative signs associated with the exponent in scientific notation and the exponent with the y x function are the same as with the HP 17BII Example 2 Now consider the solution of another relatively complex equation 3 1 4 3 2 6 4 6 1 0 3 2 1 3 5 4 4 2 108 63 1 71428 57142 9 which has been used to illustrate one of the disadvantages of the 4 level RPN mode offered in many HP calculators because its solution is not possible without storing an intermediate result Actually the real disadvantage 1s NOT the necessity to store an intermediate result but rather the necessity to recognize that the intermediate result must be stored before it 1s pushed up and off the stack The keystrokes for solution on a true algebraic machine can be 3 1 x 4 3 2 6 x 4 6 2 3 x 2 1 3 5 x 4 2 which involves 52 keystrokes as written However features provided by some machines may make a smaller key count poss ble On an HP 35s the entry sequence is PP gt 3 12x P4
20. to be shocked when I pulled a copy from my briefcase 3 When I asked if an HP Style Manual existed most of the people in the room were hesitant to talk about it They implied that they did but when I asked if I could get a copy I was stone walled Privately a writer later explained that each writer used his or her own references and that they did not have an official HP Style Manual This older HP confident writer explained that there was an official HP Style manual but it hadn t been updated in many years at that time and the younger writers didn t know about it This was during the CVD days and much has changed Perhaps there is an HP Style Manual being required at HP today My CVD photocopy really looks old 4 In the famous numerical calculation philosophy of Prof William Kahn there are avoidable errors and unavoidable errors Perhaps the lack of examples could be justified to be in the latter category but having a poor index is most certainly in the former category An index value of three or higher is most desirable and it greatly extends the usefulness of the User s Guide at a very low production cost HP Solve 22 Page 40 Page 3 of 3 Fundamentals of Applied Math Series 5 HP Solve 22 page 41 Article 5 in Fundamentals of Applied Math Series Logs Richard U Nelson Introduction What are logs ISSN Which item does not belong on this list Ir Amplifier Earth quake Antenna Loud Een n
21. which must be added when solving with an AMA machine The parentheses labeled 1 cause the value 0 2 to be multiplied by the square of 350 661 5 rather than be added to the 1 The parentheses labeled 2 cause the value 6 875E 6 to be multiplied by 25500 rather than be subtracted from the 1 The parentheses labeled 3 avoid the problem with y x acting like an arithmetic operator and multiplying the result from the solution to the left hand set of brackets by the solution to the right hand set of brackets rather than simply raising the contents of the right hand set of brackets to the 5 2656 power Of course it may not be entirely fair to use the solution of the Mach Number equation as a test of a machine that Wlodek s book describes as a low cost business model For the HP 17BII the input will be 5x 1 2 x 350 661 5 SQ 3 5 1 x 1 6 875 E 6x 25500 5 2656 1 286 1 SORT where 79 steps are required However E x 2 y x and the square root of x are second functions so the number of keystrokes required 15 85 The negative sign for the exponent of 6 876E 6 inside the second bracket must be entered by using the minus arithmetic key before the value 6 15 entered not with after the 6 is entered because after the 6 changes the sign of the mantissa not the sign of the exponent The negative sign for the exponent 5 2656 at the end of the second bracket may be input either by using the minus arithmetic key before
22. 00 X Fig 6c Y axis Linear and X axis Log semi log Solve 22 Page 45 ein ofA PARERE Fig 6b Y axis Log and X axis linear semi log ando Log Plot 88 es EE el ed macar 500 L m 2 Hh exp IIO oa aan Y FEET C ee LESSER L Wl J ous Zee jez p os pon P 1 5 10 100 500 1000 Fig 6d Standard log log plot Page 4 of 9 1N4000 Series Rectifier Diode CELL RESISTANCE VS ILLUMINANCE 13750 Zener Diode 10 1000 E 5000 lt SE 5 _ 2000 me b 500 Vz 3 3V 10 i 00 te 10 100 a goes LL RII j 10 LEE z li E SS SE 5 2 mE S en st EN ee Photoresistor 01 02 05 1 2 5 10 20 001 Cadmium Sulfide 1 ZENER CURRENT 0 6 0 8 1 0 1 2 1 4 1 6 Ve INSTANTANEOUS FORWARD VOLTAGE V Cell Zener Current vs Zener Impedence Fig 7 Electronic semiconductor component specifications frequently utilize log and semi log scales The apparent magnitude of stars measures the brightness logarithmically because the eye responds approximately logarithmically to brightness Semitones in music e g music intervals cent minor second major second and octave are measured logarithmically Compounding of interest uses logarithmic relationships Radioactive decay is measured logarithmically Measuring the efficiency of comput
23. 1 2 Euler s identity http en wikipedia org wiki Euler o27s identi a logarithmicnumber 183 o base 3 Excellent free 109 page logarithm reference in a PDF file lt exponents 725738784280 C http www mathlogarithms com MEDEN Decimal place Insert two values the third will be calculated 4 Also see the errata for 3 printed version at http www mathlogarithms com images ErrataBoundCopyMay2008 pdf Fig 11 Website Log calculator Advanced topics For the reader who wants to expand their logarithmic knowledge further here are a few terms that are suitable to search on the internet Complex logarithm Logarithm of a matrix 1s the inverse of the matrix Discrete logarithm theory of finite groups and exponential difficult to calculate Logarithm of a quaternion Double logarithm iterated logarithm Logarithm of a octonion Imaginary base logarithm logarithm derivative Indefinite logarithm Richter Scale Iterated logarithm Super logarithm Log normal distribution Weber s law Logarithms of complex numbers Zech s logarithms Reader challenge Suppose you take the average of a set of logarithmic values What is the antilog of this result called 5111301095 JY SI IIQUINU I 3seq E 01 YIVQ PIMIAUOD san eA IIUIYIIVSO IY JO 5 IYL Solve 22 Page 49 Page 8 of 9 Summary and conclusion A logarithm 15 a simple concept that most people have heard of or remember from high s
24. 717 0 012121824 0 214440152 0 083573062 0 375879626 0 30583898 0 12232013 2 144177774 0 831348223 of 15 0 115248156 0 300285767 0 361728784 0 601958039 0 723073044 1 571792692 3 178523712 4 404651736 6 02397928 5 856138001 11 45957919 Duplicate 20 0 312344571 0 681751198 1 017550377 1 833659839 2 413900034 3 078 166377 6 28979807 9 794744761 18 33662058 24 14111473 30 78155686 Roots 25 0 52104063 0 818744126 1 953900504 2 606713423 3 207609566 5 081298458 9 801778486 19 53871975 26 06771717 32 07754071 50 29599742 30 0 640049866 1 487085026 2 560174349 3 366875598 4 272291389 6 400264524 14 89794146 25 60169507 33 66914546 42 72342353 64 19602825 35 0 783125845 1 90840792 3 058388786 3 933409313 4 962886843 7 950693395 19 08430516 30 58419254 39 33407894 49 62855435 79 75180325 40 0 427179263 2 280244037 3 463778361 4 548226861 5 813349385 9 184036127 22 80205451 34 63779514 45 48218349 58 1334889 91 29228563 Figure 1 shows a linear plot for the NormErr results The graph includes a legend Series 1 refers to 5 duplicate roots series 2 refers to 10 duplicate roots and so on The lines are not perfectly linear because of the rounding errors The curve for 10 duplicates shows the most variations These variations or Series1 Series ie Series3 Series4
25. 73 2 TR2 32x P2 1 gt 2 ENTER Fig 3 Keystrokes for the example 2 problem on the HP 35s where as before I used the letter P to indicate the use ofthe combined parentheses key and gt to indicate the use of the right arrow cursor key to move past a closing parenthesis The keystroke count s 52 Ona machine like the HP 10s which offers implied multiplication the number of keystrokes can be reduced to 48 as in 3 1 4 3 2 6 4 6 2 3 2 1 3 5 4 2 With the almost algebraic mode on HP 33s the solution can be f f 3 1f f 4 3f f 2 6 4 6 f 2 3 f 2 1 f 3 5f f 4 2f f ENTER Fig 4 Keystrokes for the example 2 problem on the HP 33s where the machine accepts adjacent parentheses 1 e implied multiplication from the keyboard and inserts the multiply sign into the display of the equation in the upper line The keystroke count is 68 which illustrates how severe the penalty can be for having parentheses as second functions Users who are familiar with the feature where an or an ENTER closes the outstanding parentheses can reduce the number of keystrokes for the HP 10s and HP 35s by two keystrokes and for the HP 33s by four keystrokes The keystroke sequence for the AMA machines is G 1 x 4 3 2 6 x 4 6 2 3x 2 1 3 5 x 4 2 7 HP Solve 21 Page 16 Page 6 of 10 where additional parentheses must be inserted to circumvent the lack of precedence 60 k
26. All purpose Symbolic Instruction Code is a family of high level programming languages The original BASIC was designed in 1964 by John George Kemeny and Thomas Eugene Kurtz at Dartmouth College in New Hampshire USA to provide computer access to non science students CLI Command Line Interface typically used by the majority of graphing calculators FORTRAN per Wikipedia Fortran previously FORTRAN blends derived from IBM Mathematical Formula Translating System is a general purpose procedural imperative programming language that is especially suited to numeric computation and scientific computing Operand The values that an operator operates with RPN Reverse Polish Notation A legacy calculator logic system as advocated by Hewlett Packard RPL RPN Reverse Polish Lisp Reverse Polish Notation A calculator logic system as implemented by Hewlett Packard on newer RPL machines Pressing ENTER on these machines does not replicate X into Y Level one into level two of the stack HP Algebraic Notes 0 There is however a very strong case made by Joseph K Horn for the HP 71B as being one model that most closely approaches the ideal See the excellent article The HP 71B Math Machine that describes the HP 71B in HP Solve Issue 17 Joseph makes the case for the ideal scientific calculator user interface 1 My Dear Aunt Sally is a popular memory aid taught in schools and textbooks to remember Multiply Division Addition a
27. C N COMPLEX R N 1 R1 N 1 50 MAT R1 R0 0 60 ZER C N 1 70 FOR L 1 TON 80 FOR K N L TON 1 90 CIN C K 1 C K RO 100 NEXT K 110 EAN C LN 120 NEXT L 130 R PROOT 140 MAT RI R RI 150 E FNORM R1 SQR N 160 DISP E PAUSE 170 MAT DISP R Store the above program object in the variable PC This program takes the value of the duplicate root and the number of duplicates from levels 2 and 1 respectively The program performs the following tasks 1 Prompts the user to enter the polynomial order and the duplicate root value HP Solve 22 Page 29 Page 3 of 7 2 Declares the real and complex arrays needed by the program The array C stores the polynomial coefficients The complex array R stores the calculated polynomial roots The complex array R1 stores the value of the duplicate root 3 Assigns the values of the duplicate root to all the elements of array R1 in line 50 4 The statements in lines 60 to 120 calculate the polynomial coefficients and store them in array C 5 Calls the function PROOT line 130 to calculate the roots of the polynomial storing them in array R 6 Subtracts the array of calculated roots from the array of supplied roots in line 140 This task uses a MAT command to subtract the values in arrays Rand R1 single swoop 7 Calculates the norm of the array of root and divides it by the square root of the number of duplicated roo
28. Florida and half in Brevard North Carolina HP Solve 21 Page 20 Page 10 of 10 Better Problem Solving Part HP Solve 22 page 21 Article Better Problem Solving Part I Richard J Nelson Introduction The basic nature of human beings 1s to do things in the simplest and easiest way possible One measure of societies advancement is its use of tools We move about using our vehicles and we eat our food with utensils Practically everything we do everyday involves the manipulation of the physical objects that make up our lives The tools and methods we use represent our ability to solve problems If I break a leg I am able to still be mobile by using crutches or a wheel chair The tools we use must provide two functions be truly useful 1 They must do the required job easily and 2 they must do the job quickly We humans are always in a hurry to get the job done so we are able to tackle the next job especially if it is one that we really enjoy doing Our brains work so fast that we often feel limited by our tools in terms of getting the job done Often this limitation is measured by the amount of time we spend in the process This series of articles will address the various aspects of problem solving related to calculators This series should not be confused with the Math Review series found elsewhere in this 1ssue of HP Solve The objective of this series 1s to review how we solve problems with our calculators and
29. HP Solve Calculating solutions powered by HP From the Editor Learn more about current articles and feedback from the latest Solve newsletter including RPN tips One Minute Marvels and Ap Issue 22 January 2011 Welcome to the twenty Math problem challenges Your articles bh HP Algebraic Palmer Hanson What is an algebraic calculator How do HP algebraic calculators compare with other calculators RPN and Algebraic Learn insights and answers in Palmer s latest article PROOT A Blast From the Past Namir Shammas Finding the roots of an equation is one of the most difficult mathematical challenges Namir has been studying HP s methods for many years and provides his valuable insight into this topic HP s Calculator Manuals Richard J Nelson Here 1 a review of nearly four Better Problem Solving second edition of the Solve newsletter Learn calculation concepts get advice to help you succeed in the office or the classroom and be the first to find out about new HP calculating solutions and special offers Download the PDF version of newsletter articles Part Hichard J Nelson This is the first part in a series that will review how we solve problems with our calculators and how we might change the method or process to solve them more efficiently Contact the editor Cal
30. Number equation on the AMA machines will suggest that the need for a third set of additional parentheses to avoid an error being introduced by the y x at the end of the right hand brackets could be avoided by interchanging the material in the left and right hand set of brackets That turned out to be true for the HP 17BII and the HP 19BII But with the HP 10B the calculation would proceed satisfactorily until after the entry of the 350 value where an Error Full message would occur when the divide was entered On the HP 10B that problem can be avoided by moving the multiplication by 5 from the beginning to the end of the calculation Of course rearranging the equation for entry into the machine defeats the claimed advantage of algebraic modes namely the ability to enter the equation in textbook form The Error Full message 15 associated with a limit on what are called pending operations with algebraic implementations The classic test to determine that limit for a given machine is to enter the sequence 1 2 3 4 5 6 When did that with my 10B the Error Full message appeared when the plus after the 4 was entered indicating that the limit on pending operations was four I did the test with my HP35s HP 17BII and HP 19BII When I did not receive any error when 20 was entered I stopped the test and pressed which closes all the open parentheses and completes the calculation All three machines yielded the correct sum of 210
31. When I did the test with my HP33s I received the following display 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 where the machine stopped accepting parentheses after thirteenth one The old adage says When all else fails read the instructions so that is what I did Page C 5 of the HP33s Scientific Calculator User s HP Solve 21 Page 17 Page 7 of 10 Guide says In ALG mode you can use parentheses to 13 levels If I press after the entry of 13 I get the correct sum of 91 indicating that at least 13 pending operations are available I did the test with my HP 10s and did not receive any error when the entry reached 20 But when I pressed the message Stack ERROR appeared I pressed the left arrow and the sign after the 11 in the series flashed I deleted the entries after the 11 pressed and the correct sum of 66 appeared This all suggests that the HP10s has a limit of ten pending operations Unhappily the machine lets the user go merrily on his way with the entry of more pending operations and does not indicate a problem until the sign 15 pressed How do they Compare One measure of calculator usage is efficiency based on the number of keystrokes required for solving a problem Table 3 summarizes the keystroke counts from solutions of the problems described above Table 3 Keystroke Counts for the Two Examples Discussed HP33s 37 68 Business Models HP 17BM 85 60 HP 19B 85
32. ber of Crickets From a Single Pair Crickets Fig 4 Cricket growth plotted using a linear scale Fig 5 Cricket growth plotted using a log Y scale The purpose of any plot is to provide a meaningful visual representation of the data and the log scale used for the cricket count does a better job than the linear scale See figures four and five Note that the cricket plots are very close to the same size vertically Which better represents the data Benford s Law is a related reason this 1s especially useful for easier data plotting and reading An additional consideration for a useful plot aside from the ability to read the data values from the plot is to be able to extend the plot predict values when additional data 1s not available Figure six on the next page shows all four combinations of linear and logarithmic x and y scales for three function plots The choice of scales will strongly influence the shape of the plot It is easier and more accurate to extend a straight line plot than a curved line plot Electronics semiconductor components will often have specifications plotted using a semi log or log log plot Figure seven shows three examples Ohm s law i e I E R assumes that the resistance doesn t change with current or applied voltage Figure 7a shows that the resistance of a semiconductor does effectively change with a change in current because the voltage drop changes Logarithms have applications in fie
33. calculate the roots for the above polynomial Your input appears in bold and underlined characters The command keys appear in bold and are enclosed in square brackets For example BNDES refers to the ENDLINE key RUN ENTER POLYNOM ORDER 3 1 1 2 3 4 ESSI 17468540428 1 54686888723 17468540428 1 54686888723 1 65062919144 0 1 DONE The key sequence f triggers the CONT command that allows you to resume program execution The program calculates and displays two complex roots and a real root for the given polynomial Analyzing PROOT Errors How are the errors in calculating the duplicate roots affected by their values and their count This a question that I raised and answered for the graphing calculators PROOT functions in a previous HP Solve issue In this article I ask the same question for the HP 71B Math Pac First I will define a domain of values for the duplicate roots and the number of duplicates I studied the roots of 1 2 3 4 5 10 20 30 40 50 and 100 As far as the number of duplicates I considered the values in the range of 5 to 40 in increments of 5 This is the same domain of numbers I used for the graphing calculator analysis I used the following BASIC program to calculate a single value that summarizes the errors in the obtained roots 5 REM OPTION BASE 0 IS IN EFFECT BY DEFAULT 10 INTEGER N 20 INPUT ORDER N 30 INPUY DUPLICATE ROOT RO 40 DIM
34. chool The log of a number x is the exponent value p the base b is raised For example log 10010 2 because 10 100 In general terms Log x b The log values of numbers are used for computation convenience measurement values and plot scaling Examples of each of these primary applications are provided The three most common bases are 2 e and 10 Logarithms are used in fields as diverse as astronomy chemistry computer science economics engineering music physics and statistics Logarithms are especially useful for dealing with exponents equations The normal textbook mathematics equalities of logarithms is also described Reference links including an Internet calculator and a research topics list are provided for further study HP Solve 22 Page 50 Page 9 of 9
35. culator Display Richard J Nelson In Richard s latest article he explains the basic purposes of the comma delimiter in the display of an HP Calculator Fundamentals of Applied Math Series 5 Hichard J Nelson decades of HP calculator The fifth installment explains manuals with a personal list of and explores all of the desired topics for their content important applications of logs A comparison of the most lt is also the first serious complete User s manual ever calculator function to be written for an HP calculator is reviewed in this series with the made to illustrate the resource preceding installments issue involved dedicated to numbers and pre calculator problem solving A Update Profile Change Email Unsubscribe HP Home Support amp Drivers Po If you received this e mail from an associate and or would like to receive email of this type directly from HP please click here HP respects your privacy If you d like to discontinue receiving these type of e mails from HP please click here For more information regarding HP s privacy policy or to obtain contact information please visit our privacy statement or write to us at Privacy Mailbox 11445 Compaq Center Drive W Houston Texas 77070 ATTN HP Privacy Mailbox Hewlett Packard website Sign up for Driver and Support Alerts From Editor HP Solve 22 page 3 Article From The Editor Issue 22
36. d it a total mystery or at least profess to do so I do not really believe it when HP 35A calculators were introduced in the university where I studied I found it took about 5 minutes to teach other students how to use an HP 35 That is the way I was introduced to RPN There was a reference to how easily RPN could be used to solve a complicated problem such as the Mach number equation There was no mention of stack limitations that could lead to an incorrect answer to a problem as innocuous as the one discussed here HP Solve 21 Page 18 Page 8 of 10 Of course Wlodek s comment was written from the viewpoint of a proponent of RPN I am a long timeproponent of algebraic For eight years I was the editor of a newsletter for algebraic machines such as the TI 59 TI 95 the TI CC 40 and the Casio fx 7000G As a result of that experience I suggest that an appropriate corollary to Wlodek s statement would be Many users prefer algebraic notation including the use of parentheses because it allows them to enter textbook equations directly into their machines Many RPN users profess to find the use of parentheses a total mystery or profess to do so I do not really believe that either I do admit the proponents of algebraic have a long standing prejudice favor of direct textbook entry e g page 1 8 of TI s manual for the use of the TI 30 The Great International Math on Keys Book 1976 presents an algebraic expression and states You can
37. duct Here is a review of the nearly four decades of HP calculator manuals with a personal list of desired topics for their content A comparison of the most complete User s manual ever written for an HP calculator is made to illustrate the resource issue involved S07 5 in the Fundamentals of Applied Math Series Logs We all know what logs are but if you are like me you may not remember all of the important applications of logs This is the first serious calculator function to be reviewed in this series with the preceding four installments dedicated to numbers and pre calculator problem solving That 1s it for this 1ssue I hope you enjoy it If not tell me Also tell me what you liked and what you would like to read about lt gt Richard Email me at hpsolve hp com Solve 22 Page 4 Page l of 1 There are still a few ideas left to illustrate how RPN works so we will have a few more RPN tips Here is a reprint of an article that was published in The Hewlett Packard Calculator Catalog and Buying Guide Spring 1975 As with all things RPN it is timeless lewiett Packard began designing a family of computer All Hewlett Packard pocket ME eiu to solve complex engineering calculators feature RPN computer logic the most sensible logic system a pocket calculator can have works with numbers his effort HP thoroughly evaluated the strengths and foi the various logic systems which a person migh
38. e traits of wanting to keep life and problem solving as simple as possible I will suggest that there 15 a better way Because this better way 15 a different way you may resist its approach I guarantee however that your basic nature will take over if you actually use this approach for a while After ten years of using this method of calculating A I will bet you lunch that you will want to have the delta percent function work this way Here 15 the process as described in HP Solve Issue 14 HP Solve 22 Page 22 Page 1 of 4 Step 1 Key in the first number the order doesn t matter Step 2 Key in the second number Step 3 Press the A key That is all there is to solving A problems It is as easy as One two three No thinking is require to solve the problem none This A however is different in that it gives you two answers and you will easily and naturally choose the one you want On an RPN or RPL machine there are two numbers on the stack See fig 1 below 12 15 00 X L1 13 04 Fig 1 A calculation example The numbers for our example tax problem are 15 and 13 04 The negative number 1s on the lower level X register or level one This represents the percentage the smaller number is compared to the larger number and the next level up Y register or level two is the percentage the larger number is over the lower number the increase in taxes These two values will always be in this order and you w
39. ed in Fig 3 Y 12 1 00 oz is 0 91 ozt X 1 1 00 ozt is 1 10 oz Fig 3 New oz conversion proposal HP Solve 22 Page 23 Page 2 of 4 Problems With Two Possibilities Function amp Inverse z ET Ki Und E SN I gt d T E ae Le Ts ee a IE doc ae cos Adding labeling to answers to take full advantage of an alphanumeric display may also use the both answers solution to other functions such as Sin Cos Tan LOG LN etc See typical key designations in Fig 4 30 degrees as an input to TAN would result in the display of Fig 5 This application is less fg 4 HP 55s Trig Keys appealing than the examples above and it is included only to further illustrate the possibilities Y L2 ATAN 30 00 is 88 09 X L1 TAN 30 00 is 0 58 Fig 5 Tan inverse function example TAN What about chain calculations If the answer you need is on level 2 or in the Y register SWAP or X 2Y may be used Alternately a key that eliminates the bottom answer only leaving the top answer could be used This function would be used for all two answer results Keyboard clutter would be reduced using this approach because a single function essentially replaces two functions Problems With Two Possibilities Function amp Inverse alternate Another aspect of making problem solving simpler and easier is making the keyboard as simple and uncluttered as possible Have you ever been i
40. ed real and or complex roots HP developers added support to complex polynomial coefficients when they ported the PROOT function to the graphing calculators The PROOT function is based on the Laguerre method I have discussed this method 1s an earlier HP Solve issue The PROOT function has the following features 1 Works with polynomials that have real coefficients 2 Solves for all real and complex roots 3 There is no need to supply guesses for the roots or tolerance limits for the answers The function PROOT internally determines the initial guesses and works to maximize the accuracy of the answers Why Should I Care About the HP 71B The HP 71B is one of the vintage HP computing machines that has a dual aspect It is a calculator and a handheld BASIC computer The HP 71B spearheaded other handheld computers by implementing the IEEE floating point math standard still in proposal stage at the time which included such new concepts as infinity and NAN not a number Armed with a Math Pac and a Curve Fitting Pac which also offers nice optimization engine the HP 71B 1 a formidable machine that works with an HP version of legacy BASIC In general coding in BASIC has been and remains easy to follow and read If you feel nostalgic in reading this article and have no HP 71B at hand you can download the EMU71 software from the internet http www jeffcalc hp41 eu emu l index html which emulates the HP 71B in a DOS box This emulator also
41. er algorithms especially sorting is done logarithmically Logarithms are used to describe the fractional dimensions for fractals Figure eight is a plot of the log of x This is a similar 1 version of the plot shown in figure 5a see next section log X and box 8 below for the differences and it better shows 0 the values of the log function between zero and one Will the blue curve cross 1 at x 10 What is the value 1 of the curve at 2 What would be a better set of scales for this plot Hint See similar blue plot of figure six c The curve is not very clear for the value of 2 What is 20 1 2x 3 4 5 the antilog value of 2 What is the value at 10 Does the anti log of x ever reach zero Logs that use different bases Fig 6 Log x plot for values near zero Ifn base 8 the base may be a number that provides convenience or mathematical meaning The logs of base ten numbers have been used to illustrate how logs are used for computation measurement values and plot scaling Figure nine shows a plot of x vs log x for three different bases for comparison The red plot is to base 10 the black plot is to base e see footnote 6 and the blue plot 1 to base 2 Fig 9 Plot of logs of three of the most commonly used bases Logarithms of all bases pass through the point HP Solve 22 Page 46 Page 5 of 9 1 0 because any non zero number raised to the power 0 is 1 The plots also pass through the
42. er order About the Author Richard J Nelson has written hundreds of articles on the subject of HP s calculators His first article was in the first issue of HP 65 Notes in June 1974 He became an RPN enthusiast with his first HP Calculator the HP 35A he received in the mail from HP on July 31 1972 He remembered the HP 35A in a recent article that included previously unpublished information on this calcul ator See http holyjoe net hhc2007 Remembering 20The 20HP35A pdf ga He has also had an article published on HP s website on HP Calculator Firsts Bae See http h20331 www2 hp com Hpsub cache 392617 0 0 225 121 html Solve 22 Page 25 Page 4 of 4 PROOT Blast From the Past HP Solve 22 page 26 Article PROOT A Blast From the Past Namir Shammas The PROOT function in the graphing calculators starting with the HP 48G G GX models has its roots in the HP 71B handheld BASIC computer To be exact the PROOT function first appeared in the Math Pac for the HP 71B This ROM offered support for the following features Complex math functions variables and arrays Hyperbolic functions Matrix vector operations Solution to systems of equations Root calculations for nonlinear functions Root calculations for polynomials Numerical integration Finite Fourier transforms pe cH dl ae des The PROOT function in the HP71B Math Pac handled polynomials with only real coefficients The function return
43. erstand the answers to these questions you will know why SREV is an OMM SREV lt lt DEPTH 2 SWAP FOR j j ROLL NEXT gt gt commands 34 0 Bytes 3115h Timing I to 100 in 0 767 seconds cc Note If you don t get the same HP48 check sum verify that J is lower case and not upper case J HP Solve 22 Page 7 Page of 1 HP Solve Math Problem Challenge 3 This problem 15 the third in a series of real world practical or teaching problems offered as a challenge to HP Solve readers Send your solution to the editor and if your solution is thought to be the most practical clear and using minimal math it will be published in HP Solve This problem is an equivalent resistance problem The math involved is not very complex but if you have any basic electrical experience you will understand Ohms law and equivalent resistance Here is the equivalent resistance idea Series Connected RI R Two resistors in series amp R2 have an RT RI R2 equivalent resistance 1 R2 This means that the resistor may be replace with a single RI R2 R3 resistor of the value of Rr If RI 13 R2 12 RI R2 R3 OQ R3 25 50 The current is the same through all series resistors Parallel Connected Two resistors in parallel R1 amp R2 have RT R2 equivalent resistance R1 R2 R1 R2 AD R1 R2 This is their product over their sum If R1 13 Q R2
44. ery similar to that of the early machines manufactured by Texas Instruments and an adding machine arithmetic mode which is characterized by a lack of operator precedence similar to that offered by old mechanical calculators True Algebraic The algebraic mode machines such as the HP 10s and HP 35s operate in a manner which 1 consistent with the time honored My Dear Aunt Sally form of precedence Parentheses are allowed and even encouraged if needed for clarity There is no intermediate output during equation entry I call this mode true algebraic because an algebraic equation from a textbook can typically be entered directly step by step working from left to right Richard Nelson prefers to call this Command Line Interface CLI Whatever the nomenclature this mode 1 very similar to the old higher order languages such as FORTRAN or BASIC and to the equation entry system in graphical calculators manufactured by Texas Instruments and Casio A true algebraic mode 1s also available as an equation entry mode with an HP 33s and as the equation entry for the Solve modes in machines such as the HP 17BII and HP 19BII True algebraic will evaluate 2 3x5 as 17 because the multiply sign takes precedence over the plus sign If the user wants to perform the addition before the multiplication then appropriate parentheses may be inserted so that 2 3 x5 will be evaluated as 25 Now consider another equation 55725 which will be evaluated a
45. examine intermediate results A different keystroke sequence is required with a true algebraic machine such as the HP 35s which uses a single parentheses key to simultaneously enter an opening and a corresponding closing parenthesis and does not offer implied multiplication A similar methodology was used for parentheses brackets and braces in earlier machines such as the HP48 HP49 and 50 The keystroke sequence 1s SQR5xPPPPP1 2xfSQ 350 661 5 gt gt 3 5 1 gt 1 6 875 E 6 gt 25500 gt y x 5 2656 1 gt y x 286 1 gt gt ENTER where I used the letter P to indicate the entry of the combined parentheses key gt to indicate the use of the right arrow key to move past a closing parenthesis and f to indicate the use of a second function 75 HP Solve 21 Page 13 Page 3 of 10 keystrokes are required Note that the SQR and SQ functions automatically insert necessary parentheses so that the expression is evaluated properly An experienced user will reduce that to 71 by eliminating the parentheses surrounding the 6 875E 6 entry and eliminating the last two closing parentheses by using the characteristic that an ENTER closes all open parentheses I admit that I struggled with this because of unfamiliarity with the double parenthesis methodology There 15 no easy ability to examine intermediate results The almost algebraic mode of the HP 33s combines a left to right entry of the textbook equation except t
46. eystrokes are required with an HP17BII or HPI9BII 28 additional keystrokes are required with an HP10b where parentheses are second functions The perception that the problem can not be solved on a 4 level RPN machine without storage of an intermediate result is incorrect Without even using the LAST X function the following sequence will solve the problem 2 ENTER 3 2 ENTER 1 x 3 ENTER 5 4 ENTER 2 x 3 ENTER 1 4 ENTER 3 x 2 ENTER 6 RollDown x lt gt y RollDown x lt gt y RollDown 4 ENTER 6 x Fig 5 Keystrokes for the example 2 problem using RPN which is hardly a credible solution for most users The solution requires only 45 keystrokes on an HP 41 The solution requires 50 keystrokes on an HP 17BI or HP 19BII where RollDown and x lt gt y are second functions This problem is an interesting exercise but I think that a knowledgeable RPNer who even suspected that there might be any difficulty with stack overflow could be expected to solve the denom inator in a straightforward manner store the result solve the numerator recall the denominator and divide That can easily be done in 45 keystrokes on an HP 41 and in only 43 keystrokes on an HP 17BII or HP 19BII The problem can be solved directly without intermediate storage on machines such as the HP 28S HP 48 series HP 49 and HP 50G which do not use a limited stack Limitations on the Number of Parentheses and Pending Operations A little study of the solution of the Mach
47. gth 5 of 10 of the 259 mm it would be 129 5 mm On the log scale the corresponding value of 1 181 mm Note how the log scale tends to expand the lower values and compresses the higher values in each decade in terms of scale lengths From this exercise it is clear that the distance spacing of the slide rule numbers is based on the logarithm and proportioned to the total scale length Many slide rules have 10 inch scales The spacing of 2 15 the 4 Personal note When I used my slide rule in high school I often wondered why someone didn t make the scales on motor driven Mylar tape that could be very long and greatly increase the number of calculated digits How many digits would be readable if a nine foot scale 10 times longer than the average slide rule were used Solve 22 Page 43 Page 2 of 9 log of 2 and equal to 3 01 inches 5 is 6 99 inches etc Note that the half value of the decade range is about 70 percent of the of the scale length The greatest difference is that the first 10 of the scale value which is 30 1 of the scale length This property of logs may be used to advantage many situations Here is an another example of logarithmic scaling An experiment was performed to determine the growth rate of crickets and the following data was recorded from a starting pair of crickets under ideal conditions See table two Table 2 Cricket Growth from a Single Pair Number of Crickets From a Single Pair Num
48. hat the square root operator must come after the entry of the operand not before The mode also offers visibility into many of the intermediate results however recognition of the results is more difficult than with an RPN system The following table shows the steps in a left to right solution of the Mach Number problem and the contents of the lower display at each step with the machine FIX 2 mode The comments in quotation marks are the same as those in the HP 67 Owner s Handbook and Programming Table 2 Keystrokes Used to Solve the Mach Number Equation in Algebraic on the HP 33s Press S I I II WN gt 661 5 Xov X Y 3 5 Display 5 5 00 5 00 5 00 5 00 5 00 5 00 1_ 1 00 0 2 0 20 0 20 350_ 350 00 661 5_ 0 53 0 28 1 06 1 06 3 5 1 21 1_ 0 21 Solve 21 Page 14 Comments RPN RPN Square of bracketed quantity RPN RPN RPN Contents of left hand set of brackets are the stack Press 25500 Y 5 2656 po Y 286 1 ENTER sqrt Page 4 of 10 Display 0 21 1_ 1 00 6 875 _ 6 875 _ 6 875E6 6 975 6 6 88E 6 25500 8 82 0 82 5 2656 5 2656 0 58 0 58 1 1 58 1 58 0 286 1 14 1 0 14 0 70 0 84 Comments RPN Mach Number of Dacdalus Harrier Guide The notation RPN in the comments column indicate displayed values which are the same as those indicated as appearing in the display of
49. he Slimline Series Table 1 summarizes the radix mark and where it is most commonly used HP Solve 22 Page 35 Page 1 of 2 Table 1 Radix and 1 000 s Separator Usage Examples Display Example Examples of where used 1 618 033 988 75 Period USA UK South Africa languages of Interlinguas and Esperanto computer languages such as C Java and FORTRAN 1 618 033 988 75 Comma Asia France Italy most of mainland Europe and ISO international blueprints 1 The thousands separator is opposite to the Radix mark 2 This is not an extensive or complete list Some newer standards specify the use of a thin space half space as a group separator Examples are SI ISO 31 0 and the International Bureau of Weights and Measures 3 Default for most HP calculators Look again at the differences in the numbers shown in fig 1 Which are easier to read Did you immediately recognize that the two largest numbers are millions There are three aspects to this issue 1 The reading of numbers from the source that is reason for using the calculator Reading the number is much easier and less error prone if the comma delimiter 1s used 2 Keying in the number Keying errors will be reduced if the display shows commas as you key them in as later the pre RPL machines do 3 The answer is easier to read during and after the calculation e g 1234 x 5678 7006652 vs 1 234 x 5 678 7 006 652 If you usually calculate with
50. here s no restructuring to do ho rearranging of the equation as is so often necessary with other calculators to conform to algebraic logic So there s less confusion and less chance for error 2 The function is immediately calculated With an HP calculator pressing the function key initiates the desired action so you get your answer immediately For example to find the square root of 16 simply press three keys and your answer immediately appears on the display And it s just as fast and easy to calculate squares cosines factorials or other functions 3 The intermediate answer is displayed This enables you to check your calculation every step of the way 0 you can do something about it if it doesn t look right 4 The intermediate answer is automatically stored So there s no need to store it manually by keying in each digit if the number is needed in the next calculation Obviously this saves keystrokes and helps prevent errors And you can easily recall the intermediate answer if need be Four major advantages to give you confidence in your computations Just four simple steps To use any Hewlett Packard pocket calculator just follow these four simple steps 1 Key in the first number 2 Enter it into the stack press the ENTER key 3 Kev in the second number de Press the function key And if your numbers are already stored in the calculat
51. how we might change the method or process so that we might solve them more easily and faster Problems with Two Possibilities A A regular column in HP Solve is One Minute Marvels OMMs These are short being able to key them into your machine in a minute or less HP48 49 50 programs or routines that are examples of an elegant solution or the solution of an unusual problem In HP Solve issue 14 an OMM program was described to calculate what is often called delta percent Given two values e g your taxes for last year and your taxes for this year you are either to calculate the percentage increase or the percentage decrease To keep the math simple let s assume that the two numbers are 1 000 and 1 150 for last year and this year respectively First you have to key the two numbers and then subtract the two values Next you need to divide the difference by either last year 1 000 or this year 1 150 depending upon which you wanted the decrease or the increase The result of the division must then be multiplied by 100 to convert the value into percent The basic mechanics of the solution are as described above You the problem solver have to do a bit of thinking in order to get the right answer Suppose however you didn t pay as much this year as you did last year and the two tax values were reversed Some HP calculators have a delta percent function built in and this function will save you a little time Because I am human with th
52. ill quickly and instinctively know which number you need for your problem This approach 1 so unconventional that most mathematics types will immediately rebel I know however that 1f you use the program a few dozen times with real world data that you will soon realize how much easier A 6 problems are solved with this proposed approach Problems with Two Possibilities conversions The same approach may be used for conversions Suppose you want to convert temperatures How many degrees is 72 degrees Fahrenheit in degrees Celsius Some HP calculators have this conversion on a key labeled F or Most modern calculators use a dot matrix display which facilitates alphanumeric characters so this suggestion for temperature conversion should also take advantage of this capability The input is 72 and a single function converts this value to both Fahrenheit and Celsius The order will always be the same as shown below Y L2 72 00 is 22 22 C X 11 72 00 C is 161 60 Fig 2 New Temperature Conversion This proposed approach is suggested as a possible improvement for entry level or midrange machines The more advanced unit conversion systems of the high end 48 49 50 machines with their large number of conversions may not benefit from this approach Unit conversions may be compared to provide perspective using this approach An example is a unit conversion 1 such as the avoirdupois ounce and the troy ounce is illustrat
53. ing System It is clear that manuals will become ever smaller with the primary User s guide available on a CD and or downloaded from HP s web site Manuals will always be a topic of discussion by users young and old ignorant and thoughtful newbie and experienced What is most important however is to understand the issues involved sharing your desires with HP and being prepared to reach ever deeper into your wallet for printed good quality manuals handbooks or guides What is missing in HP s manuals How could the manuals be improved Send your comments to Email HP Solve at hpsolve hp com Notes 1 Technology changes the way we use information Manuals are provided on CD s Examples as mentioned here are usually associated with a manual printed downloaded or provided by HP on a CD What is an example It is a step by step procedure process or algorithm that shows you how to solve a problem or use a process Today HP is using video technology in the form of Training Guides for their machines The Training Guides may be found on their web site and they are being produced by experienced users of their machines 2 One of the comments I had written regarding poor HP documentation was the fact that at least the pages should be numbered on all documents over four pages in length I was severely taken to task for saying that HP didn t number their pages I then mentioned a 72 page document that wasn t page numbered They appeared
54. iter but it 1s certainly not convenient for the reader The argument that it is easier to revise a section and not impact the whole document very much isn t justified because so few updates are ever produced 5 Examples practical real world timely and meaningful are important and this is discussed above 6 Consistent Writing Style Manuals should be written following a published style that dictates what must be covered in an owner s manual handbook or guide 7 A two tier index While examples are omitted because of necessity the lousy indexes in most calculator manuals are a classic example of simple ignorance Using modern computer software it is an easy task to produce a document with an index index of 3 or higher Number of index entries divided by the number of text pages Every technical document that is to be used as a reference has at least three words terms or important ideas per page The document must be proofed Simply write down these index items with the page number as the proofing 1s being done Word and most other document software have an index capability 8 Format Most of HP s recent User s Guides follow a standard format Section one is usually Getting Started and the last part is Appendixes and References If the manual 1s printed it is a soft cover using a perfect bound style of binding HP calculator teams have reinvented themselves three times and while a historic review 1 interesting
55. key the above problem directly left to right into your TI 30 type calculator with AOS and you ll get the correct answer Not all calculators will do this Similar words appear in their manuals for all of the calculators manufactured by TI An unsuspecting user will be unhappily surprised when first experiencing an error condition due to limitations on the number of open parentheses or pending operations A comparison of the advantages of RPN and algebraic onpages 12 4 of the HP 33s User s Guide lists the strengths of RPN to Use less memory amp Execute a bit faster and the strengths of Equations and ALG operations as Easier to write and read amp can automatically prompt The same material appears on page 13 5 of the HP35s User s Guide The idea that RPN uses less memory than algebraic has not been seriously challenged as far as I know Even the days of the so called friendly competition the TI community admitted that 100 steps inn an HP program offered more computing power than 100 steps TI AOS program I obtained a current comparison by entering the Mach Number equation into the HP33s as a program and reading out the program length The RPN version used 270 bytes The algebraic version used 303 bytes Relative speed of the two methodologies was more difficult to assess in the days of the friendly competition and was strongly associated with the algorithm and processor speed The newer machine was typically faster A cha
56. ks l 8 Sets the Option Base to 1 so that the lowest index for arrays and matrices is and not the default of 0 This indexing scheme is merely convenient for the above program Prompts you to enter the polynomial order N Creates the array A to have N 1 elements The program uses this array to store the polynomial coefficients Creates the complex array B to have N elements The program uses this array to store the polynomial roots Prompts you for the polynomial coefficients using the MAT INPUT command You can enter each coefficient individually as prompted or type multiple coefficients on the input line In the latter case you need to separate the values using commas Invokes the PROOT function to calculate the polynomial roots The program stores the result of PROOT in the array Displays the elements of complex array B using a FOR loop Each loop iteration has a PAUSE statement allowing you to examine the roots at your own pace as the program pauses Invoke the CONT command from the keyboard to resume program execution and view the next root You can replace the FOR loop lines 80 to 100 with the single command MAT DISP However this command tends to display the array elements rather quickly Displays the word DONE when the program reaches the end Let s use the above BASIC program to find the roots of the following polynomial Y X 3 t 2x2 44 HP Solve 22 Page 28 Page 2 of 7 Here the session to
57. lator manufacturer spending these kinds of resources on any similar project At a value of 20 hr for the time that would be 17 5 million dollars or 35 000 dollars per document page to produce the content Even the world s largest technology company does not have that level of resources for a calculator product I mention these things because of the chatter often seen on the various HP web sites that users complain about HP s manuals I don t disagree with many of these complaints because I have been writing about and complaining about HP s manuals for 35 years I have probably met with in person face to face most of the manual writers at HP for most of their machines Back in the days when HP had all of their operations engineering marketing and manufacturing in one location in Corvallis I would visit the factory for three days meeting with many different teams dedicated to calculators I especially remember one meeting with the documentation group at a mutual request because of some of the things I had written that mentioned that they thought that their manuals were some of the best in the industry After all they had won awards for their manuals Of course I was HP Solve 22 Page 38 Page 1 of 3 coming from a perspective of what 1s desired from the user s perspective and these experiences drove the decision to include extensive examples and a formal format of section headings extensive index etc in
58. lds as diverse as astronomy chemistry computer science economics engineering music physics and statistics Logarithmic scale A scale of measurement in which an increase or decrease 3 of one unit represents a tenfold increase or decrease in the quantity measured 5 See One Minute Marvels in HP Solve Volume 15 for an illustration of Benford s law which describes the distribution of numbers used for natural data Solve 22 Page 44 Page 3 of 9 Bels decibels and nepers are used to measure sound intensity because the ear responds approximately logarithmically to sound pressure If you need to adjust something related to sound intensity loudness you should do it at the lowest possible level in order to hear and detect the smallest change Electronics technicians learn this when adjusting tuned circuits that carry an audio signal Amplifiers and antennas use a ratio of logs to calculate the Db gain value A logarithmic scale is used for pH measurements The Richter scale measures earthquake intensity using a logarithmic scale Logarithms are used in information theory as a measure of quantity of information Continued on page 5 Lin Lin Plot E VA pe ATIT Fig 6a Standard linear plot Linear Y amp X Lin Log Plot U II el Il mo p LU AE EI LLLA z BI EN a ALM Tare LIT mL LET LL BR ALL EI LLL LIT TIL LIE 500 10
59. llenge by Gene Wright in the Forum section of The Museum of HP Calculators gave me an idea for a more direct comparison again using the HP 33s I added a simple GTO loop to the Mach Number programs and counted the number of times the programs could be completed in one minute The RPN version completed 191 loops The algebraic version completed 171 loops But it s all about what the user is used to Richard Nelson reminded me of Dr Wicke s question How do you differentiate between friendly and familiar A user who has learned a keystroke sequence by using it many times may be able to find and press several keys in less time than with a less familiar fewer keystroke sequence A similar effect 15 an important aspect of speed in touch typing Familiar and frequently used words are typed in response to the appearance of the word without really thinking about the individual letters in the word Conclusions and Recommendations During the preparation of this article I had an epiphany of sorts I have always been amused and bemused by the insistence of the RPN community that solution of relatively complex equations such as the Mach Number equation was difficult if not downright impossible with algebraic machines I now suspect they were using machines with AMA algebraic As discussed in this article with AMA version of algebraic it is NOT POSSIBLE to simply enter the Mach Number equation from left to right but IS NECESSARY to insert additional parentheses t
60. n a very cluttered room looking for something There is so much stuff everywhere that your eyes are not able to process every detail in a reasonable amount of time The person who lives in the room knows where things are and will find things easily This aspect of calculator usage is familiar vs friendly experienced vs first look Reducing keyboard clutter has been addressed with ideas such as soft menus shift keys and even audio feedback Ideas that seem strange today may become commonplace in the future as more people get used to a particular way the user interacts with the machine One method is to use the timing of key pressing This may be implemented in at least two ways 1 Press and hold the key when another key is pressed This 15 an old shift key concept going back to the mechanical typewriter days 2 Press the same key twice in succession If the time between pressings 1s too long the alternate function is not executed Double pressing a key for its inverse function is a simple idea that reduces keyboard clutter See example of Trig function keys in Fig 6 The blue notations could be omitted and replaced with other functions If you need the arcsine function you simply press the SIN key twice within 1 4 of a second If the time exceeds the 1 4 second time the SIN function is executed Fig 6 HP 15C Trig Keys What I am proposing is to replace the shifted inverse function keystroke with a 5 keystroke
61. nd Subtraction as the order of precedence for algebraic expressions 2 See HP Solve Issue 4 RPN Tips for a description of the four basic calculator logic systems user interfaces 3 The words in the table are quoted from page 107 of the HP 67 Owner s Handbook and Programming Guide 4 A Guide to HP Handheld Calculators and Computers 5 Ed by W A C Mier Jedrzejowicz Ph D for additional details see http www hpcalculatorguide com About the Author Palmer Hanson worked on autopilot and bombing systems on F 100 and F 101 aircraft and on inertial navigation systems for aircraft such as the A 11 and YF 12 Blackbirds the X 15 the B 52 and the AV 8 He authored six technical papers on inertial navigation He met his first computer in the Navy in 1952 the electromechanical MK 1A which was part of the Mk 37 Gun Fire Control System He met his first digital computer at the University of Minnesota in 1960 the RemRand 1103 He met his first programmable calculator at Honeywell the TI 59 He authored five articles and numerous letters to the editor on the use of portable computers and programmable calculators in publications such as TRS 80 News Byte and PPX Exchange From 1983 through 1991 he was the editor and publisher of TI PPC Notes a newsletter for users of hand held programmable calculators That work was recognized by inclusion in the Who s Who in the South and Southwest Since retirement he spends about half the year in Largo
62. number equation which appears in many of the manuals for the earlier HP RPN machines 2 3 5 5 2656 0 286 3i 6 875 x 105 25 500 la 1 Fig 1 Mach number equation with values entered and ready to solve How does your calculator stack up The manuals contend that a solution to a complex equation such as this demonstrates an important advantage of a system such as 1 e because you calculate one step at a time you don t get lost within the problem You see every intermediate result and you emerge from the calculation confident of your final answer The following is the listing of the keystrokes needed to solve the Mach Number problem in RPN on the HP33s Table 1 Keystrokes used to Solve the Mach Number Equation Keys Pressed Display Comments 350 ENTER 350 00 661 5 0 53 x2 0 28 Square of bracketed quantity 2X 0 06 HP Solve 21 Page 12 Page 2 of 10 Keys Pressed Display Comments 1 1 06 3 5 y x 1 21 1 0 21 Contents left hand set of brackets in the stack 1 ENTER 1 00 6 875 E 6 ENTER 6 88E 6 25500 x 0 18 0 82 5 2656 2 76 Contents of right hand set of brackets are in the stack X 0 58 I 1 58 286 y x 1 14 I 0 14 5x 0 70 SORT 0 84 Answer Mach number of Daedalus Harrier 0 835724536 ten digit machine answer That is a total of 61 keystrokes on an HP 33s or HP 41C I think that is the minimum possible on any HP RPN calculator because there a
63. o circumvent the peculiarities at least to scientific people of the AMA mechanization I struggled with that when preparing for this article If I hadn t had the incentive of preparing for the article I most certainly would have given up HP Solve 21 Page 19 Page 9 of 10 The major deficiency of the algebraic mode in the HP33s and HP35s 1s the handling of the input of parentheses Part of the problem 1s a shortage of keyboard space RPNers complained about the omission of keyboard functions that they wanted in order to provide parentheses input at all Algebraic users complained about parentheses as shifted functions or as shared on a single key Here s a possible solution for calculators which offer both RPN and algebraic RPNers frequently use RollDown and x lt gt y Algebraic users rarely if ever do Why not share the RollDown function with the left parenthesis function and the x lt gt y function with the right parenthesis function with the software deciding which option to use depending on which mode the calculator 1 in Glossary Algebraic The algebraic mode in machines such as the HP 10s and HP 35s which operate in a manner which is consistent with the time honored My Dear Aunt Sally form of precedence AMA Adding Machine Arithmetic No user logic implemented as exemplified by a mechanical adding machine Math operations are implemented as they occur Also known by ATH BASIC per Wikipedia an acronym for Beginner s
64. of numbers In an STD display of a multi line stack of numbers neither the decimal points nor the separators necessarily will line up vertically so the display will be a bit of a jumble anyway and adding digit separators wouldn t improve matters In FIX modes there are always the same number of digits to the right of the decimal point with right justification of the numbers the vertical columns correspond to powers of ten and the separators line up nicely So one would use FIX for a formatted nicely aligned stack of numbers with consistent significant digits or STD for an unformatted show all digits stack With commas Without commas Fig 1 Comma delimiter use example Because many of the current crop of HP s machines are based on the RPL operating system most newer HP calculator users don t have the nice feature of the thousands separators in the display The HP48 49 50 series of machines will only display the thousands separators in the FIX mode So will the HP35s There are current machine examples of the old way of using the thousands separators in the display See Fig 2 below 10C 11C 12C 12Cpl I2Cpr jo 15C 16C Fig 2 The long reigning Voyager Series in model number order The financial versions HP 12C s are current If you look closely you will see commas in the displays of these HP stock photos The thousands separator commas are used by all Voyager series calculators sometimes called the 10 Series or t
65. oise Catenary Redwood Slide rule ea _ cm Stellar brightness Weber s law Hint rer s Fig 1 LOG Keys Think about Fig 2 Other items that should not be on the list are Holland s Rule Fabian s Rule and The Bangor Rule Log is short for logarithm Most students learn about logarithms as part of their algebra class when they study powers e g Ye Many algebraic and RPN calculators have a key marked BM Y More strongly algebraic oriented calculators such Fig 2 A precursor to learning about logs the HP39gs or HP40gs will calculate powers with a key When we use the power key for logs we assume that we are working with normal or common numbers 1 base ten numbers See Table 1 below Table 1 Powers of Ten From a practical perspective logarithms are most often used to express numbers that span a very large range of values Table 1 illustrates how logarithms are related to the power of a number Logarithms having a base of ten are called common logarithms simply expressed as log More on the name later From table 1 the log of 100 is 2 the log of 10 1s 1 and the log of 1 1s 0 The log value is the exponent of the power of ten that the number must be raised to The log of 3 must be between 0 and 1 Calculating the log of 3 1s a difficult task and most people couldn t do it We didn t learn this procedure in school when we learned to multiply and divide numbers Fortunately we have calculato
66. or as intermediate answers all you have to do is hit the function key Could anything be easier or faster Here s an example Let s take a simple problem 2 5 x 4 and solve it with an HP calculator using the four steps shown above 1 Key in the first number 5 2 Enter it into the stack 3 Key in the second number HAE 4 Press the function key Your answer appears on the display Now let s try a slightly more difficult problem 2 6 x 8 3 5 If you were working this out with paper and pencil you d probably work from left to right and first solve for 2 6 Then you d solve for 9 3 5 Finally you d multiply the two answers 8 x 5 5 and get 44 Well with an HP pocket calculator you work the problem the same way Or if you prefer you could work it right to left or even with more complex problems from the middle outwards Working from left to right press Damm The display shows the intermediate answer To solve for 9 3 5 press Moma The display shows To multiply the two intermediate answers which have been automatically stored press And the displays shows Even if your problem were as complex as converting indicated air speed to the true mach number Ceo ees you would still be able to solve it quickly easily and without confusion if you used an HP calculator thanks to
67. ot use arithmetical precedence Since these machines perform arithmetic a manner similar to the old mechanical adding machines I call this mode Adding Machine Arithmetic AMA Richard Nelson prefers to call it ATH Others call it chain algebraic AMA machines evaluate 2 3 x 5 as 25 because the arithmetic 1s performed in the order entered If the user wants to perform the multiplication before the addition appropriate parentheses may be inserted so that 2 3 x 5 will be evaluated as 17 or the equation can be changed to 3 x 5 2 which will be evaluated as 17 Now consider the equation 5 3 2 which will be evaluated as 64 because AMA treats the exponentiation operator in the same manner as the arithmetic operators Thus when the exponential operator is entered the 3 1s added to the 5 yielding 8 which is then squared To obtain the sum of 5 and 3 2 the user can enter appropriate parentheses 5 3 2 which will be evaluated as 14 or you can alter the equation to 3 2 5 which will also be evaluated as 14 TI business oriented machines such as the Business Analysts and Money Manager and the Sharp EL 733A operate in the same manner This effect is not illustrated in the owner s manual for the 19 or Sharp EL 733A It is illustrated in the manuals for the TI machines The Mach Number Equation For further demonstration of the relative capabilities of the various modes of calculation I will look at the venerable old Mach
68. pp All of these manuals were printed most spiral bound and heavy User s Guide makes sense because it 1s simple short and descriptive After 16 years of HP calculator manuals the eleven Pioneer Series of machines 1988 1991 came with an Owner s Manual larger size 250 pp More recent machines now come with a User s Guide or Quick Start Guide The name has changed from manual to handbook to guide Another often remembered feature of HP s manuals is the use of humor when illustrating examples that include people Especially memorable are the unusual names for people used in the example One reason that users tend to remember these things is that they desperately need examples in manuals The PPC ROM Manual demonstrated this need by providing extensive examples for every one of the 153 routines described in the 500 page 8 1 2 x 11 US letter size 2 1 2 pound 1 1 Kilo tome Why is this completely obvious aspect of the user s need ignored by HP when it produces its manuals After over 35 years of writing about this issue especially after producing the PPC ROM Manual I believe that I know the answer The resources required to include lots of examples for any publication is simply beyond the resources that are available The PPC ROM Manual was able to provide the much needed examples because of the donated 100 man years 876 528 man hours that the User Community put into the project Can you imagine HP or any calcu
69. re no requirements for use of a shift key 62 keystrokes are required on an HP 35s because x 2 is a second function 66 keystrokes are required for an RPN solution on an HP 17 and HP 19BII because x 2 and square root of x are second functions The advocates of algebraic tend to describe the solution to a textbook equation such as the Mach Number Equation as no more than simply entering the equation from left to right pressing the equals key and voila without any agonizing or analysis to decide where to start the problem the solution appears in the display Actually it is typically a little more complicated than that because of idiosyncrasies of individual machines The true algebraic modes will permit the equation to be entered in a manner almost identical to that in the textbook equation with one exception namely that an additional set of parentheses must be entered 1f the square root is to be taken of the entire expression With an HP 10s the keystroke sequence is SORT 5 1 2 350 661 5 x 2 y x 3 5 1 1 6 875 EXP 6 x 25500 y x 5 2656 1 y x 286 1 where the notation is the unary minus sign That requires 74 keystrokes as written Users who are familiar with the machine will reduce that to 70 by eliminating the parentheses surrounding the 6 875E 6 entry and eliminating the last two closing parentheses by using the characteristic that an equal sign closes all open parentheses There 15 no easy way to
70. rs that are able to do this with a key identified as HE Calculating logs To calculate the log of a number use the LOG key 1 To calculate the inverse or antilog use the 10 key See figure one above The log of 3 is approximately 0 47712125472 Enter 10 and 0 47712125472 and use the power key to verify that 109277121227 is indeed 3 The log of a base 10 number is the exponent of ten that represents the number I According to a USDA Forrest Service General Technical Report there are at least 95 log timber rules bearing 185 different names 2 Logarithm is a term coined by John Napier of Merchistoun 1550 1617 His 1614 book Mirifici Logarithmorum Canonis Descripti provided log tables and methods for numerical calculation 3 Mathematics building of the University of Maryland HP Solve 22 Page 42 Page 1 of 9 Logarithm The log value is the exponent of the power of ten that the number must be raised to The log of 3 is approximately 0 47712125472 which means that 2 10047712125472 15 equal to 3 Only positive real numbers have real valued logarithms Where logs are used One of the classical applications of logarithms 1s the slide rule The primary means of making calculations for over 350 years the early 1600 s to the mid 1970 s were logarithmic tables Three place log tables were printed onto a convenient slide rule calculator This device was the standard means of making numerical computations before
71. s C to D This is a 20 ohm resistor R2 connected in parallel to three series connected resistors R1 R2 amp R4 The equivalent of the series resistors is their sum or 70 ohms 70 ohms in parallel with 20 ohms 15 16 ohms Now that you have the basics of equivalent resistance here 1 the problem Twelve equal resistors are connected together as if they were the edges of a cube The equivalent resistance to be calculated 1s across the internal corners of the cube i e from a to g in Fig 2 The value of each resistor is 2 982 ohms The best solution idea is one that gives a clear step by step explanation of the solution The best solution is based on the guidelines listed below 1 The printed solution 15 the decision of the editor and multiple solutions may be published The description and clarity of the solution 1s most important 3 The use of graphics 1f needed should be used to make understanding the solution easier and Fig 2 Resistors along the edges of a cube Find the clearer Equivalent resistance from a to g across the internal 4 The use of minimal mathematics 1 algebra diagonal of the cube instead of calculus 5 The use of an HP Calculator if helpful Extra points are possible if multiple solutions are provided or if derivations of the solution equation ratio are provided Remember just getting the answer is not enough to distinguish your result from everyone else Send your entry to
72. s 14 because exponentiation takes precedence over the plus sign Almost Algebraic The algebraic mode the HP 33s uses arithmetical precedence and accepts parentheses but operators such as the trigonometric functions the square root and the logarithmic functions are entered after the operand rather than before Thus I don t consider it to be a true algebraic system because a textbook equation typically cannot be entered directly from left to right The mode 1s very similar to that which was used in TI scientific calculators from the TI 30 through the TI 95 but with important differences With the TI machines and with the algebraic mode in the HP95LX all the user sees during the solution is the calculated result at each intermediate stage of the solution With the HP 33s the sequence of steps is HP Solve 22 Page 11 Page 1 of 10 accumulated in the upper register and the calculated result at the current stage of the solution appears in the lower register The HP 33s also offers implied multiplication However the implementation of the entry of parentheses as shifted functions is a substantial deterrent to the use of the machine in algebraic mode This partly algebraic mode will evaluate the two examples 2 3x5 and 5 3 2 inthe same manner as a true algebraic machine since the rules of precedence are similar Adding Machine Arithmetic The algebraic modes in business oriented machines such the HP 10B HP 17BII and HP 19BII do n
73. s identity HP Solve tt 22 Page 47 Page 6 of 9 Let s do a mental experiment you have a calculator key that is supposed to be a logarithm key but the base is unknown How would you determine its base Logarithm Base Determination 1 Key a number press ENTER twice Try 153 2 Press the unknown log key use LN 5 03043792139 7 3 Press the 1 x key 0 198789850034 4 Press the Y key 2 71828182846 This method may also be used to test a programming language log function to verify the base You could save steps assuming you also had an anti log key for the unknown base log key by pressing 1 and the anti log key e g 1 2 71828182846 1 10 10 If you had a base two log key you would press 1 and the log key to return 2 The mathematics equalities of logarithms The equations that follow may be found in most math text or reference books and only the basic equalities are given for reference Proofs and derivations are not included here 8 Logarithms are especially useful in solving equations in which exponents are unknown For the equation b x b can be determined with radicals n can be determined with logarithms and x can be determined with exponentials Logarithm tables were calculated by John Napier Henry Briggs Jost Burgi Adriaan Vlacq Jurij Vega Leonhard Euler Francois Callet and Gaspard de Prony in the early 1600 s to make calculations easier Multiplication To multiply two numbers take
74. scientific handheld calculator is 38 years old Many of the basic functions haven t changed very much since the HP 35A started this product category in 1972 Technology has changed especially in terms of display quality and running speed Convergence is ever lurking in terms of competition for the personal handheld calculator The primary advantage in my opinion of the handheld scientific or graphing calculator is convenience and cost Otherwise a calculator in your smart phone laptop or desktop computer will suffice Reader feedback for some of these strange suggestions is solicited S04 PROOT A Blast From the Past by Namir Shammas Finding the roots of an equation 15 one of the most difficult mathematical challenges Namir has been studying HP s methods for many years and he provides his valuable insight into this topic S05 Commas in the HP Calculator Display I was reading one of my electronically emailed newsletters and I was reminded of the changes in the way various HP calculators display numbers with commas I consulted one of the experts in RPL Dr William Wickes for some official insight into this topic Personally I miss having numbers displayed like those in an HP 12C or HP 15C or HP A1C This is especially important to me because my favorite machine is the HP48GX S06 HP s Calculator Manuals Users of technical instruments need a User s Manual in order to best apply the many complex and not always obvious features of a pro
75. t use tate with an electronic calculator igic for example had many advantages hut was y except by computer programmers logic was in more common use and worked well with lations but when working complex problems it was ry to restructure the equation ich testing and evaluation Hewlett Packard selected free but unambiguous logic system derived from the mathematician named Jan Lukasiewicz It s computer logic and all HP pocket calculators use it it efficient most consistent way to solve because it reduces even the most complex atively few easily handled steps It gives you computations HP Solve 22 Page 5 Page 1 of 2 No kev is needed If you will look at the keyboard of any HP calculator shown in this catalog you will see that none has an key Nor are there any keys for parentheses None are needed Instead all HP calculators have a key like this CENTER Thanks to this key and RPN logic you get four major advantages you don t get with most other calculators 1 You work with only two numbers at a time just as if vou were solving the problem with paper and pencil Only incredibly faster Even the most complex problems are broken down into a series of easily handled two number problems which you can solve in any order that s convenient left to right right to left or from the middle of the equation outwards No matter what kind of problem it is t
76. the Editor The dead line is before the next issue 15 posted There is a long publishing lead time so send in your solution as soon as possible Send you email solution to the editor Richard J Nelson at hpsolve hp com A complete solution will be provided in this column in HP Solve Issue 23 Solve 22 Page 9 Page 2 of 2 Algebraic Solve 22 10 Article HP Algebraic Palmer Hanson Ed Note I have seen many different calculator logic systems since the first HP RPN calculator started the product category we call a scientific or financial calculator Each new model attempts to address a particular use in the market place One observation I can make is simply that no machine is pure anything All calculator designs are practical tradeoffs of features functions logic systems and applications What does Algebraic Mean HP uses the term algebraic in a very generic way in the definition of operating modes if it isn t RPN then it is algebraic unless it 15 RPL Thus the typical options for the operating mode as offered in a menu on a non RPL HP calculator will be RPN or ALG A user will find that calling the ALG option can yield one of several different operating modes depending upon which machine 15 being used For example there is a true algebraic mode in which equations can be entered in a format very similar to the way they appear a textbook an almost algebraic mode which 1s v
77. the anti log of the sum of their logs log cd log c log d el Division To divide two numbers take the anti log of the difference of their logs log c d log c log d e2 Raise to a power To raise a number to a power take the anti log of the power multiplied by the log of the number log c 9 log c e3 Extracting a root To extract the d root of a number c take the anti log of the log of the number divided by the desired root log Ve log cYd 4 Combining the division logtlo g Nc log log c logid e5 log b log b log a Changing the base base c to base a Solve 22 Page 48 Page 7 of 9 Equation 6 is useful to evaluate logarithms using other bases on calculators Most calculators have LOG and LN keys but none for logy To find log2 153 you calculate LOG 153 LOG 2 7 2573878427 As reference box 8 reminds us the LN keys may also be used place of the LOG keys Log of 1 0 regardless of the base log 1 0 7 Log of base 1 logy b 1 e8 Logarithms and exponentials are inverse operations similar to multiplication and division 9 p 9 x and logy b x References links for logarithms 1 Calculator for calculating logarithms of any base see figure 11 Calculate pe rb m re htt rechneronline de lo arithm The logarithm far an user defined base can be calculated as well as all inverse calculations Formula logp a x E JUN q 4 0
78. the first scientific calculator the HP 35A appeared in January 1972 See figure 3a below Fig Typical slide rule used by science students and engineers prior to the mid 1970 s The slide uses scales that are spaced according to the logarithm of the scale value The image in figure three b was cropped from the photograph of a slide rule fig 3a and printed landscape for maximum size The C and D scale values used to multiply numbers were measured and compared with the log of their respective scale values a decimal number and multiplied by 259 These values corresponded to the printed image measured values as shown to two decimal places plus or minus 2 counts In most cases they were identical to three places I just didn t take the time to measure to a fraction of a millimeter For example the scale length for the number 2 on the scale is 78 0 mm The ratio of 78 0 mm 259 mm is 0 301 The log of 2 1s approximately 0 301029995664 1 2 3 4 5 6 7 8 9 10 78 0 12 3 TT Ue ed ea 155 18 1 200 2 19 234 248 e 259 mm 0 259 Fig 3b C slide rule scale of figure 3a cropped and annotated with printed image measurements millimeters Another example of how logarithms are used is to plot data using a log scale instead of the normal linear scale The slide rule scale of fig 3b shows how linear amp log scales compare If the linear scale were used half the len
79. ts The result stored in variable 15 a measure for the square root of the mean squared errors I will call this result NormErr 8 Displays the NormErr value in variable E and then pauses program execution 9 Displays the array of calculated roots after you invoke the CONT command You are not obligated to view the roots You can instead skip this task and start again with task 1 to study another duplicate root Here is a sample session with the above BASIC program Let s run the program to calculate the roots of the polynomial x 1 5 and its errors This is polynomial of order 5 with 1 as the duplicated root RUN ORDER 5 NEE DUPLICATE ROOT 1 7 54717951647E 7 999998938682 0 UOUDODOZ653 3 uU UOUDUDOUZ26D223 0 4092912 42 ETT D0000026533 70 4052013742E T The NormErr value is 7 547E 07 and the five roots are very close to with the imaginary parts being 0 or very small values Table 1 shows the results for the domain of values I chose the square root of the mean squared errors in a two dimensional table of duplicate root values and number of duplicate roots HP Solve 22 Page 30 Page 4 of 7 Table 1 The resulting square roots of the mean squared errors NormErr 5 7 54718E 07 1 11055E 06 2 13388E 06 3 14669 E 06 1 87971E 06 6 16712E 06 7 83154E 06 1 82493E 05 3 11929E 05 4 27488 05 7 54718E 05 Number 10 0 0083082 0 047485168 0 104815

HP Solve: Issue 22 (January 2011)

Contents

Download Pdf Manuals

Related Search

Related Contents