Expanding the HP 12C...Page 1 of 29 Expanding the HP 12C Edward Shore Introduction The HP 12C calculator is an excellent financial calculator. The programming capability allows us

of 29

Expanding the HP 12C Edward Shore

Introduction The HP 12C calculator is an excellent financial calculator. The programming capability allows us to enter algorithms for various applications. Even without the trigonometric functions, we can still use the HP 12C to use it in a wide variety of applications, including scientific. This talk will explore programming tips and other applications.

Contents Assumptions .................................................................................................................................................. 2

Quick Mathematical Tips .............................................................................................................................. 2

Arithmetic ................................................................................................................................................. 2

Mathematical Functions ........................................................................................................................... 2

How to Enter π .......................................................................................................................................... 3

Example: Surface Area and Volume of a Sphere ..................................................................................... 3

How Many Inputs do I have? ........................................................................................................................ 5

Flag and Choice Setting with the x=0 test ..................................................................................................... 5

Example: US to SI Conversions ................................................................................................................. 5

Use of Statistic Variables ............................................................................................................................... 7

Example 1: Sum of f(x) ............................................................................................................................. 7

Example 2: Simple Logistic Curve Fitting ................................................................................................. 8

Two-Round Subroutines ............................................................................................................................... 9

HP 12 Subroutine Example ....................................................................................................................... 9

HP 12C Derivative Approximation .......................................................................................................... 10

Applications and Examples ......................................................................................................................... 12

Solving 2 x 2 Systems .............................................................................................................................. 12

Quadratic Equations ............................................................................................................................... 13

Ulam’s Conjecture ................................................................................................................................... 15

Greatest Common Divisor ....................................................................................................................... 16

Taxes: Gross Up Calculations ................................................................................................................. 17

Cash Flows: Net Present Value, Net Future Value, and Net Utility Stream ........................................... 18

Projectile Motion, No Air Resistance: Maximum Distance (U.S. Units) ................................................. 20

Geometric Areas: Right Triangle, Circle, Ellipse ..................................................................................... 22

Conversion from Binary to Decimal ........................................................................................................ 23

Conversion from Decimal to Binary ........................................................................................................ 24

Coordinate locator using length (r) and pitch (% of slope) ..................................................................... 25

of 29

Combination/Binomial Distribution/Negative Binomial Distribution ..................................................... 27

Conclusion ................................................................................................................................................... 29

Assumptions The presentation and these notes assume that we are using the classic (regular) edition. For the most part, the classic HP 12C and the HP 12C Platinum are the same. However, there are several differences that need to be addressed: Programming Capacity. The classic HP 12C has 99 programming steps and 10 storage registers. In

programming each step has two digits (01 to 99). The HP 12C Platinum has a capacity of 400 steps (the step numbers have three digits) with up to 20 registers. However, the more program steps, the less memory registers available (the minimum is 7).

Arithmetic Operation Setting. The classic HP 12C has RPN (reverse polish notation), while the HP 12C Platinum has both RPN and algebraic. All programs in this presentation are set in RPN mode.

Step Markers. The classic HP 12C has two digit step markers (00-99), while the HP 12C Platinum has three digit step markers (000-400). The difference comes into play when using the GTO (go to) command.

Last X. The location of last X command is in different locations. On the classic HP 12C, LST X is located on the ENTER key (code 43, 36), while the HP 12C Platinum is located on the plus key (code 43, 40).

No trigonometric functions are used. Everything is rounded to 2 decimal places, with Fix 2 mode set ([ f ], [ 2 ]). The accuracy of the programs were based on Fix 2 mode.

Quick Mathematical Tips

Arithmetic Multiplying by 100: [ENTER], 1, [xy], [%T] Example: 7.28, [ENTER], 1, [xy], [%T] returns 728 One key stroke less than [ENTER], 1, 0, 0, [ * ] Dividing by 100: [ENTER], 1, [ % ] Example: 283, [ENTER], 1, [ % ] returns 2.83 Two key strokes less than [ENTER], 1, 0, 0, [ ÷ ]

Mathematical Functions Absolute Value: 2, [y^x], [ g ] ( √ ) Example: -13.3, [ENTER], 2, [y^x], [ g ] ( √ ) returns 13.03

of 29

Sign Function: [ENTER], [ENTER], 2, [y^x], [ g ] ( √ ), [ ÷ ] Example 1: -28, [ENTER], [ENTER], 2, [y^x], [ g ] ( √ ), [ ÷ ] returns -1 Example 2: 28, [ENTER], [ENTER], 2, [y^x], [ g ] ( √ ), [ ÷ ] returns 1 Modulus of two positive numbers (a mod b): a, [ENTER], b, [ ÷ ], [ g ] (LSTx), [ xy ], [ g ] (FRAC), [ * ] Example: a = 43, b = 7, 43 mod 7. 43, [ENTER], 7, [ ÷ ], [ g ] (LSTx), [ xy ], [ g ] (FRAC), [ * ] returns 1 Number of Digits of an integer: [ g ] (LN), [EEX], 1, [ g ] (LN), [ ÷ ], [ g ] (INTG), 1, [ + ] Example: 12345, [ g ] (LN), [EEX], 1, [ g ] (LN), [ ÷ ], [ g ] (INTG), 1, [ + ], returns 5. Note that I used [EEX] 1 to create 10. We could type 10 and the number of key strokes would be the same.

How to Enter π The HP 12C does not have a π button, we’ll have to enter π manually or use an approximation. Using the 10 digit approximation 3.1415965359.

This will take 11 steps, one for each digit and the decimal point.

We can use the approximation π ≈ 355/113. 355/113 = 3.14159292035 355/113 approximates π to 5 decimal places. This takes 8 steps: 3, 5, 5, [ENTER], 1, 1, 3, [ ÷ ] Richard Nelson suggests this memory aid: take the first three odd digits twice (113355), divide

last three by first three (355/113).

Also, we can use the approximation π ≈ 22/7 22/7 = 3.14285714286 22/7 approximates π accurately to 3 places. This takes 5 steps: 2, 2, [ENTER], 7, [ ÷ ]

In this presentation, I use the 355/113 approximation for π. I find this the perfect balance between accuracy and the number of steps. One word of caution, using a fractional approximation can maintain accuracy for a few decimal places. Also the more complex the operation, the more round off error is encountered.

Example: Surface Area and Volume of a Sphere This program calculates the surface area and volume of a sphere give the radius r. Again we take 355/113 as an approximation for π. The well-known formulas: S = 4*π*r^2 V = 4/3*π*r^3 = S * r/3

Step Code Key Notes

01 44, 0 STO 0 Store radius in R0

of 29

02 2 2 Calculate surface volume

03 21 Y^X

04 4 4

05 20 *

06 3 3 Enter π

07 5 5

08 5 5

09 36 ENTER

10 1 1

11 1 1

12 3 3

13 10 ÷

14 20 *

15 31 R/S Display surface volume

16 3 3 Calculate volume

17 10 ÷

18 45, 0 RCL 0

19 20 *

20 43, 33, 00 GTO 00 Display volume, stop program execution

Registers used: R0 = r Input: Enter radius, r, and press [R/S]. Output: Obtain the approximate surface area. Press [R/S] for the volume. Examples: Radius = 2. Surface area ≈ 50.27, Volume ≈ 33.51 Radius = 8.64. Surface area ≈ 938.07, Volume ≈ 2701.65 Fun fact: A sphere of radius 3 will have the same surface area and volume, at approximately 113.09734.

of 29

How Many Inputs do I have? There are three ways to generate inputs:

Loading the stack before executing the program. This is useful when you have only one input.

Prompt for the inputs using the [R/S].

Store all the required inputs in the registers before execution. This is my preferred method when I have 3 or more inputs.

Flag and Choice Setting with the x=0 test You can branch a program by designating a flag variable. You can designate a register of your choice. For this technique, I prefer R0. This is useful for providing several functions without the need to memorize specific line numbers, setting monthly vs. yearly payments, or deciding which conversion is needed. Depending on the size of the program, 2 to 4 choices could fit within program memory of the classic HP 12C. General algorithm: RCL [register number] [test value] - X=0 GTO [appropriate line number] (repeat until you are down to the last two choices) RCL [register number] [test value n-1] - X=0 GTO [line number for choice n-1] GTO [line number for choice n]

Example: US to SI Conversions The program will require two inputs: the value to be converted and a number that designates the following conversion: Choice 0: lb to kg, ÷ 2.2046 Choice 1: in to cm, * 2.54 Choice 2: gal to L, * 3.7854 Choice 3: mi to km, * 1.6093 For example, to convert 18 miles to kilometers, type 18, [ENTER], 3, [R/S]. 3 is stored in the choice variable. (Result: 28.97 km). In this program the choice variable (flag) is stored in R0.

of 29

Program:

Step Code Key Notes

01 44, 0 STO 0 Store choice variable in R0

02 34 XY

03 44, 1 STO 1 Store value to be converted in R1

04 34 XY

05 43, 35 X=0 Conversion: lb to kg?

06 43, 33, 18 GTO 18

07 45, 0 RCL 0 Conversion: in to cm?

08 1 1

09 30 -

10 43, 35 X=0?

11 43, 33, 27 GTO 27

12 45, 0 RCL 0 Conversion: gal to L?

13 2 2

14 30 -

15 43, 35 X=0

16 43, 33, 34 GTO 34

17 43, 33, 43 GTO 43 Conversion: mi to km?

18 45, 1 RCL 1 Conversion: lb to kg, flag = 0

19 2 2

20 48 .

21 2 2

22 0 0

23 4 4

24 6 6

25 10 ÷

26 43, 33, 00 GTO 00

27 45, 1 RCL 1 Conversion: in to cm, flag = 1

28 2 2

29 48 .

30 5 5

31 4 4

32 20 *

33 43, 33, 00 GTO 00

34 45, 1 RCL 1 Conversion: gal to L, flag = 2

35 3 3

36 48 .

37 7 7

38 8 8

39 5 5

40 4 4

41 20 *

42 43, 33, 00 GTO 00

43 45, 1 RCL 1 Conversion: mi to km, flag = 3

44 1 1

45 48 .

46 6 6

47 0 0

48 9 9

49 3 3

50 20 *

51 43, 33, 00 GTO 00

of 29

Use of Statistic Variables We are able to use statistical functions, such as x^ and Σ+ in programs. This will allow us to take advantage of statistical registers. The cost is that six variables will be used (R1 to R6). This leaves only four variables for inputs and outputs (R0, R7, R8, R9) Statistics Variables

R1: n R4: Σy

R2: Σx R5: Σy^2

R3: Σx^2 R6: Σxy

Example 1: Sum of f(x) Σ f(x) R0 = a/x, R7 = b; result R2 = Σx, sum Function starts at step 15 and ends with GTO 02. Template:

Step Code Key Notes

01 43, 33, 15 GTO 15 Calculate f(x)

02 49 Σ+ Add result, return point

03 1 1 Increase R0 by 1, counter

04 44, 40, 0 STO+ 0

05 45, 0 RCL 0 Recall counter (R0)

06 45, 7 RCL 7 Recall upper limit (R7)

07 1 1

08 40 +

09 30 -

10 43, 35 X=0 Test: Is counter = upper limit + 1 Has the counter exceed the upper limit

11 43, 33, 13 GTO 13 Yes: Go to the end of the program

12 43, 33, 01 GTO 01 No: Start the loop again

13 45, 2 RCL 2 Recall Σ

14 43, 33, 00 GTO 00 Stop Execution

15 f(x), use R0 … Start f(x) here, use R0 as x

(last step) 43, 33, 02 GTO 02

Example: Σ x^2 – 2*x = Σ x * (x – 1)

Step Code Key Notes

01 43, 33, 15 GTO 15 Calculate f(x)

02 49 Σ+ Add result, return point

03 1 1 Increase R0 by 1, counter

04 44, 40, 0 STO+ 0

05 45, 0 RCL 0 Recall counter (R0)

06 45, 7 RCL 7 Recall upper limit (R7)

07 1 1

08 40 +

09 30 -

10 43, 35 X=0 Test: Is counter = upper limit + 1 Has the counter exceed the upper limit

11 43, 33, 13 GTO 13 Yes: Go to the end of the program

12 43, 33, 01 GTO 01 No: Start the loop again

of 29

Step Code Key Notes

13 45, 2 RCL 2 Recall Σ

14 43, 33, 00 GTO 00 Stop Execution

15 45, 0 RCL 0 Start f(x) here, use R0 as x

16 36 ENTER

17 36 ENTER

18 2 2

19 30 -

20 20 *

21 43, 33, 02 GTO 02 Finish with GTO 02.

Example 2: Simple Logistic Curve Fitting Curve: y = 1 / (b + m*e^(-x)) Transformation: X’ = e^(-X), Y’ = 1/Y L.R. function on the 12C: Intercept: 0, [ g ] (y^) Slope (from intercept): 1, y^, xy, R↓, xy, - Instructions: 1. Clear statistical registers. Press [ f ], [CLx] (CLEAR REG) 2. Data: y, [ENTER], x, [R/S] 3. For the intercept and slope: GTO 08, R/S, A is displayed, R/S, B is displayed

Step Code Key Notes

01 16 CHS Enter data

02 43, 22 e^x Adjust X: e^-x

03 34 xy

04 22 1/x Adjust Y: 1/y

05 34 xy

06 49 Σ+ Enter data point, n is displayed

07 43, 33, 00 GTO 00 Stop program execution

08 0 0 Calculate b

09 43, 2 y^

10 44, 7 STO 7

11 31 R/S Display b

12 1 1 Calculate m

13 43, 2 y^

14 34 xy

15 33 R↓

16 34 xy

17 30 -

18 44, 8 STO 8

19 43, 33, 00 GTO 00 Display m

Example:

X 0.5 1.0 1.5 2.0 2.5

Y 0.384 0.422 0.450 0.468 0.480

Results: b = 2.00, m = 0.99 (Fix 2). y = 1/(2 + 0.99*e^(-x))

of 29

Two-Round Subroutines The HP 12C does not have subroutine commands, but you can simulate subroutines. Here is one way to do it, if you know the subroutine will be used twice: It is very important you plan your program in advance, with line numbers. Marking the line numbers will facilitate where the subroutine and return markings will be located.

1. Designate a register as a flag variable. Store 1 in the flag variable. 2. Enter the main portion of the program. Determine where the subroutine will be called. 3. Follow the subroutine call with the roll down command (R↓). 4. At the end of the subroutine, recall the flag variable. 5. Determine which round it is by recalling the flag variable. First round: flag = 1. Second round:

flag = 0. Use the test x=0. If it passes, send the pointer to the second R↓. If it fails, subtract 1 from the flag variable and send the pointer to the first R↓.

The following examples will illustrate this method.

HP 12 Subroutine Example Let f(x) = g^2(x) / (1 + g(x)) Where g(x) = x^2 – 1

Step Code Key Notes

01 44, 0 STO 0 Store x in R0

02 1 1

03 44, 2 STO 2 Store flag variable

04 45, 0 RCL 0 Set up for subroutine round 1

05 43, 33, 19 GTO 19

06 33 R↓ 1st return point

07 2 2

08 21 Y^X g^2(x)

09 44, 1 STO 1 Store immediate result

10 45, 0 RCL 0 Set up for subroutine round 2

11 43, 33, 19 GTO 19

12 33 R↓ 2nd return point

13 1 1

14 40 + 1 + g(x)

15 45, 1 RCL 1

16 34 XY

17 10 ÷

18 43, 33, 00 GTO 00 Stop execution

19 2 2 Start subroutine, g(x)

20 21 Y^X

21 1 1

22 30 -

23 45, 2 RCL 2 Recall flag variable

24 43, 35 X=0 Are we in round 2?

25 43, 33, 12 GTO 12 If yes, go to 2nd return point

26 1 1 Otherwise, decrease flag variable

27 44, 30, 2 STO- 2

28 33 R↓ Set up display

29 43, 33, 06 GTO 06 Go to 1st return point

of 29

Variables used: R0 = x (input), R1 = flag variable, R2 = temp storage/result Example 1: x = 1.00, Result: 0.00 Example 2: x = 8.36, Result: 67.90 Example 3: x = -2.81, Result: 6.02

HP 12C Derivative Approximation Approximate the derivative: f’(x) ≈ (f(x + h) – f(x))/h where h = 10^-5 Customize f(x). The function f(x) starts at line 22 with x+h or x on the stack. The GTO statements remain constant regardless of f(x). Up to 71 steps are available for f(x). In the program listing provided below, I provide f(x) = x^2*e^x.

Step Key Code Notes


02 1 1

03 44, 3 STO 3 Set up flag variable, 2 subroutine rounds

04 1 1

05 26 EEX

06 5 5

07 16 CHS

08 44, 1 STO 1 Store h in R1

09 45, 0 RCL 0

10 45, 1 RCL 1

11 40 + x+ h

12 43, 33, 23 GTO 23 Subroutine: 1st round

13 33 R↓ Return: 1st round

14 44, 2 STO 2 Store temp result, f(x + h)

15 45, 0 RCL 0 x

16 43, 33, 23 GTO 23 Subroutine: 2nd round

17 33 R↓

18 44, 30, 2 STO- 2 f(x+h) – f(x)

19 45, 2 RCL 2

20 45, 1 RCL 1

21 10 ÷ (f(x+h) – f(x))/h


23 36 ENTER Subroutine begins here, f(x)

24 36 ENTER

25 2 2

26 21 Y^X

27 34 XY

28 43, 22 e^x

29 20 * End of f(x)

30 45, 3 RCL 3 Need the next 7 steps to the end of f(x) – Check flag variable

31 43, 35 X=0

32 43, 33, 17 GTO 17 Are we in round 2?

33 1 1 If yes, go to 2nd return point

of 29

34 44, 30, 3 STO- 3 Otherwise, decrease flag variable

35 33 R↓ Set up display

36 43, 33, 13 GTO 13 Go to 1st return point

Example 1: x = 1.00. Result: 8.16 Example 2: x = 3.34. Result: 503.68 (approximate) Example 3: x = 6.10. Result: 22030 (should be 22029.83) Template for the derivative below:

Step Key Code Notes


02 1 1

03 44, 3 STO 3 Set up flag variable, 2 subroutine rounds

04 1 1

05 26 EEX

06 5 5

07 16 CHS

08 44, 1 STO 1 Store h in R1

09 45, 0 RCL 0

10 45, 1 RCL 1

11 40 + x+ h

12 43, 33, 23 GTO 23 Subroutine: 1st round

13 33 R↓ Return: 1st round

14 44, 2 STO 2 Store temp result, f(x + h)

15 45, 0 RCL 0 x

16 43, 33, 23 GTO 23 Subroutine: 2nd round

17 33 R↓

18 44, 30, 2 STO- 2 f(x+h) – f(x)

19 45, 2 RCL 2

20 45, 1 RCL 1

21 10 ÷ (f(x+h) – f(x))/h


23 Subroutine begins here, f(x)

… …

End of f(x)

nn 45, 3 RCL 3 Need the next 7 steps to the end of f(x) – Check flag variable

nn 43, 35 X=0

nn 43, 33, 17 GTO 17 Are we in round 2?

nn 1 1 If yes, go to 2nd return point

nn 44, 30, 3 STO- 3 Otherwise, decrease flag variable

nn 33 R↓ Set up display

nn 43, 33, 13 GTO 13 Go to 1st return point

of 29

Applications and Examples

Solving 2 x 2 Systems R4 * x + R5 * y = R6 R1 * x + R2 * y = R3 Calculations: R0 = R4 * R2 – R5 * R1 x = (R2 * R6 – R5 * R3)/R0 y = (R4 * R3 – R1 * R6)/R0 Program:

Step Code Key Notes

01 45, 4 RCL 4 Start with all the values stored in R1 – R6. Calculate determinant which will be stored in R0.

02 45, 2 RCL 2

03 20 *

04 45, 5 RCL 5

05 45, 1 RCL 1

06 20 *

07 30 -

08 44, 0 STO 0

09 34 R/S Display determinant of [[R4, R5][R1, R2]]. If the determinant is 0, then stop. There is no solution.

10 45, 2 RCL 2 Calculate x

11 45, 6 RCL 6

12 20 *

13 45, 5 RCL 5

14 45, 3 RCL 3

15 20 *

16 30 -

17 45, 0 RCL 0

18 10 ÷

19 44, 7 STO 7

20 31 R/S Display x

21 45, 4 RCL 4 Calculate y

22 45, 3 RCL 3

23 20 *

24 45, 1 RCL 1

25 45, 6 RCL 6

26 20 *

27 30 -

28 45, 0 RCL 0

29 10 ÷

30 44, 8 STO 8

31 43, 33, 00 GTO 00 Display y

Example: Solve the system 2 * x + 4 * y = 18 and 3* x – 2* y = 3 Results: Determinant of [[2, 4] [3, -2]] = 16, x = 3, y = 3

of 29

Quadratic Equations R1*x^2 + R2*x + R3 = 0 If R0 < 0, then R4 = real part, R5 = +/- imaginary part If R0 > 0, then R4, R5 are the two real roots R0 = R2^2 – 4 * R1 * R3

Step Code Key Notes

01 45, 2 RCL 2 Start with values stored in R1, R2, R3. Calculate discriminant.

02 36 ENTER

03 20 *

04 45, 1 RCL 1

05 45, 3 RCL 3

06 20 *

07 4 4

08 20 *

09 30 -

10 44, 0 STO 0

11 31 R/S Display discriminant which is stored in R0.

12 0 0

13 43, 34 X≤Y Is 0 ≤ R0?

14 43, 33, 32 GTO 32 If so, go to step 32.

15 45, 1 RCL 1 Complex Roots: real part

16 36 ENTER

17 40 +

18 45, 2 RCL 2

19 16 CHS

20 34 XY

21 10 ÷

22 44, 4 STO 4

23 31 R/S

24 43, 36 LSTx Complex Roots: imaginary part

25 45, 0 RCL 0

26 16 CHS

27 43, 21 √

28 34 XY

29 10 ÷

30 44, 5 STO 5


32 45, 2 RCL 2 Real root calculations

33 16 CHS

34 45, 0 RCL 0

35 43, 21 √

36 40 +

37 45, 1 RCL 1

38 36 ENTER

39 40 +

40 10 ÷

41 44, 4 STO 4

42 31 R/S Real root 1

43 43, 36 LSTx

of 29

Step Code Key Notes

44 45, 2 RCL 2

45 45, 0 RCL 0

46 43, 21 √

47 40 +

48 16 CHS

49 34 XY

50 10 ÷

51 44, 5 STO 5

52 43, 33, 00 GTO 00 Real root 2

Example 1: -2 * x^2 + 3 * x + 3 = 0 (R1 = -2, R2 = 3, R3 = 3) Outputs: R0 = 33; real roots: 2.19, -0.69 Example 2: -2 * x^2 + 3 * x - 3 = 0 (R1 = -2, R2 = 3, R3 = -3) Outputs: R0 = -15; real part: .75, imaginary part: -.97 (roots: .75 +/- .97i)

of 29

Ulam’s Conjecture Determine how many steps it takes an integer n to get to 1 using the following steps: If n is even: divide n by 2 If n is odd: multiply n by 3 and add 1 Input: n (stored in RO) Output: number of steps (stored in R1) Program:

Step Code Key Notes

01 44, 0 STO 0 Store n in R0

02 0 0

03 44, 1 STO 1 Initialize the number of steps in R1

04 45, 0 RCL 0 Check to see if n is even, main loop starts here

05 2 2

06 10 ÷

07 43, 24 FRAC

08 43, 35 X=0

09 43, 33, 16 GTO 16 If n is even, go to step 16

10 3 3 If n is odd

11 44, 20, 0 STO* 0

12 1 1

13 44, 40, 0 STO+ 0 Store result in n for the next loop

14 44, 40, 1 STO+ 1 Increment the counter

15 43, 33, 20 GTO 20

16 2 2 Divide even n by 2, store the result

17 44, 10, 0 STO÷ 0

18 1 1 Increment the counter

19 44, 40, 1 STO+ 1

20 45, 0 RCL 0 Is the result = 1?

21 1 1

22 30 -

23 43, 35 X=0

24 43, 33, 26 GTO 26 If the result = 1, go to the end of the program

25 43, 33, 04 GTO 04 If not, start the loop again

26 45, 1 RCL 1

27 43, 33, 00 GTO 00

Examples: The path from 69 to 1 takes 14 steps. The path from 84 to 1 takes 9 steps.

of 29

Greatest Common Divisor This program calculates the greatest common divisor of two integers. You can enter the two integers in either order. Program:

Step Code Key Notes

01 43, 34 X≤Y Determine which integer is greater

02 34 XY

03 44, 1 STO 1 R1 = max(X,Y)

04 34 XY

05 44, 0 STO 0 R0 = min(X,Y)

06 45, 1 RCL 1 Begin Euclidian division routine

07 45, 1 RCL 1 Recall R1 twice

08 45, 0 RCL 0

09 10 ÷ Divide R1 by R0

10 43, 25 INTG int(R1/R0)

11 45, 0 RCL 0

12 20 *

13 30 - Calculate remainder: remainder = int(R1/R0) * R0 – R1

14 43, 35 X=0 Is remainder = 0?

15 43, 33, 21 GTO 21 If yes, go to the end

16 34 XY If not, set up for the next loop

17 44, 1 STO 1 Old min becomes new max, R0 → R1

18 34 XY

19 44, 0 STO 0 Store new min

20 43, 33, 06 GTO 06 Start Euclidian division routine again

21 45, 0 RCL 0 End Result: display GCD


Input: integer [ENTER] integer [R/S] Result: GCD Test 1: GCD(142,25) Input: 142 [ENTER] 25 [R/S] Result: 1 Test 2: GCD(2555, 1365). Result: 35

of 29

Taxes: Gross Up Calculations We have to pay a person, and the person lives outside of your tax jurisdiction (outside of your state or the United States). Tax withholding may be required. However, you or someone else that is requesting the payment wants the face amount. In order to give the face amount and fulfill the tax requirement at the same time, the payment is must be grossed up. Formulas: Gross up Amount = Face Amount/(1 – Tax Rate%) Amount of Tax = Gross up Amount – Face Amount Program:

Step Code Key Notes

01 44, 0 STO 0 Store Tax Rate

02 34 XY

03 44, 1 STO 1 Store Face Value

04 45, 0 RCL 0

05 1 1

06 25 %

07 1 1

08 30 -

09 16 CHS

10 45, 1 RCL 1

11 34 XY

12 10 ÷

13 31 R/S Display Gross up Amount

14 45, 1 RCL 1

15 30 -

16 43, 33, 00 GTO 00 Display Tax

Input: face value, [R/S], tax rate, [R/S] Result: gross amount, [R/S], tax Part 1: Face Value: $1,000, Tax Rate: 30%. Input: 1000 [R/S], 30 [R/S] Result: 1428.57 [R/S], 428.57 [R/S] The gross up amount is $1,428.57 (total amount to be paid), and corresponding tax is $428.57. Part 2: Face Value: $2,500, Tax Rate: 7%. Input: 2500 [R/S], 7 [R/S] Result: 2688.17 [R/S], 188.17 [R/S] The gross up amount is $2,688.17 (total amount to be paid), and corresponding tax is $188.17.

of 29

Cash Flows: Net Present Value, Net Future Value, and Net Utility Stream Initialization Steps:

1. Clear the registers/finance registers 2. Press [ g ] (GTO) 00 to get the pointer to line 00. (000 for the HP 12C Platinum

editions. Remember, the Platinum editions have up to 400 steps and line numbers are in triple digits)

3. Store -1 in [ n ] (don’t worry, n is incremented in the main loop) 4. Store periodic interest rate in [ i ]. 5. Store 0 in register 0 (0 [STO] [ 0 ]). Register 0 is use for calculating NPV

Enter each cash flow one at a time, and press [R/S]. Ready to calculate:

1. Press [ g ] (GTO) 10 (or 010) to get the pointer to line 10. 2. Press [R/S] to get NPV (net present value) 3. Press [R/S] to get NFV (net future value) 4. Press [R/S] to get NUS (net utility stream)

Program:

Step Code Key Notes

01 16 CHS Main loop starts here, input data

02 15 FV Calculate PV of each cash flow

03 45, 11 RCL n

04 1 1

05 40 +

06 11 N

07 13 PV

08 44, 40, 0 STO+ 0

09 43, 33, 0 GTO 00 Finish main loop

10 45, 0 RCL 0 Calculate NPV

11 31 R/S Display NPV

12 1 1 Calculate NFV

13 16 CHS

14 13 PV

15 0 0

16 14 PMT

17 15 FV

18 45, 0 RCL 0

19 20 *

20 31 R/S Display NFV

21 45, 0 RCL 0 Calculate NUS

22 1 1

23 16 CHS

24 14 PMT

25 0 0

26 15 FV

27 13 PV

28 10 ÷

of 29

Step Code Key Notes

29 43, 33, 0 GTO 00 Display NUS

Example: Rate: 15%

Cash Flow # Flow

0 -10,000.00

1 2,500.00

2 3,500.00

3 5,000.00

4 5,000.00

Input: [ f ] CLEAR FIN, [ f ] CLEAR PRGM -1 [ n ] // do this on each new problem 0 [STO] [ 0 ] // do this on each new problem 15 [ i ] 10000 [CHS] [R/S] 2500 [R/S] 3500 [R/S] 5000 [R/S] 5000 [R/S] Results: [GTO] 10 [R/S]: NPV = 966.76 [R/S]: NFV = 1690.88 [R/S]: NUS = 338.62 Input slowly and carefully.

of 29

Projectile Motion, No Air Resistance: Maximum Distance (U.S. Units) For an object that travels in a projectile motion, we can track its range (distance traveled from the beginning) and height by: R = v^2 * sin (2 * θ)/g H = (v^2 * (sin θ)^2) / (2 * g) Where: v = initial velocity θ = initial angle g = Earth’s gravity. For in US units, g = 32.1740468 ft/s^2. This program uses the approximation g ≈ 32.174 ft/s^2 The projectile will have maximum range (distance) if we aim the object at 45°. Aside: Why? Let’s let range (R) be a function of angle (θ): R = v^2/g * sin(2 * θ) Find the critical points by finding the zero of the first derivative: dR/dθ = 2 * v^2/g * cos (2 * θ) 0 = 2 * v^2/g * cos (2 * θ) 0 = cos (2 * θ) arccos 0 = 2 * θ π/2 = 2 * θ θ = π/4 Now we can use the second derivative to test whether the function is at a maximum (less than 0) and minimum (more than 0) at the crucial point. d^2 R/dθ^2 = -4 * v^2/g * sin(2 * θ) Let θ = π/4 -4 * v^2/g * sin(2 * π/4) = -4 * v^2/g * sin(π/2) = -4 * v^2/g < 0 (We are assuming the initial velocity is positive, and g ≈ 32.174 >0) Since the second derivative at θ = π/4 is negative, the range is at its maximum. Note that calculus is done when the angles are measured in radians. π/2 in degrees is 90° and π/4 in degrees is 45°. (We are only concentrating on angles between 0° and 90°) To find the maximum range and height, substitute at θ = 45° and range and height are: R = v^2 /g H = v^2 / (4 * g) The time this certain projectile lasts is: T = (v * √2) / (2 * g) Program: Keep in mind: this is done on the HP 12C (regular). For the HP 12C Platinum, the code for Last X is 43, 40 ([ g ] [ + ])

of 29

Step Code Key Notes

01 44, 1 STO 1 Input Velocity, store in R1

02 2 2 Calculate time of projectile

03 10 ÷

04 43, 36 LST x Recall 2

05 43, 21 √x

06 20 *

07 3 3 g ≈ 32.174 ft/s^2 (= 9.80665 m/s^2)

08 2 2 Earth’s gravitational constant

09 48 .

10 1 1

11 7 7

12 4 4

13 44, 0 STO 0 Store g in R0

14 10 ÷

15 31 R/S Display time of projectile (s)

16 45, 1 RCL 1 Calculate range

17 36 ENTER

18 20 *

19 45, 0 RCL 0

20 10 ÷

21 31 R/S Display range (ft)

22 4 4 Calculate height

23 10 ÷

24 43, 33, 00 GTO 00 Display height (ft)

Input: velocity in ft/s (convert from mph to ft/s by multiplying it by 22/15) Output: time of projectile in seconds, [R/S] range of projectile in feet, [R/S] height of projectile in feet Example: V = 25 mph = 36.6666667 ft/s (110/3) Output: Time: 0.81 sec, Range: 41.79 ft, Height: 10.45 ft Source: Rosenstein, Morton. Computing With the Scientific Calculator Casio: Tokyo, Japan. 1986. ISBN-10: 1124161430

of 29

Geometric Areas: Right Triangle, Circle, Ellipse Here is a program where a choice variable (flag) is used to determine which area is to be counted. R0 = choice, R1 = see below, R2 = see below Choice 0: Triangle, R1 = A, R2 = B, area = 0.5 * A * B Choice 1: Circle, R1 = A, area = π * A^2 Choice 2: Ellipse, R1 = A, R2 = B, area = π * A * B

Step Code Key Notes

01 45, 0 RCL 0 Input and store the choice variable

02 43, 35 X=0 Is choice = 0? (Triangle)

03 43, 33, 10 GTO 10

04 45, RCL 0 Is choice = 1? (Circle)

05 1 1

06 30 -

07 43, 35 X=0

08 43, 33, 16 GTO 16

09 43, 33, 28 GTO 28 If not, assume choice = 2. (Ellipse)

10 45, 1 RCL 1 Calculate area of a triangle

11 45, 2 RCL 2

12 20 *

13 2 2

14 10 ÷

15 43, 33, 00 GTO 00

16 45, 1 RCL 1 Calculate area of a circle

17 2 2

18 21 Y^X

19 3 3 Enter approximation of π

20 5 5

21 5 5

22 20 *

23 1 1

24 1 1

25 3 3

26 10 ÷

27 43, 33, 00 GTO 00

28 45, 1 RCL 1 Calculate area of an ellipse

29 45, 2 RCL 2

30 20 *

31 3 3

32 5 5

33 5 5

34 20 *

35 1 1

36 1 1

37 3 3

38 10 ÷

39 43, 33, 00 GTO 00

of 29

Conversion from Binary to Decimal Maximum value: 1023 (1111111111 (10 ones)) R0 = binary, R1 = decimal, R2 and R3 are used

Step Code Key Notes

01 44, 0 STO 0 Store the binary integer, assume no sign bit

02 43, 23 LN Determine the number of digits

03 26 EEX

04 1 1

05 43, 23 LN

06 10 ÷

07 43, 25 INT

08 1 1

09 40 +

10 44, 2 STO 2 Store number of digits in R2

11 0 0 Initialize decimal conversion, store 0 in R1

12 44, 1 STO 1

13 44, 3 STO 3

14 45, 0 RCL 0 Conversion process

15 45, 3 RCL 3

16 1 1

17 40 +

18 1 1

19 0 0

20 34 XY

21 21 Y^X

22 10 ÷

23 43, 24 FRAC

24 26 EEX

25 1 1

26 20 *

27 43, 25 INTG

28 2 2

29 36 ENTER

30 45, 3 RCL 3

31 21 Y^X

32 20 *

33 44, 40, 1 STO+ 1

34 1 1

35 44, 40, 3 STO+ 3

36 45, 3 RCL 3 Determine the number of digits left to work with

37 45, 2 RCL 2

38 30 -

39 43, 35 X=0

40 43, 33, 42 GTO 42

41 43, 33, 14 GTO 14

42 45, 1 RCL 1 Display decimal integer

43 43, 33, 00 GTO 00

Examples: BIN 1101, DEC 13 BIN 11000, DEC 24

of 29

Conversion from Decimal to Binary Maximum number to be converted depends on setting: 1023 (Fix 0), 511 (Fix 1), 255 (Fix 2), etc. R0 = decimal, R1 = binary, R2, R3, and R4 are used

Step Code Key Notes

01 44, 0 STO 0 Store decimal in R0 and R4 (temp)

02 44, 4 STO 4

03 43, 23 LN Determine the number of digits

04 2 2

05 43, 23 LN

06 10 ÷

07 43, 25 INTG

08 44, 2 STO 2 Store the number of digits in R2 and R3

09 44, 3 STO 3

10 0 0 Initialize binary conversion, store 0 in R1

11 44, 1 STO 1

12 45, 4 RCL 4

13 2 2

14 45, 3 RCL 3 R3: digit counter

15 21 Y^X

16 43, 34 X≤Y

17 43, 33, 19 GTO 19

18 43, 33, 33 GTO 33

19 43, 30, 4 STO- 4

20 1 1

21 43, 30, 3 STO- 3

22 1 1

23 0 0

24 45, 3 RCL 3

25 1 1

26 40 +

27 21 Y^X

28 44, 40, 1 STO+ 1

29 45, 3 RCL 3 Are there any more digits to work with?

30 43, 35 X=0

31 43, 33, 36 GTO 36

32 43, 33, 12 GTO 12

33 1 1

34 44, 30, 3 STO- 3

35 43, 33, 29 GTO 29

36 45, 4 RCL 4

37 44, 40, 1 STO+ 1

38 45, 1 RCL 1 Display binary conversion

39 43, 33, 00 GTO 00

Example: Turn the calculator to floating mode (FIX 0) Decimal: 128, Binary: 10000000 Decimal: 463, Binary: 111001111

of 29

Coordinate locator using length (r) and pitch (% of slope) The coordinates x and y are determined using similar triangles.

Step Code Key Notes

01 45, 1 RCL 1

02 1 1

03 2 2

04 44, 2 STO 2

05 44, 20, 1 STO* 1

06 45, 1 RCL 1

07 36 ENTER

08 20 *

09 45, 2 RCL 2

10 36 ENTER

11 20 *

12 40 +

13 43, 21 √ √(R1^2 + R2^2)

14 44, 3 STO 3

15 45, 1 RCL 1 Solve the ratio R4/R1 = R3/R0

16 45, 0 RCL 0

17 20 *

18 45, 3 RCL 3

19 10 ÷

20 44, 4 STO 4 Display y

21 31 R/S

22 45, 2 RCL 2 Solve the ratio R5/R2 = R3/R0

23 45, 0 RCL 0

24 20 *

25 45, 3 RCL 3

26 10 ÷

27 44, 5 STO 5 Display x

of 29

Step Code Key Notes

28 43,33,00 GTO 00

Store the following values prior to running the program: R0: r R1: pitch (as a decimal amount) Pitches vs. Angles

Angle Pitch Angle Pitch

5° 0.08749 30° 0.57735

10° 0.17633 45° 1.00000

15° 0.26795 60° 1.73205

20° 0.36397 75° 3.73205

25° 0.46631

14.03624° 0.25 36.86990° 0.75

26.56505° 0.5

Example 1: r = 17 (R0), pitch = 0.33 (R1). Result: y = 5.33 (R4), x = 16.14 (R5)

of 29

Combination/Binomial Distribution/Negative Binomial Distribution Introduction and Formulas Combination: Find the number of groups out of a possible set of objects. The order of objects obtained does not matter. Store n in R1, x in R0, and p in R2. Press [ f ] [ R↓] (CLEAR PRGM), [ R/S ] Formula: COMB(n, x) = n!/(x! * (n-x)!) Binomial Distribution: Find number of successes (x) in a fixed number of trials (n). Store n in R1, x in R0, and p in R2. Press [ g ] [ R↓ ] (GTO) 26, [R/S] Formula: COMB(n, x) * p^x * (1 – p)^(n – x) Negative Binomial Distribution: Find the number of trials (n) needed to obtain a fixed amount of successes (x). Store x in R1, n in R0, and p in R2. Press [ g ] [ R↓ ] (GTO) 43, [R/S] Formula: COMB(x – 1, n – 1) * p^(n -1) * (1 – p)^((x - 1) - (n – 1)) In the distribution calculations, p is the probability where 0 ≤ p ≤ 1. Note: R3 is used as a flag, which will allowed for branching.

Step Code Key Notes

Combination

01 0 0

02 44, 3 STO 3

03 45, 1 RCL 1

04 43, 3 N!

05 45, 0 RCL 0

06 43, 3 N!

07 10 ÷

08 45, 1 RCL 1

09 45, 0 RCL 0

10 30 -

11 43, 3 N!

12 10 ÷

Flag Testing

13 45, 3 RCL 3

14 1 1

15 30 -

16 43, 35 X=0

17 43, 33, 29 GTO 29

18 45, 3 RCL 3

19 2 2

20 30 -

21 43, 35 X=0

22 43, 33, 49 GTO 49

23 33 R↓

24 33 R↓

25 43, 33, 00 GTO 00

Binomial Distribution

26 1 1

of 29

Step Code Key Notes

27 44, 3 STO 3

28 43, 33, 03 GTO 03

29 33 R↓

30 45, 2 RCL 2

31 45, 0 RCL 0

32 21 Y^X

33 20 *

34 1 1

35 45, 2 RCL 2

36 30 -

37 45, 1 RCL 1

38 45, 0 RCL 0

39 30 -

40 21 Y^X

41 20 *

42 43, 33, 00 GTO 00

Negative Binomial Distribution

43 1 1

44 44, 30, 1 STO- 1

45 44, 30, 0 STO- 0

46 2 2

47 44, 3 STO 3

48 43, 33, 03 GTO 03

49 33 R↓

50 33 R↓

51 45, 2 RCL 2

52 45, 0 RCL 0

53 21 Y^X

54 20 *

55 1 1

56 45, 2 RCL 2

57 30 -

58 45, 1 RCL 1

59 45, 0 RCL 0

60 30 -

61 21 Y^X

62 20 *

63 43, 33, 00 GTO 00

Examples: Find the number of combinations of groups of 2 out of possible 12 objects. 12 [STO] 1, 2 [STO] 0, [ f ] [ R↓ ] (CLEAR PRGM) Result: 66 Binomial Distribution: Toss a coin 25 times. (trails) What is the probability of tossing 10 heads? (successes) Assume a fair coin. The variables n = 25, x = 10, p = 0.5 25 [ STO ] 1, 10 [ STO ] 0, 0.5 [ STO ] 2, [ g ] [ R↓ ] (GTO) 26 [ R/S ] Result: 0.10 (0.0974166393)

of 29

Negative Binomial Distribution: Assume a fair coin. What is the probability that the 15th tossed of heads comes on the 25th toss of the coin? x = 15, n = 25, p = 0.5 25 [STO] 1, 15 [STO] 0, 0.5 [ STO ] 2, [ g ] [ R↓] (GTO) 43 [ R/S ] Result: 0.12 (0.1168999672)

Conclusion As we can we see, we can use basic programming and mathematical techniques to expand the HP 12C beyond the financial applications. The HP 12C is one of the few RPN calculators being sold in the open market and remains a favorite. Contact Eddie Shore Email: [email protected] Blog: Eddie’s Math and Calculator Blog, edspi31415.blogspot.com
mailto:[email protected]://www.edspi31415.blogspot.com/

Documents

Expanding the HP 12C...Page 1 of 29 Expanding the HP 12C Edward Shore Introduction The HP 12C calculator is an excellent financial calculator. The programming capability allows us