Happy Halloween and Happy Birthday, HP 42S

Happy Halloween and Happy Birthday, HP 42S

The HP 42S calculator was released to the market on October 31, 1988.  

I also want to wish you a safe and Happy Halloween.  Thank you.

Eddie 

All original content copyright, © 2011-2022.  Edward Shore.   Unauthorized use and/or unauthorized distribution for commercial purposes without express and written permission from the author is strictly prohibited.  This blog entry may be distributed for noncommercial purposes, provided that full credit is given to the author. 

Paco Arjonilla: HP 35S Program Suite

 Paco Arjonilla:  HP 35S Program Suite

On today's blog post, I am going to highlight a recent suite of programs, written by Paco Arjonilla, is designed to compliment the rich function set of the HP 35S calculator.

Each category will have a set of programs of functions.  Stack diagrams are provided in the documentation.  Each category belongs to a single label.

Label P:  vector functions (includes norm, cross product, 3 x 3 determinant) - 149 steps

Label G:  complex numbers (conjugate, polar/rectangular conversions, construct and break down complex numbers) - 48 steps

Label Q:  quaternions (construction, angle, rotation) - 80 steps

Label K:  quadratic equation solver - 28 steps

Label H:  save stack data to variables - 55 steps

Label I:  recall saved stack - 55 steps

Label J:  stack operations (push and pop the x stack, delete the x stack, clear reserved memory registers, copy memory registers) - 159 steps

Paco's website also has a cheat sheet which can be downloaded as a PDF file.

For the programs and for license information, click here:  

http://www.pacoarjonilla.es/HP-35s.html

Special thanks to Paco Arjonilla.

Eddie 

All original content copyright, © 2011-2022.  Edward Shore.   Unauthorized use and/or unauthorized distribution for commercial purposes without express and written permission from the author is strictly prohibited.  This blog entry may be distributed for noncommercial purposes, provided that full credit is given to the author. 

HP 71B and TI-84 Plus CE: Binary Decimal Conversions Revised

HP 71B and TI-84 Plus CE:  Binary Decimal Conversions Revised

The programs presented can easily convert integers without the needs of lists or strings in input.  

HP 71B Programs:  BINDEC, DECBIN

No modules are required.

BINDEC (137 bytes)

100 DISP "BIN>DEC" @ WAIT .5

105 DESTROY A,B,C,D,E

110 INPUT "BIN? ";A

115 B=1 @ D=0 @ C=A

200 E=A/10 @ A=INT(E)

205 D=D+10*B*(E-A) @ B=2*B

210 IF A THEN 200

300 DISP "(2)";C;"="D

DECBIN (130 bytes)

100 DISP "DEC>BIN" @ WAIT .5

105 DESTROY A,B,C,D

110 INPUT "DEC? ";D

115 B=0 @ A=D @ C=1

200 E=D/2 @ D=INT(E)

205 B=B+C*(D<E)

210 C=C*10

215 IF D THEN 200

300 PRINT A;"=(2)";B

TI-84 Plus CE Programs:  BINDEC2, DECBIN2

BINDEC2

"V.2"

Disp "BIN>DEC"

Input "BIN? ",A

1→B:0→D:A→C

Lbl 2

A/10→E:iPart(E)→A

D+10*B*(E-A)→D

2*B→B

If A:Goto 2

Disp "(2) "+toString(C)+"= "+toString(D)

DECBIN2

"V.2"

Disp "DEC>BIN"

Input "DEC? ",D

0→B:D→A:1→C

Lbl 2

D/2→E:iPart(E)→D

B+C*(D<E)→B

10*C→C

If D:Goto 2

Disp toString(A)+"=(2) "+toString(B)

This is recommended for only integer values of 512 (decimal base) or less.  

Example

Decimal: 476  (DEC>BIN)

Result Binary:  111011100

Binary:  10111011 (BIN>DEC)

Result Decimal:  187

Source

Craig, John Clark.  119 Practical Programs For The TRS-80 Pocket Computer  TAB Books Inc.  Blue Ridge Summit, PA.  1982 pg. 133

Happy computing,

Eddie 

All original content copyright, © 2011-2022.  Edward Shore.   Unauthorized use and/or unauthorized distribution for commercial purposes without express and written permission from the author is strictly prohibited.  This blog entry may be distributed for noncommercial purposes, provided that full credit is given to the author. 

Plus42 Solver: Derivatives

Plus42 Solver: Derivatives 

Introduction

The Technical Applications book for the HP 27S and HP 19B (and can apply to the HP 17B outside of trigonometry) shows the numerical first and second derivative can be calculated by the formulas:

f ' (x) = (f(x+h) - f(x-h)) / (2 * h)

f ' ' (x) = (f(x + h) - 2 * f(x) + f(x + h)) / h^2 

where h is sufficiently small, like 10^-5 to 10^-12.

Legacy Formulas vs. Plus42 Formulas 

The formulas suggested by the Technical Applications Book are:

First Derivative:

Second Derivative:

Depending on the function FX, this above can turn the above into long equations.  With the ability of user functions, this allows us to use the original definitions.

FX(X): f(x) (insert f(x)

First Derivative:

F'X=(FX(X+H)-FX(X-H))÷(2×H)

Second Derivative:

F''X=(FX(X+H)-2×FX(X)+FX(X-H))÷SQ(H)

SQ:  press by the key sequence [(shift)] (x^2)

':  (ALPHA) [ ↓ ] (PUNC) [ ↓ ] ( ' )

Note:  Radians mode 

Examples 

FIX 5 mode is set.

Example 1:

f(x) = 0.5 * cos(3*x)

x = π/4

FX(X):0.5×COS(3×X)

f'(x) ≈ -1.06066

f''(x) ≈ 3.18198

Example 2:

f(x) = (x^2 + 3*x + 5) / (4*x - 1)

x = 2

FX(X):(X^2+3×X+5)÷(4×X-1)

f'(x) ≈ -0.22449

f''(x) ≈ 0.54227

Functions with Variable Constants

It is easy to expand the user function FX to include variable constants.  For example:

f(x) = -ln(cos(√(a*x)))

Calculate the value and first derivative at x = 0.11 and a = 0.46

Attach variable constants at the end of FX:

FX(X:A):-LN(COS(SQRT(A×X)))

F'X=(FX(X+H:A)-FX(X-H:A))÷(2×H)

f(x:a) ≈ 0.02552

f'(x:a) ≈ 0.23396

The user function makes the calculating numerical derivatives easier.  

Source:

Technical Applications: Step-by-Step Solutions for Your HP-27S or HP-19B Calculator Hewlett Packard.   Edition 2.  Corvallis, OR.   November 1988.  pg. 44

Have any Halloween plans?  Wishing you a great day,

Eddie 

All original content copyright, © 2011-2022.  Edward Shore.   Unauthorized use and/or unauthorized distribution for commercial purposes without express and written permission from the author is strictly prohibited.  This blog entry may be distributed for noncommercial purposes, provided that full credit is given to the author. 

Retro Review: Texas Instruments Financial Investment FIA-10

Retro Review:   Texas Instruments Financial Investment FIA-10

Possibly a predecessor to the well known BA-II Plus, the FIA-10 introduced financial worksheets for various financial applications.  

Quick Facts

Model:  FIA-10

Company:  Texas Instruments

Years:  1988 - 1990

Type:  Finance

Batteries: 1 x CR-2032

Operating Modes:  Chain

Number of Registers: 10

Display:  large display, 5 lines for variables, 1 line for calculations 

A Collector's Calculator

The FIA-10 is a collector's calculator! The FIA-10 is a financial calculator has a large screen with five soft keys located to the right side of the screen.  There is a sixth button, Compute, is used to solve for variables.  

The FIA-10 has ten worksheets.  Each worksheet has up to four columns, in which each column can have a separate list of data.

1.  Time Value of Money/Amortization:  4 columns

2.  Cash Flow including net present value, net future value, internal rate of return:  2 columns

3.  Bond Calculations:  yields and prices:  4 columns

4.  Depreciation:  including straight line, sum of the year's digits, declining balance:  1 column

5.  Statistics with Linear Regression: 1 column

6.  Interest Conversions between APR and EFF/NOM:  1 column

7.  Percent Change: 1 column

8.  Profit:   cost vs. selling price vs. margin:  1 column

9.  Date Calculation:  for dates from October 15, 1982 to February 28, 4000:  1 column

10.  Memory:  up to ten memory registers R0- R9 can be seen in this worksheet:  1 column

Storage arithmetic is allowed  (STO+, STO-, STO×, STO÷, STO^).  Any variable can be recalled by using the [ RCL ] key.   

In short, this rare calculator is worth collecting.  If you are familiar with the BA II Plus, the FIA-10 will be easy to work with.  

Eddie 

All original content copyright, © 2011-2022.  Edward Shore.   Unauthorized use and/or unauthorized distribution for commercial purposes without express and written permission from the author is strictly prohibited.  This blog entry may be distributed for noncommercial purposes, provided that full credit is given to the author. 

Happy Kibibyte Day! (October 24)

 

Unlike a kilobyte, that is 1,000 bytes, a kibibyte has 2^10 bytes.   

Happy programming,

Eddie

HP 71B, TI 84 Plus CE: Egyptian Fractions to Unit Fractions by the Greedy Algorithm

HP 71B, TI 84 Plus CE:  Egyptian Fractions to Unit Fractions by the Greedy Algorithm

Introduction 

An Egyptian fraction consists of a sum of unit fractions.  A unit fraction is a fraction with 1 in its numerator and a positive integer in its denominator.   

Yesterday, we calculated the Egyptian fraction given unit fractions, full simplified.   Today, we will attempt the reverse process, breaking down an Egyptian fraction into a sum of unit fractions.

N/D = 1/A1 + 1/A2 + 1/A3 + 1/A4 + ....

The Algorithm

The Greedy Algorithm is used.  

The Greedy Algorithm attempts to start the expansion by using the division of ceiling(D/N) = int(D/N) + 1.   Please be aware that this may not be the fastest or more efficient way.  

We also have the problem of a lot of calculators inability to display long integers, that is longer than 10 (usually) before switching to scientific notation mode.  In order to accommodate for memory and the calculator's ability to display integers, I set the algorithm to stop should the denominator for the next step exceed 10^8.  In that case, the remainder fraction is displayed.  

TI-84 Plus CE Program EGYPTUNT

"2022-09-04 EWS"

Disp "EGYPT>SUM UNIT FRACS"

Input "NUM? ",N

Input "DEN? ",D

N→M:D→C

Disp "SUM: ":Wait .5

While M≠1

int(C/M)+1→B

Disp "1/"+toString(B)

Pause 

M*B-C→M

B*C→C

If C>1E8:Goto 2

End

Disp "1/"+toString(C)

Stop

Lbl 2

Disp "REMAIN:"

Disp toString(M)+"/"+toString(C)

* This assumes your TI-84 Plus CE is at least has the 5.5 operating system.

HP 71B Program EGYPTUNT

100 DISP "EGYPT>SUM UNIT FRACS" @ WAIT .5

105 DESTROY N,D,C,M

110 INPUT "NUM? ";N

115 INPUT "DEN? ";D

120 M=N @ C=D

125 DISP "SUM=" @ WAIT .5

200 B=INT(C/M)+1

205 DISP "1/"&STR$(B) @ PAUSE

210 M=M*B-C @ C=C*B

215 IF C>10^8 THEN 400

220 IF M#1 THEN 200

300 DISP "1/"&STR$(C)

305 STOP

400 DISP "REMAIN: " @ WAIT .5

405 DISP STR$(M)&"/"&STR$(C)

410 STOP

Examples:

6/7

NUM = 6

DEN = 7

Results:  1/2, 1/3, 1/42

6/7 = 1/2 + 1/3 + 1/42

181/360

NUM = 181

DEN = 360

Results: 1/2, 1/361, 1/129961, REMAIN: 2/33779463120

Note:  Please be aware that this Greedy Algorithm  is only one approach and just because we get a REMAIN message does not conclude that the fraction cannot be broken into a sum of unit fractions.  For example:  181/360 = 1/5 + 1/8 + 1/9 + 1/15

See the source below for more information.

"Greedy algorithm for Egyptian fractions" Wikipedia.  Edited December 10, 2021.  Last Accessed September 4, 2022.  https://en.wikipedia.org/wiki/Greedy_algorithm_for_Egyptian_fractions

Eddie

Halloween is just around the corner!

All original content copyright, © 2011-2022.  Edward Shore.   Unauthorized use and/or unauthorized distribution for commercial purposes without express and written permission from the author is strictly prohibited.  This blog entry may be distributed for noncommercial purposes, provided that full credit is given to the author. 

HP 71B, TI 84 Plus CE: Sum of Egyptian Fractions

HP 71B, TI 84 Plus CE:  Sum of Egyptian Fractions

Introduction 

An Egyptian fraction consists of a sum of unit fractions.  A unit fraction is a fraction with 1 in its numerator and a positive integer in its denominator.   

Today's blog entry will calculate the Egyptian fraction given unit fractions, full simplified.   On tomorrow's entry, we will attempt the reverse process, breaking down an Egyptian fraction into a sum of unit fractions.

N/D = 1/A1 + 1/A2 + 1/A3 + 1/A4 + ....

The Algorithm

Start out with two unit fractions 1/A1 and 1/A2.  

Let N2/D2 = 1/A1 + 1/A2

Then:

N2/D2 = 1/A1 + 1/A2

N2/D2 = (A1 + A2)/(A1*A2)

Let N3/D3 = 1/A3 + N2/D2

Then:  N3/D3 = (A3*N2 + D2)/(N2*D2)

For any general point:

N_x+1/D_x+1 = (N_x * A_x + D_x)/(A_x * D_x)

This allows the user to input as many unit fractions as allowed.  

TI-84 Plus CE Program EGYPTSUM

Disp "EGYPT FRACTION SUM"

"2022-09-03 EWS"

Input "NO. TERMS? ",S

Input "1/A1? ",A

Input "1/A2? ",B

A+B→N

A*B→D

For(I,3,S)

"1/A"+toString(I)+"? "→Str1

Input Str1,A

A*N+D→N

A*D→D

End

gcd(N,D)→G

N/G→N

D/G→D

Disp "NUM= "+toString(N)

Disp "DEC= "+toString(D)

Disp "FRAC="+toString(N/D)

* This assumes your TI-84 Plus CE is at least has the 5.5 operating system.

HP 71B Program EGYPTSUM 

100 DISP "EGYPT FRACTION SUM" @ WAIT .5

105 DESTROY S,A,B,I,N,D,G

110 INPUT "# TERMS? ";S

115 INPUT "1/A1? ";A

120 INPUT "1/A2? ";B

125 N=A+B @ D=A*B

130 IF S=2 THEN 300

200 FOR I=3 TO S:

205 DISP "1/A"&STR$(I)&"?" @ WAIT .5

210 INPUT "1/A? ";A

215 N=A*N+D

220 D=A*D

225 NEXT I

300 A=N @ B=D

305 G=MOD(A,B)

310 A=B @ B=G

315 IF G THEN 305

320 N=N/A @ D=D/A

400 DISP "NUM= ";N

405 DISP "DEN= ";D

410 DISP "FRAC= ";N/D

415 STOP

The 300-325 section calculates the common greatest divisor, based off the "Greatest Common Divisor" program from 119 Practical Programs for the TRS-80 Pocket Computer (see source below).

Examples:

1/2 + 1/6

# TERMS = 2

Result:  2/3 (NUM = 2, DEN = 3)

1/7 + 1/8 + 1/2 

# TERMS = 3

Result:  43/56 (NUM = 43, DEN = 56)

1/9 + 1/5 + 1/15 + 1/8

# TERMS = 4

Result: 181/360 (NUM = 181, DEN = 360)

Source

Craig, John Clark.  119 Practical Programs For The TRS-80 Pocket Computer  TAB Books Inc.  Blue Ridge Summit, PA.  1982 pg. 133

All original content copyright, © 2011-2022.  Edward Shore.   Unauthorized use and/or unauthorized distribution for commercial purposes without express and written permission from the author is strictly prohibited.  This blog entry may be distributed for noncommercial purposes, provided that full credit is given to the author. 

TI-65: Relativity Applications

TI-65:  Relativity Applications

Introduction

The programs on today's blog entry were programmed on the Texas Instruments TI-65 calculator.   One of the features of the TI-65 is the set of physical constants.  

The programs call up the speed of light, c = 299,792,458 m/s, by the [ 3rd ] [ 2 ] key combination.  If you are using a TI-58, TI-59, or TI-66, where there are no physical constants, please store 299792458 in a memory register first, and replace the key sequence by recalling that register.  

These programs will use the shortcut:

cos(arcsin x) = √(1 - x^2) for all -1 ≤ x ≤ 1

This allows us to cut down on the number of bytes required.  

Redshift vs. Blueshift

f0 = fs * √((1 - v/c) / (1 + v/c))

v = velocity of the source/star in m/s

blueshift:  v < 0

redshift:  v > 0

fs = frequency given off by the source/star in Hz

f0 = observed frequency in Hz

v/c is stored in register 0.

Program 

Keys:

LBL F1

1

-

(

R/S

÷

[c]

)

STO 0

=

÷

(

1 

+

RCL 0

)

=

√

×

R/S

=

RTN

Key Codes:

2nd 53.53

1

49

16

51

28

3rd 46

17

12.0

39

28

16

1

59

13.0

17

39

33

38

51

39

2nd 52

Instructions:

1.  Press  [ F1 ]. 

2.  Enter v when execution stops, press [ R/S ].

3.  Enter fs when execution stops, press [ R/S ].

4.  f0 is calculated.

Example:

v = 210,000,000 m/s

fs = 450 * 10^6 Hz  (450 [ EE ] 6)

Result:  f0 = 1.888581542E8 Hz  (188.8581542 * 10^6)

Relativistic Doppler Effect

f0 = (fs * √(1 - v^2/c^2)) / (1 - v/c * cos Θ)

v = velocity of the source

Θ = direction of source's motion relative to the observer (angle from the observer's perspective)

fs = frequency given off by the source/star in Hz

f0 = observed frequency in Hz

Program

Key:

LBL F2

(

R/S

÷

[c]

)

STO 0

INV SIN

COS

×

R/S

=

÷

(

1

-

RCL 0

×

R/S

COS

)

=

RTN

Key Code:

2nd 53.54

16

51

28

3rd 46

17

12.0

-22

23

38

51

39

28

16

1

49

13.0

38

51

23

17

39

2nd 52

Instructions:

1.  Press  [ F2 ]. 

2.  Enter v when execution stops, press [ R/S ].

3.  Enter fs when execution stops, press [ R/S ].

4.  Enter Θ when the execution stops, press [ R/S ].

5.  f0 is calculated.

Example:

v = 196 * 10^6 m/s

fs = 500 * 10^6 Hz  

Θ = 40°  (degrees mode)

Result:  f0 = 7.579362482 * 10^8 Hz = 757.9362482 * 10^6 Hz

Time Dilation

t' = t0 * √(1 - (v/c)^2)

t0 = source's time 

v = velocity of source in m/s

t' = observer's time

Key:

LBL 0

R/S

÷

[ c ]

=

INV SIN

COS 

×

R/S

=

RTN

Key Codes:

2nd 53.0

51

28

3rd 46

39

-22

23

38

51

39

2nd 52

Instructions:

1.  Press  [ SBR ] [ 0 ]. 

2.  Enter v, press [ R/S ].

3.  Enter t0 when execution stops, press [ R/S ].

4.  t' is calculated.

Example:

v = 164 * 10^6 m/s

t0 = 1.5 yr

Result:  t' = 1.255654687 yr

Source:

"Sinclair Enterprise Programmable: Physics Engineering Electronics Program Library"  Sinclair Radionics Inc.  New York, NY and St. Ives, Huntingdon, Cambridgeshire UK.  1976

All original content copyright, © 2011-2022.  Edward Shore.   Unauthorized use and/or unauthorized distribution for commercial purposes without express and written permission from the author is strictly prohibited.  This blog entry may be distributed for noncommercial purposes, provided that full credit is given to the author. 

Swiss Micros DM42, Free42, Plus42: FUNC command

Swiss Micros DM42, Free42, Plus42:   FUNC command

Turning Programs into User Functions

The FUNC command saves the stack contents and treats the program like it is a function.  This allows the program to work with a temporary stack which is "deleted" when the program reaches either END or RTN (and RTNYES, RTNN), while storing the original contents of the X stack in the L (LAST X) stack.  

FUNC requires a two digit argument.  The first digit is the stack levels used in input, while the second digit is the stack levels used in output.

FUNC 11:  Takes the argument from the X stack and returns the answer in X stack, while leaving the rest of the stack intact.

X:  f(x),  Y: y0,  Z: z0,  T:  t0,  Last X:  x0

FUNC 21:  Takes the arguments from the X and Y stacks and returns the answer in the X stack.

X:  f(x,y),  Y:  z0, Z: t0, T: t0, Last X: x0

FUNC22:  Takes the arguments from the X and Y stacks and returns the answers in the X and Y stacks.

X:  x from f(x,y),  Y:  y from f(x,y), Z:  z0,  T:  t0, Last:  x0

Example 1:  f(x,y) = (y + x)/(y - x)

Stack set up:

T = 24

Z = 15

Y = 45

X = 21

PR1:  Program 

00 { 17-Byte Prgm }

01▸LBL "PR1"

02 -

03 ENTER

04 ENTER

05 LASTX

06 +

07 LASTX

08 +

09 X<>Y

10 ÷

11 RTN

12 END

BEFORE PR1:

T=               70.0000

Z=               15.0000

Y=               45.0000

X=               21.0000

AFTER PR1:

T=               15.0000

Z=               15.0000

Y=               15.0000

X=                2.7500

FN1:  Using FUNC21

00 { 20-Byte Prgm }

01▸LBL "FN1"

02 FUNC 21

03 -

04 ENTER

05 ENTER

06 LASTX

07 +

08 LASTX

09 +

10 X<>Y

11 ÷

12 RTN

13 END

BEFORE FN1:

T=               70.0000

Z=               15.0000

Y=               45.0000

X=               21.0000

AFTER FN1: 

T=               70.0000

Z=               70.0000

Y=               15.0000

X=                2.7500

Example 2:  f(x,y) = x * y * √(x^2 + y^2)

Stack set up:

T = 64

Z = 11

Y = 15

X = 25

Note: The →POL command calculates that  x^2 + y^2 on the x stack

PR2:  Program 

00 { 17-Byte Prgm }

01▸LBL "PR2"

02 STO ST Z

03 X<>Y

04 STO× ST Z

05 →POL

06 RCL× ST Z

07 RTN

08 .END.

BEFORE PR2:

T=               64.0000

Z=               11.0000

Y=               15.0000

X=               25.0000

AFTER FN2:

T=               64.0000

Z=              375.0000

Y=                1.0304

X=           10,933.0348

FN2:  Using FUNC21

00 { 20-Byte Prgm }

01▸LBL "FN2"

02 FUNC 21

03 STO ST Z

04 X<>Y

05 STO× ST Z

06 →POL

07 RCL× ST Z

08 RTN

09 END

BEFORE FN2:

T=               64.0000

Z=               11.0000

Y=               15.0000

X=               25.0000

AFTER FN2:

T=               64.0000

Z=               64.0000

Y=               11.0000

X=           10,933.0348

As we can see, the FUNC command is a powerful command that allows programs to act like user functions, and is a great function for the DM42, Free42, and Plus42.  Unfortunately, FUNC is not available on the original HP 42S.

Until next time, happy programming,

Eddie 

All original content copyright, © 2011-2022.  Edward Shore.   Unauthorized use and/or unauthorized distribution for commercial purposes without express and written permission from the author is strictly prohibited.  This blog entry may be distributed for noncommercial purposes, provided that full credit is given to the author. 

Swiss Micros DM42: Time Dilation

Swiss Micros DM42:  Time Dilation 

Introduction

The program DILATE calculates the time passage from the traveler's perspective:

t' = t * √(1 - (v/c)^2)

where c = Speed of Light = 299,792,458 m/s

We can also state the equation as:

t' = t * cos(arcsin (v/c)) 

because cos(arcsin(x)) = √(1 - x^2) for |x| ≤ 1

Why cos(arcsin(x)) = √(1 - x^2)?

Let Θ = arcsin x.  Then x = sin Θ.  Assume that -1 ≤ x ≤ 1.

From the trigonometric identity:

cos^2 Θ + sin^2 Θ = 1

(cos Θ)^2 +  (sin Θ)^2 = 1

With sin Θ = x,

(cos Θ)^2 + x^2 = 1

(cos Θ)^2 = 1 -  x^2

cos Θ = √(1 - x^2)

cos(arcsin x) = √(1 - x^2)

Swiss Micros DM42 Program:  DILATE

Also:  Free42, Plus42

00 {52-Byte Prgm}

01  LBL "DILATE"

02  "TIME?"

03  PROMPT

04  "VEL (M/SEC)?"

05  PROMPT

06  299792458

07  ÷

08  ASIN

09  COS

10  ×

11 "T'= "

12  ARCL ST X

13  AVIEW

14  RTN

15  END

Example

A spaceship traveling at a velocity of 186,000,000 m/s (about 416,070,150 mph) for 1 year from our perspective.

XEQ DILATE

TIME?  1

VEL (M/SEC)? 186000000

 

Result:  0.78426 year would have passed on the spaceship, which is about 9 months and almost 13 days

Until next time,

Eddie 

All original content copyright, © 2011-2022.  Edward Shore.   Unauthorized use and/or unauthorized distribution for commercial purposes without express and written permission from the author is strictly prohibited.  This blog entry may be distributed for noncommercial purposes, provided that full credit is given to the author. 

Drawing Random Numbers with Playing Cards 

An Easy Way to Get Random Numbers

Need an easy and fun way to draw random is to use a playing deck of cards.  No electricity or potentially expensive electronic devices are needed, and your shuffling skills will improve dramatically.  

Take a regular deck of cards, and you will only need 40 cards.  Get all the Aces, ones through nines, and one set of face cards.   Assign the following cards as the following values:

Aces:  1

Twos:  2

Threes: 3

Fours: 4

Fives: 5

Sixes: 6

Sevens: 7

Eights: 8

Nines: 9

Any set of face card or tens: 0  (I use Jacks, see the pictures above.  You can use Queens, Kings, or Tens)

To get a random number:

1.  Shuffle the deck

2.  Draw up to four cards 

3.  That card becomes the random number.  Insert a decimal point anywhere if appropriate.  

Why only up to four cards?  There are only four digits available per deck.  We are generating a samples with no replacement.  

Sample four digit numbers:

8-clubs, Ace-hearts, 2-clubs, 5-diamonds:  8125

2-diamonds, Ace-clubs, 3-clubs, 5-spades:  2135

4-hearts, 8-spades, 3-diamonds, 9-spades: 4839

4-hearts, 3-diamonds, 9-slides, 8-clubs: 4398

Jacks-clubs, Ace-clubs, 2-diamonds, 2-spades: 0122

How many possible permutations are possible with this deck?

2 digits:  nPr(40, 2) = 1560

3 digits:  nPr(40, 3) = 59,280

4 digits:  nPr(40, 4) = 2,193,360

Can we create a program to generate random numbers using this method?  

Yes.   I am using the TI-84 Plus CE to for this program.   The TI-84 has a randIntNoRep function, which generates a list of random integers without replacement.   The nice part of the is we are working with base 10, so the use of a remainder function will give the results we want: 0 through 9.  

Syntax:

Random Sample of Integers Without Replacement:

randIntNoRep(low, high, size of the sample)

Remainder function, which can be used as the modulus function.

remainder(x, y):  returns the remainder of x ÷ y

The program RANDDECK will generate a list and draw a histogram.

TI-84 Plus CE Program:  RANDDECK

"2022-08-16 EWS"

ClrHome

Disp "MAKE A 4 DIGIT,","NUMBER FROM A DECK 0-9"

Input "NO OF TRIALS: ", N

For(I,1,N)

randIntNoRep(1,40,4)→L6

remainder(L6,10)→L6

sum(L6*{1000,100,10,1})→N

If I=1

Then

{N}→L5

Else

augment(L5,{N})→L5

End

End

Pause L5

ClrHome

Disp "PRESS [ENTER] TO"

Pause "DISPLAY HISTOGRAM"

Plot1(Histogram,L5,1,BLUE)

PlotsOn 1

ZoomStat

Here is an example.

Until next time,

Eddie

All original content copyright, © 2011-2022.  Edward Shore.   Unauthorized use and/or unauthorized distribution for commercial purposes without express and written permission from the author is strictly prohibited.  This blog entry may be distributed for noncommercial purposes, provided that full credit is given to the author.