** TI-Eighties Graphing Calculators: A Timeline**

A blog is that is all about mathematics and calculators, two of my passions in life.

## Sunday, September 26, 2021

### TI-Eighties Graphing Calculators: A Timeline

## Saturday, September 25, 2021

### Sharp EL-5500III & PC-1403: Spin a Wheel and Random Samples

** Sharp EL-5500III & PC-1403: Spin a Wheel and Random Samples**

**Spin a Wheel**

**Sharp EL-5500III/PC-1403 Program: Spin a Wheel**

**Random Sample**

**Sharp EL-5500III/PC-1403 Program: Random Sample**

**Swiss Micros DM41X Month: October 2021**

## Monday, September 20, 2021

### Calculator Python: Lambda Functions

**Calculator Python: Lambda Functions**

**Introduction to Lambda Functions**

Lambda functions are a quick, one expression, one line, python function. Lambda functions do not require to be named though they can be named for future use if desired.

The syntax for lambda functions are:

One argument:

lambda argument : expression

Two or more arguments:

lambda arg1, arg2, arg3, ... : expression

The expression must return one result.

The quick, versatile of lambda functions are make lambda functions one of the most popular programming tools.

**Filter, Map, and** **Reduce**

Filter: uses a lambda function to filter out elements of a list and array using criteria. For a list, the list command must be used to turn the result into an actual list.

Syntax using Lambda and List:

list(filter(lambda arguments : expression))

Map: uses a lambda function to apply a function to each element of a list. Like filter, the list command must be used to turn the result into an actual list.

Syntax using Lambda and Map:

list(map(lambda arguments : expression))

Reduce: uses a lambda function to use two or more arguments in a recursive function.

Syntax using Lambda:

reduce((lambda arguments : expressions), list)

Note, as of August 31, 2021, that the reduce command is NOT available on any calculator, only on full version of Python 3. This may change with future updates.

HP Prime: lambda, filter, map

Casio fx-CG 50 and fx-9750GIII: lambda, map

Numworks: lambda, filter, map

TI-84 Plus CE Python: lambda, filter, map

TI-Nspire CX II Python: lambda, filter, map

**Nuwmorks Sample Python File: introlambda.py**

from math import *

from random import *

# lambda test

n=randint(10,9999)

tens=lambda x:int(x/10)%10

hunds=lambda x:int(x/100)%10

thous=lambda x:int(x/1000)%10

print(n)

print(tens(n))

print(hunds(n))

print(thous(n))

print("List:")

l1=[1,2,3,4,5,6]

print(l1)

print("Filter demonstration")

print("Greater than 3")

l2=list(filter(lambda x:x>3,l1))

print(l2)

print("Odd numbers")

l3=list(filter(lambda x:x%2!=0,l1))

print(l3)

print("Map demonstration")

print("Triple the numbers")

l4=list(map(lambda x:3*x,l1))

print(l4)

print("exp(x)-1")

l5=list(map(lambda x:exp(x)-1,l1))

print(l5)

**Sources**

Maina, Susan "Lambda Functions with Practical Examples in Python" Towards Data Science (membership blog with limited free access per month) https://towardsdatascience.com/lambda-functions-with-practical-examples-in-python-45934f3653a8 Retrieved August 29, 2021

Simplilearn "Learn Lambda in Python with Syntax and Examples" April 28, 2021. https://www.simplilearn.com/tutorials/python-tutorial/lambda-in-python Retrieved August 29, 2021

All original content copyright, © 2011-2021. 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.

## Sunday, September 19, 2021

### Differences Between HP 32S and HP 32SII

** Differences Between HP 32S and HP 32SII**

The HP 32S and HP 32SII are two classic RPN calculators which feature a rich set of scientific calculations, including logarithms, trigonometry, hyperbolic functions, integer part, fraction part, absolute value, statistics, and linear regression (y = mx + b). Both are keystroke RPN programming programs with a capacity of 390 bytes.

**What are the differences? **

**HP 32S**

* Production: 1988 - 1991

* Keyboard Colors: dark brown, almost black keys; one orange shift key

* One key deals with scrolling: down, with up shifting

**HP 32SII**

* Production: 1991 - 2002

* Keyboard Colors: dark brown keys, orange shift key, blue shift key (1st edition); black keys, green shift key, pink shift key

* One key deals with scrolling: down, with up shifting

* Has four sets of US/SI conversions: kg/lb, °C/°F, cm/in, l/gal

* Pressing the decimal key ( [ . ] ) twice will create fractions and mixed fractions. The FDISP toggles between decimal approximation and fractions.

Menu | HP 32S | HP 32SII |
---|---|---|

PARTS | [ shift ] [ x<>y ]: IP, FP, RN, ABS | [ |> ] [ √ ]: IP, FP, ABS (RN is on the keyboard) |

PROB | [ shift ] [ 3 ]: COMB, PERM, x!, R# | [ |> ] e^x |

STAT/LR | One group of menus | Split into four menus |

SHOW | [ shift ] [ . ] | [ |> ] [ ENTER ] |

SOLVE, ∫ | [ shift ] [ 1 ] | SOLVE: [ |> ] [ 7 ]; ∫: [ |> ] [ 8 ] |

LOOP: ISG/DSE | [ shift ] [ 5 ] | ISG: [ <| ] [ 9 ]; DSE: [ |> ] [ 9 ] |

TESTS | [ shift ] [ × ]: offers < = ≠ > | x?y: [ <| ] [ ÷ ], x?0: [ |> ] [ ÷ ]: offers < ≤ = ≠ > ≥ |

## Saturday, September 18, 2021

### Sharp EL-5500III & PC-1403: Complex Number Arithmetic and Vectors

** Sharp EL-5500III & PC-1403: Complex Number Arithmetic and Vectors**

**Complex Number Arithmetic**

**Sharp EL-5500III/PC-1403 Program: Complex Number Arithmetic**

**Vectors**

**Sharp EL-5500III/PC-1403 Program: Vectors**

## Monday, September 13, 2021

### Retro Review: Lloyd's Accumatic 321

**Retro Review: Lloyd's Accumatic 321**

**Quick Facts**

Model: 321

Company: Lloyd's (Japan)

Years: 1975

Memory Register: 1 independent memory

Battery: Either 4 AAA batteries or the use of a 6V DC 300mW, Series 255A AC adapter

Screen: LCD, blue-green digits, 8 digits

**Features**

The Accumatic 321 is a four function calculator with additional features:

* Parenthesis

* Square (x^2)

* Reciprocal (1/X)

The square (x^2), square root (√), and reciprocal (1/X) act immediately on the number on the display.

The 321 operates in chain mode: which means that the operations are done in the way the keys are pressed. Thankfully the parenthesis keys are there to assist us in following the order of operations, which we will have to deal with manually. Case in point with two examples:

Example 1:

6 [ + ] 3 [ × ] 7 [ = ] returns 63

Parenthesis are needed to invoke the proper order of operations:

[ ( ] 6 [ + ] 3 [ ) ] [ × ] 7 [ = ] returns 63

6 [ + ] [ ( ] 3 [ × ] 7 [ ) ] [ = ] returns 27

Example 2:

5 [ × ] 8 [ - ] 4 [ × ] 3 [ = ] returns 108

Parenthesis are needed to invoke the proper order of operations:

[ ( ] 5 [ × ] 8 [ ) ] [ - ] [ ( ] 4 [ × ] 3 [ ) ] [ = ] returns 32

**Additional Calculations**

There may be more than one keystroke sequence to tackle each problem.

1/(1/5 - 1/8) = 13.333333...

5 [ 1/X ] [ - ] 8 [ 1/X ] [ = ] [ 1/X ]

5.99 * 11 - 2.95 * 2 plus 10% sales tax. Total: 46.519

[ MC ] 5.99 [ × ] 11 [ = ] [ M+ ]

2.95 [ × ] 2 [ = ] [ M- ]

[ MR ] [ + ] 10 [ % ] [ = ]

√(4^2 + 7^2 + 10^2) ≈ 12.845232

4 [ X^2 ] [ + ] 7 [ X^2 ] [ + ] 10 [ X^2 ] [ = ] [ √ ]

**Verdict**

As suggested by the seller, the keyboard is fragile; extra care is needed. The keys are legible and have good color contrast. The keys need a solid press. However, holding down the key too long will cause a double entry. The additional reciprocal, square, and parenthesis are welcome to basic calculators. I wish modern basic calculators would add these keys plus the pi (Ï€) key.

I purchased the 321 for $13.00 which has the AC adapter. I like the fact that batteries not required (in fact, the battery case must be empty) to operate the calculator with the AC adapter. Worth the buy.

**Source**

Dudek, Emil J. "Calculators: Handheld: Lloyd's Accumatic 321 (aka E321)" 2021. Retrieved September 4, 2021. https://vintage-technology.club/pages/calculators/l/lloyds321.htm

All original content copyright, © 2011-2021. 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.

## Sunday, September 12, 2021

### Square Root Trick - by @Mathsbook7474

**TI-84 Plus CE and TI-Nspire CX II: Square Root Trick - by @Mathsbook7474**

**Introduction**

From the Instagram account @Mathsbook7474, it is stated that the square root of a number can be approximated by:

√x ≈ (x + y) ÷ (2 · √y)

where y is the number nearest to the root, preferably a perfect square. (see source for the link)

For example:

Approximate √52.

Let x = 52. Note that 49 is a perfect square near to 52. Since √49 = 7, let y = 49.

√52 ≈ (52 + 49) ÷ (2 · √49) = 7.214285714

Accuracy: 0.0031831631 (√52 = 7.211102551)

**TI-84 Plus CE Program: SQAPPROX**

The program SQAPPROX will calculate the approximation above, the actual root, and compare the results for the accuracy.

Listing:

**TI-Nspire CX II tns File: square root approximate.py**

**Python program: spapprox.py**

## Saturday, September 11, 2021

### Sharp EL-5500 III & PC-1403: Fan Laws and Voltage Drop Percentage

** Sharp EL-5500 III & PC-1403: Fan Laws and Voltage Drop Percentage**

**Fan Laws**

**Sharp EL-5500III/PC-1403 Program: Fan Laws**

**Voltage Drop Percentage**

**Sharp EL-5500III/PC-1403 Program: Voltage Drop Percentage**

## Monday, September 6, 2021

### Swiss Micros DM42: Subfactorial and Numworks Update (16.3)

**Swiss Micros DM42: Subfactorial**

**Happy Labor Day!**

This is a request by Marko Draisma and gratitude to Mr. Draisma.

**Calculating the Subfactorial**

A common, and perhaps the most straight forward, formula to calculate the subfactorial is:

!n = n! × Î£((-1)^k ÷ k!, k=0 to n)

Yes, the subfactorial is written with the exclamation point first. The subfactorial finds all the possible arrangements of a set of objects where none of the objects end up in their original position.

For example, when arranging the set {1, 2, 3, 4} the subfactorial counts sets such as {2, 1, 4, 3} and {3, 4, 1, 2} but not {1, 4, 3, 2}. For the positive integers: !n < n!.

I am going to present two programs. The first will use the formula stated above.

The second uses this formula, which will not require recursion or loops:

!n = floor[ (e + 1/e) × n! ] - floor[ e × n! ]

Note: Since the N! function on the DM42 accepts only positive integers, we can use the IP (integer part) to simulate the floor function.

integer(x) = { floor(x) if x ≥ 0, ceiling(x) if x < 0

The following programs can be used on Free42, HP 42S, or Swiss Micros DM42.

**Swiss Micros DM42 Program: Subfactorial Version 1**

This is a traditional route. Registers used:

R01: k, counter

R02: sum register

R03: n!, later !n

Program labels can start with symbols on the 42S.

01 LBL "!N"

02 STO 01

03 N!

04 STO 03

05 0

06 STO 02

07 RCL 01

08 1E3

09 ÷

10 STO 01

11 LBL 00

12 RCL 01

13 IP

14 ENTER

15 ENTER

16 -1

17 X<>Y

18 Y↑X

19 X<>Y

20 N!

21 ÷

22 STO+ 02

23 ISG 01

24 GTO 00

25 RCL 02

26 RCL× 03

27 STO 03

28 RTN

**Swiss Micros DM42 Program: Subfactorial Version 2**

I only put 2 in the label to distinguish the two programs.

01 LBL "!N 2"

02 N!

03 ENTER

04 ENTER

05 1

06 E↑X

07 ENTER

08 1/X

09 +

10 ×

11 IP

12 X<>Y

13 1

14 E↑X

15 ×

16 IP

17 -

18 RTN

**Examples**

!2 = 1

!3 = 2

!4 = 9

!5 = 44

!9 = 133,496

!14 ≈ 3.2071E10

**Sources**

"Calculus How To: Subfactorial" College Help Central, LLC .https://www.calculushowto.com/subfactorial/ Retrieved September 5, 2021.

Weisstein, Eric W. "Subfactorial." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/Subfactorial.html Retrieved September 5, 2021

**Numworks 16.3 Update**

Numworks recently updated its firmware to Version 16.3. Find details of the changes and additions here:

https://my.numworks.com/firmwares

All original content copyright, © 2011-2021. 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.

## Sunday, September 5, 2021

### TI-Nspire CX II and TI-84 Plus CE: COUNTIF, SUMIF, AVERAGEIF

**TI-Nspire CX II and TI-84 Plus CE: COUNTIF, SUMIF, AVERAGEIF**

**Introduction**

Three popular spreadsheet functions are operating on elements of a list contingent of a criteria.

Let L be a list of numerical data.

COUNTIF(L, criteria): returns a count of all the elements that fit a criteria

SUMIF(L, criteria): returns the sum of all the elements that fit a criteria

AVERAGEIF(L, criteria): returns the arithmetic average of all the elements that fit a criteria

Example:

L = {0, 1, 2, 3, 4, 5, 6}

COUNTIF(L, "≥4") = 3.

Counts all the elements of L that are greater than or equal to 4.

SUMIF(L, "≥4") = 15

Sums all the element's of L that are greater than or equal to 4.

AVERAGEIF(L, "≥4") = 5

Returns the arithmetic average of L that are greater than or equal to 4.

Note that:

AVERAGEIF(L, criteria) = SUMIF(L, criteria) ÷ COUNTIF(L, criteria)

TI-Nspire CX II: The functions countif and sumif

The TI-Nspire has two built in functions countif and sumif. The averageif can easily be defined in the equation from the last section.

You can download a tns demonstration document here:

https://drive.google.com/file/d/1XfRyHSQz92TXGmzohhej0--wLYS0yxkF/view?usp=sharing

**TI-84 Plus CE Programs: LISTIF**

The program LISTIF calculates all three functions COUNTIF, SUMIF, and AVERAGEIF. There are two custom lists that are created as a result of this program:

List A: The input list.

List B: The list that meets the criteria.

To get the small "L" character, press [ 2nd ], [ stat ] (LIST), B* for the small L. The small L must be the first character of your list name. Once created, custom lists are shown under the LIST - NAMES menu.

Although I did not test this on previous calculators, this program should work on the TI-82, TI-83 family, and all of the TI-84 Plus family.

Program listing:

## Saturday, September 4, 2021

### Retro Review: Novus 4510 - Mathematician

**Retro Review: Novus 4510 - Mathematician**

Happy Labor Day! Hopefully you are safe, healthy, and sane.

**Quick Facts:**

Model: 4510, also known as Mathematician

Company: National Semiconductor

Years: 1975 - 1977

Memory Register: 1 independent memory, cleared when it is turned off

Battery: 1 9-volt battery, could be powered by a certain AC Volt plugs

Screen: LCD, 8 digits

**A RPN Calculator from the 1970s**

The Mathematician is a Reverse Polish Notation (RPN) calculator. On an RPN calculator, instead of an equals key, you have an ENTER key to separate numbers and execute an operation to complete the calculation.

Examples:

400 ÷ 25 = 16

Keystrokes:

400 ENT 25 ÷

(21 × 5) + (11 × 13) = 248

Keystrokes:

21 ENT 5 × 11 ENT 13 × +

√(3^2 + 2^3) ≈ 3.3166238

Keystrokes:

3 [ F ] (x^2) ENT 2 ENT y^x + √

Forensic Results:

3 × 1/3 returns .99999999

asin(acos(atan(tan(cos(sin(60°)))))) returns 59.25697°

**Keyboard**

The keyboard of Novus 4510 has gray and ivory keys. The font is a light gray against a black background, with shifted functions are in yellow, which gives the fonts great contrast. The keys are rubbery and soft, and thankfully, the 4510 I was purchased had working keys.

**The Basic Set of Functions**

The 4510 has a basic set of scientific functions: principal square root, square root, powers, reciprocal, logarithms and exponents, and trigonometric functions. The angle mode will always be in degrees. The deg and rad commands are not mode settings, they are conversions: deg changes angles from radians to degrees, and rad changes angles from degrees to radians.

There is no scientific notation on the 4510. You are limited to 8 digits. Anything over beyond ±99,999,999 will cause an error.

Any errors will be displayed by .0.0.0.0.0.0.0.0.

There is only one memory register. The storage arithmetic functions available are M+, M-, and M+x^2 (square the value on the x register and adds it to memory).

**The Stack of Three Levels**

The 4510 has three stack levels: x, y, z. How the stack reacts depends on which operation is executed. For example:

The ENT (Enter) Key:

x, y, z -> x, x, y

The Arithmetic Operators (+, -, ×, ÷):

x, y, z -> result, y, 0

The contents of the z stack are zeroed, which unusual for RPN calculators.

The functions x^2, √, 1/x, rad, deg:

x, y, z -> result, y, z

The Trigonometric, Exponential, and Exponential Operators (sin, cos, tan, and their inverses, log, ln, e^x):

x, y, z -> result, y, 0

The contents of the z stack are zeroed, which unusual for RPN calculators.

That makes for a very particular set up due to the usual stack operations, for example:

sin 30° sin 40° sin 60°

Key strokes:

30 sin 40 sin 60 sin × × returns the very incorrect answer 0

However:

30 sin 40 sin × 60 × returns the correct approximation .27833519

There is a swap key but there is no roll down key.

**Verdict**

I like operating the 4510. However, my biggest gripes are the way the stack is used depending on the operation, and the lack of operations and statistics. It's good for a basic RPN calculator.

**Sources**

"National Semiconductor Novus Mathematics Handled Electronic Calculator" National Museum of American History Behring Center. Washington, D.C. Accessed August 15, 2021. https://americanhistory.si.edu/collections/search/object/nmah_1305810

"Novus 4510 (Mathematician)" calculator.org: the calculator home page. Flow Simulation Ltd. 2021 Accessed August 14, 2021. https://www.calculator.org/calculators/Novus_4510.html

Eddie

### HP 15C: Error Function and Lower Tail Normal Cumulative Function

HP 15C: Error Function and Lower Tail Normal Cumulative Function Formulas Used Error Function erf(x) = 2 ÷ √Ï€ * ∫( e^(-t^2) dt, t = 0 t...