Spotlight: Aristo Multilog Nr. 970 Slide Rule

Spotlight: Aristo Multilog Nr. 970 Slide Rule

Release Year:  1954

Santa Barbara and Carpentaria, California, are two of my favorite places to visit.   While on vacation in December 2023 in Santa Barbara, I bought a Aristo Nr. Slide Rule at Antique Alley.  

Not only did the slide rule have a nice, hard case, the slide rule is large. The slide rule also has a functional description of each scale.  I also like the slide rule uses two colors, black and brown, to distinguish the scales from each other.  

Side 1:

LL00:   e^(-0.001X)

K:  X^3  (K = D^3)

A:  X^2  (A = D^2)

CF:  πX  

CIF:  1/(πX)

L:  lg X  (log X)

CI:  1/X

C:  X

D:  X

DI:  1/X

LL0:  e^(0.001X)

Side 2:

LL01:  e^(-0.01X)

LL02:  e^(-0.1X)

LL03:  e^(-X)

DF:  πX

B:  X^2

T:  ∡tg   (tan X, X is in degrees, tan D° = T)

ST:  ∡arc  (tan X, sin X)

S:  ∡sin  (arcsin X, sin D° = T) 

C:  X

D:  X 

LL3:  e^X  (LL3 = e^D)

LL2:  e^(0.1X)

LL1:  e^(0.01X)

As a bonus, the slide rule came with a reference card.  One one side is the Dietzgen Slide Rule Conversion Tables  (U.S. and SI units, 1950), and other side is a table of common areas, surface areas, volumes, and trigonometric properties of the right triangle.   I don't know if this was standard addition when the Aristo Multilog Nr. 970 was sold.  

This may be my favorite slide rule:  not only the scales are large (and as a result easier on the eyes to read), but the fact that there are function descriptions on the scales is a big plus.  

Eddie

All original content copyright, © 2011-2024.  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. 

Casio fx-5800P: Multiplying Big Numbers - Laws of Logarithms

Casio fx-5800P: Multiplying Big Numbers - Laws of Logarithms

Introduction

This post was inspired by a recent video from Scott Collins, where he discussed what happens when multiplying numbers with large valued exponents:

https://www.youtube.com/watch?v=ziMIphyd9wg

Scientific calculators handle large numbers in the form M * 10^E where M is the mantissa and E is the exponent.  For most calculators, E ranges from -99 to +99.   For most Hewlett Packard calculators, the range expands to -499 to +499.  On CAS enabled calculators, the range usually tops out at -999 to +999.

At some point, if the exponent goes too small, say 10^-12 or less, the number is approximated with 0. 

For example, entering 4 *10^-56 * 2 * 10^-76 returns 0 on a lot of calculators, which the real answer is 8 * 10^-132.  

If you work with multiplying numbers of this magnitude, it may be useful to work with the mantissa and exponent parts separately, which can be the done thanks to the laws of exponents:

( A * 10^B ) * ( C * 10^D )

=  A * C * 10^B * 10^D

= ( A * C) * 10^(B + D)

Mantissa:  A * C

Exponent:  B + D

Similarly: 

( A * 10^B )^S * ( C * 10^D )^T

= (A^S * (10^B)^S) * (C^T * (10^D)^T)

= A^S * 10^(B*S) * C^T * 10^(D*T)

= (A^S * C^T) * 10^(B*S + D*T)

The program BIGMULT can automate this process, as well as adjust the answer to scientific notation form: #.####### * 10^# 

Casio fx-5800P Program:  BIGMULT

"(A×10^B)^(S)×(C×10^D)^(T)"

"A"?→A

"B"?→B

"S"?→S

"C"?→C

"D"?→D

"T"?→T

A^(S)×C^(T)→M

B×S+D×T→E

Intg(log(M))→S

M÷10^(S)→M

E+S→E

M ◢

E

The results:

Mantissa

Exponent

Note:  

Intg is the largest/greatest integer function, which allows us to fine tune our answer into the standard format.

Intg(log(M))→S

M÷10^(S)→M

E+S→E

Using the Int/IP (integer part) function would require additional adjustments.  

Examples

Example 1:

4 *10^-56 * 2 * 10^-76 

Input:

A = 4, B = -56, S = 1

C = 2, D = -76, T = 1

Result:

8

-132

8 * 10^-132

Example 2:

(2 * 10^5)^2 * (3 * 10^6)^3

Input:

A = 2, B = 5, S = 2

C = 3, D = 6, T = 3

Result:

1.08

30

1.08^30

Example 3:

(0.076 * 10^3)^2 * (0.9 * 10^2)^3

A = 0.076, B = 3, S = 2

C = 0.9, D = 2, T = 3

Result:

4.210704

9

4.210704 * 10^9

Until the 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. 

Even and Odd Integers in Arithmetic

Even and Odd Integers in Arithmetic

Introduction

Let n and m be integers, where:  

n = ..., -3, -2, -1, 0, 1, 2, 3, ...

and 

m = ..., -3, -2, -1, 0, 1, 2, 3, ...

An even integer is any integer that can evenly divided by 2, without remainder.  Hence, if p is an even integer, then p = 2 ∙ n

An odd integer is any integer that can not be evenly divided by 2  (remainder 1).  In this case:  p = 2 ∙ n + 1

Addition

Adding two integers will result in an integer.  

even + even = even

2 ∙ n + 2 ∙ m  

= 2 ∙ (n + m)

odd + odd = even

(2 ∙ n + 1) + (2 ∙ m + 1)

= 2 ∙ n + 2 ∙ m + 2

= 2 ∙ (n + m + 1)

even + odd = odd

(2 ∙ n) + (2 ∙ m + 1)

= 2 ∙ n + 2 ∙ m + 1

= 2 ∙ (n + m) + 1

Multiplication 

Multiplying two integers will result in an integer.  

even × even = even

(2 ∙ n) ∙ (2 ∙ m)

= 2 ∙ (n ∙ m)

odd × odd = odd

(2 ∙ n + 1) ∙ (2 ∙ m + 1)

= 4 ∙ m ∙ n + 2 ∙ n + 2 ∙ m + 1

= 2 ∙ ( 2 ∙ m ∙ n + n + m) + 1

even ×  odd = even

(2 ∙ n + 1) ∙ (2 ∙ m)

= 4 ∙ m ∙ n + 2 ∙ m

= 2 ∙ (2 ∙ m ∙ n + n)

even^2 = even

(2 ∙n)^2 

= 4 ∙ n^2

= 2 ∙ (2 ∙ n ∙ n)

odd^2 = odd

(2 ∙ n + 1)^2

= 4 ∙ n^2 + 4 ∙ n + 1

= 2 ∙ (2 ∙ n ∙ n + 2 ∙ n) + 1

 

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. 

Calculos: Mathematics Calculation Game (Kickstarter)

Calculos:  Mathematics Calculation Game  (Kickstarter)

The Card Game

Xavier Wilain created the card game Calculos, which is based on quick, mental mathematics dealing with the four arithmetic operations:  addition, subtraction, multiplication, and division.  There is a mixed version that has all four arithmetic operations.  

Players start with a card and try to find a calculation that matches any of the calculations on the center card.  

For example, if the center card has:

2 × 5

3 × 6

5 × 5

4 × 4

If you have any of the calculations on your card, say 4 × 4, you shout "4 times 4 equals 16" and claim the card.  The object is for find a calculation faster than their opponents.   

Here is a video of the card game and how to play it:

https://www.youtube.com/watch?v=-PEfEVTV4wI

The card game comes in four levels: each level has a different set of calculations on the card, from four to eight.   On the XW_SeriousGames website, each level can be purchased separately or altogether for each operation.

Link:  https://www.xwseriousgames.com/

The Mobile App (Kickstarter)

Wilain is running a Kickstarter campaign to adapt the card game into a mobile app that contains all 20 levels (4 four levels for each arithmetic operation plus one for mixed) with plans to add levels in the future.  Wilain has goal to raise 33'000 CHF (Swiss Francs, which is $35,897.63 in US Dollars on 6/28/2021 - conversion provided by Morningstar).  If successful, the mobile app is set to begin development in October 2021.  The target fund date is July 27, 2021.  

Kickstarter page:

https://www.kickstarter.com/projects/xw-serious-games/calculos/description

Please check this game out, it promises to be a great way to strength our mental calculation skills.  

Eddie

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. 

Breaking Down the Factorial

Breaking Down the Factorial

Factorial: It's Not Just For Integers

Let n be a positive number, where n > 0.   n! can be rewritten as:

n! 

= n * (n - 1)!

= n * (n - 1) * (n - 2)!

= n * (n - 1) * (n - 2) * (n - 3)!

...

= n * (n - 1) * (n - 2) * (n - 3) * ... * k

where 0 ≤ k ≤ 1.   Note that 0! = 1.   Keep the loop multiplying n, n - 1, n - 2, n - 3, etc. until you a multiplying a number between 0! and 1! to the total.

For certain k:

0.25! ≈ 0.9064024771

0.50! = ≈ 0.8862269255

0.75! ≈ 0.9190625268

1! = 1

Examples

3! = 3 * 2 * 1! = 3 * 2 * 1 = 6

3.25! = 3.25 * 2.25 * 1.25 * 0.25! = 9.140625 * 0.25! ≈ 8.285085142

3.5! = 3.5 * 2.5 * 1.5 * 0.5! = 13.125 * √π ÷ 2 ≈ 11.6317284

3.75! = 3.75 * 2.75 * 1.75 * 0.75! = 18.046875 * 0.75! ≈ 16.58620654

4! = 4 * 3 * 2 * 1! = 4 * 3 * 2 * 1 = 24

4.25! = 4.25 * 3.25 * 2.25 * 1.25 * 0.25! = 38.847652625 * 0.25! ≈ 35.21161185

4.5! = 4.5 * 3.5 * 2.5 * 1.5 * 0.5! = 59.0625 * √π ÷ 2 ≈ 52.3427778

4.75! = 4.75 * 3.75 * 2.75 * 1.75 * 0.75! = 85.72265625 * 0.75! ≈ 78.78448106

Factorial Values of 0 to 1

Below is a chart are the values for 0 to 1, along with several approximation polynomials.  The value and polynomials have been determined using LibreOffice's Calc application.  

Happy Halloween, 

Eddie

All original content copyright, © 2011-2020.  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. 

Algebra: Multiplying a * b Trick (Using the Difference between a and b)

Algebra:  Multiplying a * b Trick (Using the Difference between a and b)

Can we find a formula to find products where two values are an equal-distant apart

The Values of a and b Differ by 2

Let a and b be real numbers which differ by 2, that is b – a = 2.  Here I am assuming that b > a. 

Let n be the midpoint between a and b.  That is:

n = b – 1 and

n = a + 1

Therefore:

b = n + 1

a = n - 1

Then:

a * b

= (n - 1) * (n + 1)

= n^2 - n + n – 1

= n^2 - 1

Example:  51 * 49

Notice that:

51 – 49 = 2, and

51 - 1 = 50

49 + 1 = 50

Hence:

51 * 49 = 50^2 – 1 = 2499

Can we expand this included products of a * b, where the difference is b – a = 2 * w

The Values of a and b Differ by 2*w

Let’s look at a more general case. 

Let b – a = 2*w

Then:

b = n + w and a = n – w

Then:

a * b

= (n – w) * (n + w)

= n^2 – n*w + n*w – w^2

= n^2 – w^2

Example:  37 * 43.

43 – 37 = 6

w = 6/2 = 3

Then:

n = 43 – 3 = 37 + 3 = 40

Then:

37 * 43 = 40^2 – 3^2 = 1600 – 9 = 1591

Try another example:  57 * 49

57 – 49 = 8

8 / 2 = 4

57 – 4 = 53, 49 + 4 = 53

Then:

57 * 49 = 53^2 – 4^2 = 2809 – 16 = 2793

In summary for a * b with b > a.

Let w = (b – a)/2 and n = a + w or n = b – w

Then a * b = n^2 – w^2

Eddie

All original content copyright, © 2011-2018.  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.  Please contact the author if you have questions.

HP 12C: Complex Arithmetic (Addition, Multiplication, and Division)

HP 12C:   Complex Arithmetic (Addition, Multiplication, and Division)

Let A and B represent the two complex numbers

A = R1 + R4*i

B = R2 + R5*i

Where i = √-1

Store real components in R1 and R2, and imaginary parts R4 and R5.  Choose the arithmetic calculations:

Enter 0 for addition:  A + B

Enter 1 for multiplication:  A * B

Enter 2 or otherwise for division:  A ÷ B

Result:  C = R3 + R6*i

Program:

STEP	CODE	KEY
01	44, 0	STO 0
02	43, 35	X=0
03	43, 33, 09	GTO 09
04	1	1
05	30	-
06	43, 35	X=0
07	43, 33, 18	GTO 18
08	43, 33, 35	GTO 35
09	45, 1	RCL 1
10	45, 2	RCL 2
11	40	+
12	44, 3	STO 3
13	45, 4	RCL 4
14	45, 5	RCL 5
15	40	+
16	44, 6	RCL 6
17	43, 33, 63	GTO 63
18	45, 1	RCL 1
19	45, 2	RCL 2
20	20	*
21	45, 4	RCL 4
22	45, 5	RCL 5
23	20	*
24	30	-
25	44, 3	STO 3
26	45, 1	RCL 1
27	45, 5	RCL 5
28	20	*
29	45, 2	RCL 2
30	45, 4	RCL 4
31	20	*
32	40	+
33	44, 6	STO 6
34	43, 33, 63	GTO 63
35	45, 2	RCL 2
36	36	ENTER
37	20	*
38	45, 5	RCL 5
39	36	ENTER
40	20	*
41	40	+
42	44, 0	STO 0
43	45, 1	RCL 1
44	45, 2	RCL 2
45	20	*
46	45, 4	RCL 4
47	45, 5	RCL 5
48	20	*
49	40	+
50	45, 0	RCL 0
51	10	÷
52	44, 3	STO 3
53	45, 4	RCL 4
54	45, 2	RCL 2
55	20	*
56	45, 1	RCL 1
57	45, 5	RCL 5
58	20	*
59	30	-
60	45, 0	RCL 0
61	10	÷
62	44, 6	STO 6
63	45, 3	RCL 3
64	31	R/S
65	45, 6	RCL 6
66	43, 33, 00	GTO 00

Example:

A = 4.25 + 3.16*i, B = -2.27 + 1.04*i

Option 0 (+):  1.98 + 4.20*i

Option 1 (*):  -12.93 – 2.75*i

Option 2 (÷): -1.02 – 1.86*i

Eddie

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