Apple-2-SW/apple2_russian_peasant_multiplication
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Cleanup Representation vs Presentation

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michaelangel007 2020-02-05 09:55:49 -08:00
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@ -157,12 +157,30 @@ Does this algorithm work in other bases such as 2, 8, or 16?
Consider the question:
Q. Does multipling by 2 work in other bases?
A. Yes.
Q. Why?
A. When we write a number in a different base we have the _same representation_ but a _different presentation._
Adding, Multiplying, Dividing all _function_ the same regardless of which base we use.
A. When we write a number in a different base we have the _same_ **representation** but a _different_ **presentation.**
Adding, subtracting, multiplying, dividing all _function_ the same _regardless_ of which base we use.
This is the _exact_ same reason that 0.999999... = 1.0. The exact same _represented_ number is _presented_ differently -- which
confuses the uninformed. It is a "Mathematical illusion."
Proof:
```
1 = 1 Tautology
1/3 = 1/3 Divide by sides by 3
3 * 1/3 = 3 * 1/3 Multiply by sides by 3
3 * 1/3 = 3 * 0.333333... Express integer fraction in decimal
1 = 3 * 0.333333... Simply left side (fractions cancel)
1 = 0.999999... Simply right side
```
QED.
# Efficiency
@ -194,4 +212,6 @@ Notes:
* https://tspiteri.gitlab.io/gmp-mpfr-sys/gmp/Algorithms.html#Multiplication-Algorithms
* https://en.wikipedia.org/wiki/Multiplication_algorithm
* Multiplication is associative
* Multiplication is commutative
* https://en.wikipedia.org/wiki/Order_of_operations