diff --git a/README.MD b/README.MD index 82a2727..bb25c7f 100644 --- a/README.MD +++ b/README.MD @@ -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