Fast Exponentiation, revisited


Quite a while ago, I presented a fast exponentiation algorithm that uses the binary decomposition of the exponent n to perform O(\log_2 n) products to compute x^n.

While discussing this algorithm in class, a student asked a very interesting question: what’s special about base 2? Couldn’t we use another base? Well, yes, yes we can.

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Best. Number Base. Ever.


One of the first things we learn when we begin programming is that there are different number bases: base 10, the usual one, but also binary, octal, and hexadecimal. Their expressive power is strictly equivalent, but we also notice that hexadecimal numbers are written much more compactly than binary, and so, hexadecimal is “more efficient.”


But “efficiency” begs the question: how do we define it, and, once efficiency defined, which base is the best? Let’s find out!

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