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Comment by peeters | Hacker News Reader

I agree from the perspective of how most humans could memorize pi. But that's exploiting the fact that humans treat decimal representations as having implicit denominators--3.14 is actually 314/100.

So say you had a computer that didn't treat base 10 as special. Then an apples-to-apples comparison would be that 355/113 (9/7 = 16 binary digits) has the same precision as 3141592/1000000 (22/20 = 40 binary digits). 16 is a lot fewer than 40.

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> But that's exploiting the fact that humans treat decimal representations as having implicit denominators

That's to say, we know which base we are using... which is a requirement for writing down numbers in the first place.

>So say you had a computer that didn't treat base 10 as special

Your computer treats base 2 as special. The apples-to-apples comparison would be comparing 355/113 to 11.0010010000111₂ (or a 16-bit float representation of Pi).

355/113 is a better approximation, but not as drastically as you wrote.

Writing in binary, 355/113 = 101100011₂/1110001₂, and the equivalent one is 11.0010010000111111011₂ - hardly that big of a difference.

I was using base 2 just for conveying the numbers of digits, i.e. showing that the digits in representation is related logarithmically to the magnitude of the base-agnostic integers in the fraction. I could have just as easily used natural log or log10 for evaluating the size of those numbers, because the important part is that it's not exploiting the implicit base in your denominator.

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