Pi base calling

So. You know the decimal system, right? 103.4 stands for “1 hundred, 0 tens, 3 ones, 4 tenths”, or

1\times 10^2 + 0\times 10^1 + 3\times 10^0 + 4\times 10^{-1}.

You know the binary system, too: instead of tens, hundreds, thousands, etc., it counts twos, fours, eights, sixteens, and so on; powers of 2 instead of powers of 10. In binary 1101 means

1\times 2^3 + 1\times 2^2 + 0\times 2^1 + 1\times 2^0,

which translates to 8+4+0+1 or 13 in decimal. (You don’t get a number like 102 in binary because you can only use “digits” smaller than your base number; and for powers of 2 that means using only 1 and 0.)

I came into contact with a pi-based system today, and had my biggest conceptual shock in a long time. Namely, in the pi-based system the following mind-ripping inequality holds: 0.333\ldots > 1.

Note it’s not even an equality; it’s a pure inequality in what feels like exactly the wrong direction.

This is easier to see than to accept; you can simply write and beautify pi-base 0.333\ldots or

3\times \pi^{-1} + 3\times \pi^{-2} +3\times \pi^{-3} +3\times \pi^{-4} + \ldots

into a decimal notation as

\sum\limits_{k=1}^\infty 3 \pi^{-k} = 3 \sum\limits_{k=1}^\infty (1/\pi)^{k} = \frac{3}{\pi-1} \approx 1.4;

and as 1.4 plus something is bigger than one, we have what we want. (Also a headache. I had naively thought that in any real base, integer or not, it was enough to compare the largest different “decimal”, save in tricksy cases like decimal 0.999\ldots = 1 (wiki), and even there you only got equality; not the reverse of the generally expected inequality.)

So mathematics makes fools of us all.

Freaks also: the average human has one tit, one testicle, and around 1.999998 arms, and that doesn’t describe me.

3 Responses to “Pi base calling”

  1. Bob O'H Says:

    What base is 1.999998 arms?

    Also, what does base pie look like?

  2. masksoferis Says:

    Ah, such base questions.

    And 1.999998 is based on the wild guess that 2 people out of a million lack an arm — tragic accident with a pencil, terrible lecture boredom, or the like.

    (Also, as a purely random remark, if 1.999998 was a number in base 9.99998 instead of base 10, it would be exactly equal to two in base 10. That is, 1.999998_{9.99998} = 2_{10}, using the usual notation.)

  3. Nels Says:

    you should check out base phi if you thought base pi blew your mind. EVERY rational number can be represented as a rational number in base phi, even though it’s an irrational base. ex:
    1 φ^0 1
    2 φ^1 + φ^−2 10.01
    3 φ^2 + φ^−2 100.01
    4 φ^2 + φ^0 + φ^−2 101.01
    5 φ^3 + φ^−1 + φ^−4 1000.1001

    and not only this, but each number can be represented in several ways, since φ^n = φ^n-1 + φ^n-2 which also leads to the beautiful equalities φ + 1 = φ^2 and φ – 1 = 1/φ


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