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

.

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

,

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: .

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 or

into a decimal notation as

;

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 (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.

August 8, 2009 at 4:12

What base is 1.999998 arms?

Also, what does base pie look like?

August 8, 2009 at 18:13

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, , using the usual notation.)

August 17, 2009 at 3:48

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

Enjoy!