Beyond Base 10

This week we looked at numbers written in different bases.

We usually write numbers in base 10. In base 10, the string of digits

321

represents the sum of 3 hundreds, 2 tens, and 1 one. In base 10 the places are

..., ten thousands, thousands, hundreds, tens, ones.

Another way to say this is that the places are powers of ten:

..., 10^4, 10^3, 10^2, 10^1, 10^0.

We could also write numbers in base 2, in which case we're only allowed to use the digits 0 and 1. The places in base 2 are powers of two instead of powers of ten:

..., 2^4, 2^3, 2^2, 2^1, 2^0

or in other words,

..., sixteens, eights, fours, twos, ones

In base 2, the string of digits

1101

represents the sum of 1 eight, 1 four, 0 twos, and 1 one for a total of thirteen.

We could also write numbers in base 3, or base 4, or any base we like. Computers use base 2 a lot. They also use base 16, better known as hexadecimal. In base 16 we use the digits from 0 to 9, but we also use letters:

10 A

11 B

12 C

13 D

14 E

15 F

The place values in hexadecimal are powers of 16:

..., 4096, 256, 16, 1

Thus the string

10AF

represents the sum of 1 four-thousand ninety-six, 0 two-hundred fifty-sixes, 10 sixteens, and 15 ones for a total of 4271.

The class solved the following puzzle:

If only you, I, and DEAD people know hexadecimal, how many people know hexadecimal?