Section 13-4

Different Bases

We tend to think it's perfectly natural to use 10 symbols to write out numbers: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9. But the only reason we do this is because we grow up counting on our fingers, of which we happen to have ten. There's no real reason why ten is any better for math than another number, say 2, 5, 12, or 16.

With one digit, we can count up to 9. Then we use place value to write larger numbers. "10" means one ten and zero ones. The number 5723 is really shorthand for:

5723 = (5 × 1000) + (7 × 100) + (2 × 10) + (3 × 1)

The places stand for thousands, hundreds, tens, and ones. Notice that these are all powers of 10:

5723 = (5 × 10³) + (7 × 10²) + (2 × 10¹) + (3 × 10⁰)

BASE 3

What if we restricted ourselves to only three digits, 0, 1, and 2, and used powers of 3 instead of powers of 10 as the place values? Below we count up to 27 in base 3.

BASE 3	BASE 10
1	1
2	2
10	3
11	4
12	5
20	6
21	7
22	8
100	9
101	10
102	11
110	12
111	13
112	14
120	15
121	16
122	17
200	18
201	19
202	20
210	21
211	22
212	23
220	24
221	25
222	26
1000	27

Notice that instead of the "tens", "hundreds", and "thousands" places, we have the "threes", "nines", and "twenty-sevens" places in the left column.

It may seem a little weird, but you can do math just as well in base 3 as in base 10, or any other base.

Historically, most but not all cultures have used base 10. The Yuki Indians of California used to use base 8, because they counted the spaces between their fingers rather than the fingers themselves. The Babylonians used base 60, and the Mayans used a mix of base 20 and 18. Some old base 20 terminology has crept into the French and English languages. The French say "soixante et onze" for 71, which literally means "three twenties and eleven". And US President Abraham Lincoln's Gettysburg Address began, "Four score and seven", meaning 87.

Finally, in modern times, base 2 (binary) and base 16 (hexadecimal) are used frequently in computer science. (If you have ever played around with making a web page, you might know that HTML uses a 6-digit hexadecimal code to specify colors. The 16 digits are 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E, F. The code for black is "000000"; the code for white is "FFFFFF"; "0033CC" is sort of a nice blue.)

There are also some people, like the Dozenal Society of America, who advocate changing the whole world over to a base 12 system. They claim base 12 is superior to base 10 because it is divisible by more numbers... so it is easier to learn the multiplication tables!