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:
$5723=\left(5\times 1000\right)+\left(7\times 100\right)+\left(2\times 10\right)+\left(3\times 1\right)$
The places stand for thousands, hundreds, tens, and ones. Notice that these are all powers of $10$ :
$5723=\left(5\times {10}^{3}\right)+\left(7\times {10}^{2}\right)+\left(2\times {10}^{1}\right)+\left(3\times {10}^{0}\right)$
Example: 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. To illustrate, we'll do an addition problem (in base $3$ on the left, base $10$ on the right). Notice that we have to carry when we add $1+2$ !
$\begin{array}{ccccc}\begin{array}{l}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}1\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}1\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}0\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}1\\ \underset{\_}{+\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}1\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}2\text{\hspace{0.17em}}}\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}1\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}2\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}0\end{array}& & & & \begin{array}{l}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}{1}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}{0}\\ \underset{{\_}}{{+}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}{5}\text{\hspace{0.17em}}}\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}{1}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}{5}\end{array}\end{array}$
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 even 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 $16$ -digit hexadecimal code to specify colors. The $16$ digits are $0,1,2,3,4,5,6,7,8,9,\text{A},\text{B},\text{C},\text{D},\text{E},\text{F}$ . The code for black is " $000000$ "; the code for white is " $\text{FFFFFF}$ "; " $9\text{B20DF}$ " is this sort of nice mellow purple color .)
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!