Counting in Twos and Sixteens

The number four hundred and seven, written in the usual way, means four

hundreds, no tens and seven ones. The columns are worth one, ten, a hundred, a

thousand, each ten times the one to its right. That multiplier is the base, and

nothing else about the system depends on it being ten.

Put two in instead, and the columns are worth one, two, four, eight, sixteen.

Each column holds either none or one of its value, because those are the only

digits available. Put sixteen in, and the columns are worth one, sixteen, two

hundred and fifty six, and each column holds anything from none to fifteen of

its value.

$$v = \sum_{i} d_i \, B^{i}$$

Here $B$ is the base, $d_i$ is the digit in column $i$ counting from the right

starting at zero, and $v$ is the value. The sum is the whole of place value, and

the three bases in this lesson are the same expression with two, ten and sixteen

put in for $B$.

A value written in sixteens needs sixteen digits and the familiar set runs out

at nine, so the next six are written as the letters A to F standing for ten to

fifteen. That is a typographic convenience, chosen so each digit remains one

character wide, and it carries no mathematical content at all.

Reading a value written this way is the sum from above with sixteen put in. The

three digit value written 1F4 is one lot of two hundred and fifty six, plus

fifteen lots of sixteen, plus four, which is five hundred.