The idea of a base

Around 3000 BCE the Egyptians developed a written system that used landmark numbers, like the Romans, but with a special sequence of them.

Picture making collections of pebbles. The first landmark is 1. Group ten of those together, and the size of the group is the second landmark: 10. Group ten of those together, and you have the third: 10 × 10 = 100. And so on.

So each Egyptian landmark is 10 times the one before, and since the first is 1, they're all powers of 10. Each got its own picture symbol (the sign for a million, for example, is a person with both arms raised).

Writing a number works as in Rome: count it in groups of the landmarks, starting from the largest that fits. 324 = 100 + 100 + 100 + 10 + 10 + 4, so its numeral is three hundred-signs, two ten-signs and four single strokes.

Now try the same trick with a different group size. Let 1 be the first landmark. Group five of them: the second landmark is 5. Group five of those: the third is 5 × 5 = 25. Then 5 × 25 = 125, and so on. The landmarks are the powers of 5:

Write 143 this way. The largest landmark below 143 is 125, so start there: 143 = 125 + 5 + 5 + 5 + 1 + 1 + 1. That's one 125, three 5s and three 1s.

That gives the general idea. A number system is a base-n system when its first landmark is 1 and each next landmark is the one before multiplied by the same fixed number n.

The Egyptian system is base-10; the one we just made is base-5. A base-10 system is also called a decimal system. In general, the landmarks of a base-n system are the powers of n: n⁰ = 1, n, n², n³, … So base-7 would have 1, 7, 49, 343, and so on.