Why a base makes arithmetic easy

Having a base isn't just neater to write: it changes what you can do with the numbers.

Take adding. Add two Egyptian numerals and you might end up with, say, 15 tens and 15 ones. Regroup: ten of the ones make a ten, leaving 5 ones and giving 16 tens; then ten of those tens make a hundred, leaving 6 tens. The answer falls out.

What made that work? The regrouping was by ten both times, because every landmark is ten of the one below. The book's own character says it out loud: 'I see a similarity in the method of adding numbers in the Egyptian and the Hindu system!' That similarity is carrying.

In the Roman system you can't do this without thinking. Five Is make a V, but two Vs make an X; five Xs make an L, but two Ls make a C. The regrouping size keeps changing, so every step needs care.

Multiplying shows the gain even more clearly. In the Egyptian system, what is any landmark times 10? Each landmark is a power of 10, so multiplying by 10 raises the power by 1: you simply get the next landmark. And 10² × 10² = 10⁴: multiply two landmarks and you just add the powers.

So the product of any two landmarks is another landmark. That's true in every base system, and it makes multiplying manageable: split each number into landmarks, multiply landmark by landmark using the distributive property, (a + b + c) × n = an + bn + cn, and regroup.

Try that with Roman numerals and you'd have to remember what V × L and L × D come to, one product at a time. So a system whose landmarks are powers of one number is efficient both for writing numbers and for calculating with them. That was the turning point, and our Hindu system is built on it.

One more thing about the abacus: by about the 11th century, even people writing Roman numerals had adopted a decimal counting board. It had a line for 1, then a line for each power of 10, with a counter for each one, and a counter placed above a line counted as 5. The decimal idea was being used even by people who didn't write numbers that way.

Yet the Egyptian system still had a flaw: to write bigger and bigger numbers it needed more and more new symbols for higher powers of 10. The original problem had come back in a different form.