Why the Digits of Nine Always Add Up to Nine

There is a line you were given and then walked straight past: for nine times anything, the two digits always add up to nine. Nine. One and eight. Two and seven. Three and six. All the way down.

It is true, and it was stated rather than explained. Two questions come out of it. Why does that happen at all, and does it stop the moment you run out of fingers?

The nine you are holding

Start with the fingers, because the answer is right there and it is not a coincidence.

You have ten fingers. You bend one down. Nine are left standing.

The trick then splits those nine into two groups — the ones before the bent finger and the ones after it — and reads them as the tens digit and the ones digit. Of course they add up to nine. They are the nine fingers you did not bend, counted in two halves.

The trick is not really about the nine-times table. It is about ten fingers with one folded away.

Why nine behaves like this

Now the arithmetic, which says the same thing without hands.

Nine is one less than ten. So every step up the table does two things at once: it adds a ten and takes away a one.

Adding a ten pushes the tens digit up by one. Taking away a one pulls the ones digit down by one. Watch them go: 27, 36, 45. Two becomes three becomes four on the left, seven becomes six becomes five on the right.

And here is the part worth holding on to. A total that gains one and loses one at the same moment does not change. It cannot. The sum of the digits was nine when you started, and every step moves both digits by exactly one in opposite directions, so it is still nine.

That holds for exactly as long as the ones digit has a one left to give. Which, as it happens, is exactly as far as your fingers go.

It is not a pattern that happens to keep working. It is pinned.

Where the fingers give up

Go past ten and the trick dies, exactly as you would expect. Nine times eleven is ninety-nine, and there is no eleventh finger to bend.

But look at ninety-nine. Nine and nine make eighteen, not nine. So has the rule broken too?

Check the next one. Nine times twelve is a hundred and eight, and one plus zero plus eight is nine again.

The rule did not break. It grew up. Eighteen is nine times two. What actually survives is this: the digits of a multiple of nine always add up to a multiple of nine. Below ninety-nine that always lands on nine itself, which is why it looked like the simpler rule for as long as your fingers lasted.

Why it keeps working forever

The fingers ran out because you only have ten. The rule does not run out, and this is why.

That is the rule, and now it is not a trick at all. Every place value is one more than a pile of nines, so when you break a number apart the nines look after themselves and the only thing left to check is the pile of leftover ones — which is the digits.

A number divides by nine exactly when its digits add up to a multiple of nine.

Try it on something too big for fingers

Pick any number you like. Add its digits. If the answer is still big, add those digits too.

Try 4271. Four and two and seven and one make fourteen, and fourteen is not a multiple of nine, so 4271 is not either. Divide it and you get a remainder, every time.

Now try 7254 by hand, the long way, and watch it come out at 806 exactly.

Hold up your hands one more time and bend a finger. You are not doing a memory trick. You are holding ten of something, hiding one, and reading the nine that are left in two groups — and that is the same reason the digits of a million-something add up the way they do.