1 and 0.99999….

The other day, this old familiar topic came up: how can 0.99999…. equal 1?

Now, there are plenty of sites dedicated to proving that these numbers are equal. However, these proofs clearly aren’t enough for some people. It doesn’t feel “right” in their gut that this recurring decimal should equal that whole number.

Perhaps that’s how you feel too.

If so, let me ask you this: is 2/5 the same number as 4/10?

Now, they look quite different. Sometimes, in practice, they are quite different – I can imagine it might sometimes make a difference if a pizza is sliced into five slices or ten, even if the amount of pizza eaten is still 40% of the original whole pizza.

Nonetheless, 2/5 and 4/10 are the same number. Or rather, they represent the same number – a point on the real number line, a bit less than a half, a bit more than a third… We say these fractions are “equivalent”, and that two fifths “equals” four tenths.

On the other hand, 2/5 and 4/10 look very different. The two fractions take up different amounts of pixels on your screen. Yet, we’re happy to accept that they are equal.

One way to resolve this – the correct way, actually – is to say that the fractions themselves merely represent a number. They are the language we use to talk about the number. However, the number is an abstract thing that exists quite apart from the symbols we use to write it down.

Ceci n'est pas un chat
Ceci n’est pas un chat

If I write the letters C, A and T in that order, your mind is filled with purry furry goodness. However, cats exist quite apart from the word we use for them. And words can never perfectly capture the concept of a cat. (And different words can be used to represent the same concept, but we don’t complain about that.)

Nobody thinks that a cat is the same thing as the word spelt C-A-T. Why should a number be the same thing as the text we use to write it down?

Scribbles on a page aren’t numbers, even if they represent numbers. This is true, whether they are fractions or strings of decimal digits. And just as many different fractions can represent the same fractional number, there is sometimes more than one string of decimal digits that represent it too.

We aren’t taight this in schools. We are taught “0.4” is a number. “0.3333333….” is a different number.

Then we meet “0.3999999….”, which is clearly a different string of digits from “0.4”. However, they represent the same abstract number that 2/5 and 4/10 do.

You can do arithmetic with strings of digits. If you do arithmetic with 0.39999…. and 0.4, you get no weird surprises if you believe them to represent the same number. If you insist they must be different, though, arithmetic starts to behave in strange, hard-to-understand, counterintuitive ways.