Why does 0.999… equal 1?

One mathematical question that sometimes causes some confusion is this:

“Are 0.999… and 1 the same thing?”

The answer is yes, assuming by “0.999…” we mean a zero, then a decimal point, then nines that repeat forever.

There are pages and pages of lengthy mathematical proofs available that show why this is so, but my favorite way of showing why 0.999… and 1 are the same is quite simple: divide each one by three.

A simple illustration

If we divide 0.999… by 3, we get 0.333… (zero, then a decimal point, then threes that repeat forever.)

If we divide 1 by 3, we also get 0.333…

Voilà.

If we got different results for each division, we could say that 0.999… and 1 were different somehow. But since we get the same result in both cases, we can see they represent the same number.

0.999… does look different from 1, but so do 7/7, 4-3, and 28⁰, and yet those are all legitimate ways of representing the number 1.

Need more convincing? There are plenty of pages out there that go into the topic in greater detail.