Why does 0.999 recurring equal 1? As a reminder, the “….” means the nines repeat infinitely; they never terminate. When we write numbers decimally, we have more than one way to write the same number. So, we will prove that 0.999 is just another way to write the number 1.
Everybody knows that 1/3 is equal to 0.333…. We can verify this plugging into a calculator. Then if you multiply both sides by 3, we will get 1 equals to 0.999…. If you still don’t believe it, then you can check the other proofs below.
0.999… and 1 are clearly very close but they start with different numbers. So it might seem like they represent different values. Looking at a number line might help us come to a conclusion.
Let’s start by laying out some facts about real numbers on a number line.
- If two numbers are different, there will be some space between them.
- This line has no gaps.
So if 0.999… and 1 were different numbers there would have to be some space between them. Let’s zoom in on this space. Some other number, x, would have to fit here. Can you think of a number that would fit here? Every other number is either less than 0.999… or greater than 1.
So no numbers would fit in this space. But then the line wouldn’t be continuous! So they must be the same number. Representing one value two different ways may seem strange but we do it all the time! One is the same thing as 1.0, 0.333… is 1/3 and 1/10 is the same thing as 0.1.
If you’re not convinced yet, here’s a quick algebraic proof:
And if you’re familiar with limits, it might help to think of 0.999… as a limit. Consider this sequence:
As you can see the limit is going to 1.