It may sound counter-intuitive, because the number 0.99999… has clearly a different notation than the number 1. It must be a different number then, right? Well, it is not. It’s the same number. We can prove it by using simple linear equation and a few tricks. Imagine this equation:

0.99999… = x

Now we can multiple the whole equation by ten:

9.99999… = 10x

Then we subtract 0.99999…:

9.99999… − 0.99999… = 10x − 0.99999…

Remember, we know that *0.99999… = x*, thus we can rewrite the right side as follows:

9.99999… − 0.99999… = 10x − x

Suddenly, we get:

9 = 9x

And from this we can easily derive that *1 = x*.