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:
Now we can multiple the whole equation by ten:
Then we subtract 0.99999…:
Remember, we know that 0.99999… = x, thus we can rewrite the right side as follows:
Suddenly, we get:
And from this we can easily derive that 1 = x.