.9999 repeating = 1
Any math major will not be surprised by this, but just try to make that fact click in your brain. It seems impossible, because there should be a remainder of .000…1 repeating to equal 1, right?
It is actually quite easy to prove this to yourself. First we have to consider another principle of mathematics. If two numbers, both divided by the same number, equal the same number, they must be equal to each other. So:
.999 repeating / 3 = .333 repeating AND 1 / 3 = .333 repeating
There you have it, they are the same value.
Another way to prove this to yourself is to think of a pie with nine slices. One slice of the pie is .111 repeating of the pie, two slices of the pie is .222 repeating of the pie, and so on. We reach eight slices, .888 repeating of the pie, and if we continue to nine slices of the pie, this would be .999 repeating and 1.0 of the pie. Although fractions can result in repeating decimals, they are still finite numbers. 1/9 is .111 repeating and the finite fractional value, just as .999 is the finite fraction 9/9’ths and 1.