My favourite math fact is that 0.9999999.. is equal to 1. Exactly. Not approximately. Not as a rounded number. 0.9999 (recurring) is exactly 1.
Question. How the fuck does that work?
I tried explaining it here:
Here’s another perspective on why .999... repeating is exactly equal to 1.
For any two distinct real numbers, we can always find a rational number strictly between them, i.e. that rational number must be able to be expressed as a terminating decimal or a repeating decimal. To be clear, that rational number is strictly between the two values; it is not allowed to be equal to either.
Suppose k is a rational number strictly between 1 and 0.9999.... If this is possible, then, I can write k exactly as either a decimal with finite digits, or I can write k as a repeating decimal. The problem is, there are no decimals with finite digits between 1 and 0.999... , and there is no way to write a repeating decimal that is greater than 0.999... and still less than 1. Either way, a k strictly between 1 and 0.999... does not exist. The only way this can be true is if those two numbers are not actually distinct. That is to say, 1 = 0.999.....
i truly appreciate how math seems like it’s this infallible always-true only-one-answer thing, when in reality math is just like:
