I hate these posts because they’ve gotten in my head, and when I was drunk in the passenger seat of my friends car last night I saw a cop in hi-vis directing traffic and my first thought was “ok he do be lookin kinda safe doe”
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: