oysters-aint-for-me:

secondaristh:

homo-sex-shoe-whale:

antifascistdean:

homo-sex-shoe-whale:

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: