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Cake day: June 6th, 2024

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  • Who cares about solving lost and unprovable theorems — how do these help anyone?

    I moved this one first because it’s the most important to answer. A lot of esoteric math does end up leading to useful results in science, engineering, or computer science. A lot of breakthroughs in physics, especially historically, came from breakthroughs in math. A lot of computer science, such as error correction and encryption, came from what was previously esoteric mathematics.

    Why does the 3-4-5 triangle work out cleanly, and yet π and e are irrational?

    There kinda isn’t a satisfying answer to this; it just turns out that’s how the world works. Some important questions have nice integer answers, and some don’t.

    How can 0.999… and 1 be exactly the same number?

    0.999… == 1 because there is no number in between 1 and 0.999… therefore they must be the same number.

    For any two numbers that aren’t equal, you can find numbers between them (specifically do something like a*0.5 + b*0.5). You can’t define 0.999… as something like “the largest number less than 1” because there is no such number, because if you found such a number, you could find another number between it and 1.

    However, there are some situations where the idea of “0.999…” might have some meaning, if you interpret it as “taking the limit of something as it approaches 1 from below”. See: https://en.wikipedia.org/wiki/One-sided_limit for some examples. These examples are mostly centered around 0, but if you moved it to be centered around 1, you would get a function where f(0.999…) ≠ f(1.000…1) with f(1) not having a well defined value.

    But because 0.999… is not the commonly accepted notation for that limit, some people reading your work would be confused. In the end it’s a matter of language: agreeing on a meaning for symbols so you can communicate your ideas clearly.