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It doesn’t make logical sense. Why does the 3-4-5 triangle work out cleanly, and yet π and e are irrational? How can 0.999… and 1 be exactly the same number? Who cares about solving lost and unprovable theorems — how do these help anyone?
A small neat proof about 0.9999…. = 1
1/3 =0.3333….
Multiply both sides by 3
3 x 1/3 = 3 x 0.333…
1 = 0.999…
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.
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.
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.
To add to this excellent answer, in my mind at least some of the confusion comes from trying to compare a number to a concept.
Using the 0.999... example, that doesn't have a set numeric value as such. You cant use it in an algebraic formula to calculate a fixed result. Logically, it's a number just below 1.0, but mathematically, every time you try to make that distinction it effectively shifts the goal posts.
It's like the old "count to infinity" bit, it's not impossible because it's a big number, it's impossible because it's not a number at all, it's a concept.
They do have a way of eventually coming up in something practical. It might take a long time though, that's fair.
spoiler since I want to be sensitive to you not liking math, but I have comments
I mean, you can define things a different way, but it gets messy fast. The basic idea is that there shouldn't be any gaps between the numbers - so a line is as unbroken as it looks. Without making all kinds of other adjustments 0.999... would break that, since there's no way to add further decimals in between it and 1.
As for the constants, yeah, it's weird. A lot of math is weird; the universe wasn't built for us.