I find math really interesting.
Not just because of the real world implications, but because the mindset it gives you is empowering. You follow the trail of logical breadcrumbs following rules that are so intuitive they are clearly true, and see where that leads you. Even if it’s somewhere completely unintuitive.
Here’s an example of that, that I find really satisfying and mildly exhilarating.
0.99999…. with repeating, never ending 9s, is really just “1”. Not almost 1, or “basically 1 for all practical purposes”.
0.9999… and 1 are actually the same number. Here, let me prove it to you. If you know how to use basic arithmetic, you can follow along.
So, let’s say that x equals our 0.9999...: x = 0.9999....
Now x is just a placeholder for that number. If I multiply 10 by x, clearly 10 * x = 10 * 0.999.... Which means, 10x = 9.999....
Now, I can subtract some number from both sides of the equal sign, and that equality will still hold. In this case, let’s subtract 0.999… : 10x - 0.999... = 9.999... - 0.999....
That means, 10x - 0.999... = 9.
But 0.999... is x! We said that at the beginning! Ok, so 10x - x = 9.
Now, 10x - x is obviously 9x. So, 9x = 9.
If I now divide both sides by 9, the equality still holds true.
9x/9 = 9/9, which simplifies to x = 1.
Remeber what was x? That’s right! It was 0.999...
So, 0.999 = 1!
Isn’t that amazing?