I know the Arithmetic proof and everything but how to explain this practically to a kid who just started understanding the numbers?
In: 3202
This is far, far, far simpler than it sounds.
The easy and unsatisfying answer is: “because we’ve decided that’s what infinity means.” Which sounds dumb, but it’s actually kinda deep.
Infinity doesn’t exist in the real world; it’s not an actual number. It’s just an idea. It’s the answer to a question. Or rather, infinity is the question itself.
The question is: “what happens if you never stop?” That’s infinity. Infinity is the question asking what happens when you don’t ever stop.
So, if you say: `0.999…` you’re not saying the same thing as `1`, because 1 is a number while 0.999… is an infinite series. In other words: 1 is an answer, while 0.999… is a question.
The question is: “what happens when you keep adding 9’s?” And the answer is: “you get closer and closer to 1.”
Or in more formal terms: “the infinite series 0.999… approaches 1.” And because math people like simple answers, you can write the previous statement simply as “0.999… = 1”. Which, since we know that 0.999… deals with infinity, we know that one side is the question and the other side is the answer.
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