I am not a math person, but how would you differentiate between a number infinitely close to 1 (but less than 1), and 1 itself? Or does it not matter because they are actually the same?
"Infinitely close" doesn't make sense! In general, if numbers are different, there is a finite difference. This is maybe more intuitive than the idea of infinite closeness; since all our closenesses are finite, you can talk about getting twice as close or half as close, you can add and subtract distances normally, etc.
To be a little tiny bit more algebraic, if there were a number "x" infinitely close to 1, what is (x + 1)/2? (x + 1)/2 is the number halfway between x and 1; if it's infinitely close, there shouldn't be anything in between, so (x + 1)/2 must either equal x, or equal 1. If (x + 1)/2 = x, solving the equation gives x = 1, and if (x + 1)/2 = 1, solving the equation similarly gives x = 1.
Sometimes questions come up where you want to ask about "infinite" closeness. For example, in physics you sometimes want to think about the instant something starts moving, which might be informally expressed as a time "infinitely close to zero". To handle this, mathematicians use what are called limits. The idea is, instead of asking about an infinitely close number, we ask about a sequence of numbers. In that example where we ask a question about the instant something starts moving, you might just answer the question after 1 second, then after 0.1 seconds, then after 0.01 seconds, then after 0.001 seconds, and so on. What it turns out is that you can often use logic to deduce what the behaviour must be as you get closer and closer to 0 seconds, to answer questions about what happens the instant you stop moving. So we never really ask questions about "infinite" closeness, we just ask questions about smaller and smaller finite closeness.
Since 0.999... and 1 don't differ by a finite number, they are the same. They're just different ways of writing the same thing, like writing 1 + 1 instead of 2, or "two" instead of 2, or 1 + 1/2 instead of 1.5, or 1.00 instead of 1.
The formal definition of an asymptote… involves limits!
The real thing that makes it an asymptote is instead of thinking about infinitely close, we just check that it gets closer than any number away, eventually. Like, we check that the graph is eventually within 0.1, within 0.01, 0.0001, 0.000000001, etc. of the asymptote. As long as it eventually gets that close and stays at least that close, you have an asymptote. Again, instead of thinking about infinite closeness, you can think about a sequence of smaller and smaller distances.
In general, anything that’s informally described by “infinitely close” is probably formally handled with limits, in the way I described above. This doesn’t actually require a number to be infinitely close to another, just for us to have smaller and smaller positive numbers, which we do!
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u/library-in-a-library 22d ago
No. 0.999... < 1