0.99999~ = 1 ?

Son Goku's explanation is the one that deals best with an actual mathematical understanding of what is going on.When I got on to my degree course, one of the first things we were taught was a new definition of equality. It's not the same as Son Goku's point, but it is very closely related.That is, two numbers a and b are equal if there is no number, e such that
0 < e <|a - b|(Which is the same as saying that there is no number between them, if you think about it).In fact we needed this definition to deal with the sums of infinite sequences. And if we couldn't do that, integration wouldn't work (and we'd need to find a physical solution to Zeno's paradox, too !) So it is important to mathematics that 1 and 0.999... are the same number. If they were different we'd have real problems.Edited by PaulK, : Corrected after prompting from Dr. Adequate.

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But is it wrong to say that 0.999R <1?

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I'm afraid that it is. Unless you can show that 1 - 0.999R is greater than zero.(Hint: What is the decimal expansion of 1 - 0.999R. How far do you have to go to find a non-zero digit ?)

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Can it be shown mathematically that 1 - 0.999R is equal to zero?

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Yes we can show that there is no number between it and zero, for instance (you can get that from the fact that there is no number between 1 and 0.999R)

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One step further down the infinite chain than you need to go to make 0.999R the same as 1?

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Nice attempt to turn it around. However, would you agree that 1 - 0.999R is zero to an infinite precision ?

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But does that mean that infinity squared is the same as infinity to the power of 10?

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There are degrees of infinity. For instance the infinite number of real numbers is greater than the infinite number of integers (as shown by Cantor's diagonalisation argument).Off hand I don't remember exactly how they relate mathematically, it's been a while since I read up on this stuff.

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But it still seems "wrong" that 0.999R is a whole number.

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Isn't that just a matter of presentation ? You see something that doesn't look like a whole number, but in fact it's just an odd and impractical way of writing 1 (or just wrong if you go with the finitists).

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I don't think so. I think it is because the human brian (well mine at least) cannot cope with infinity. And for this to make sense we need to think of 0.99999999999 - to infinity.

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I would say that human intuition can't handle it (with maybe a few rare exceptions). But your unease is the product of your intuition. Which is why when you see 0.999R you think of it as something different from and less than 1.But if you accept that 0.999R is just another way of writing 1 it IS a matter of presentation. Your intuition wouldn't rebel against "1 = 1" or "1 is a whole number", you'd just regard them as trivial and obvious truths.

That's because infinity is weird.Remember that we had to redefine equality to do integration ? That's because integration is - effectively - summing up an infinite number of infinitesimals. But if we use the definition that lets us work with infinitesimals, then infinitesimals - or the sum of any finite number of infinitesimals equal zero.And in that case we are still left 0.999R = 1.

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The error made in many of the above posts that try to prove that 0.99999.... is the same (is equal to) 1.0 is in confusing "converges to" and "is equal to".

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I don't think so. 0.999... IS the limit of 0.9(n) as n tends to infinity, and that is what we want to prove equal to 1. It's the series which converges to 1, not the number.
(That's not to say that there might not be similar confusion in some of the posts).