Algebraically, it makes sense somewhat, but in theoretical terms, where the number line is infinite, then an infinite amount of numbers assumes an infinite amount of space, with each number having it's own allocated slot. Therefore, no matter how large 0.999... becomes, it cannot suddenly assume a different unconnected number. Of course, in reality, nothing is infinite and only occupies a finite space, so it's not that bad.
If there's infinite space for an infinite number of numbers, how much space does each get? Is it well-defined?
If there are an infinite amount of numbers, then the number line they can be placed on must be infinite, meaning every number has its own space to preoccupy on this number line, which is why two cannot one space.
The point is that what you think is '0.999...' has no meaning. I'm guessing you think it's '0.999... where there are infinitely many 9s' but such a thing has no meaning in itself. That's like considering the number '111... where there are infinitely many 1s'. There is no ambiguity in maths.
The idea is that what we call '0.999...' is actually a limit (which has a precise mathematical meaning and does not use some vague concept of 'infinity' in its definition) of a sequence, namely the sequence 0.9, 0.99, 0.999, ... , all of which are well defined numbers. It happens that this limit is 1, and this is quite intuitive anyway.
Certainly '0.999...=1' has no meaning in itself, without preliminary definitions (as mentioned above). I must emphasise that the argument that the 'difference between the two numbers is infinitely small' is not in any way mathematics. The same goes for all the other intuitive explanations. They're by no means proofs.
It has got nothing to do with rounding. They are different representations of the same number.
If you would like a more analytical proof, using slightly more rigorous mathematics, then I'll sketch one here. As there is an infinite number of "9"s, the number is equal to the infinite series below.
As 101<1, we know that the sum of the series will converge. In very very loose Layman's terms, this means that the difference between terms and a given limit L, ie. ∣L−xn∣ becomes arbitrarily small through increasing n. You may think of this as the value added on to the sum being increasingly small to the point at which the difference is almost incomprehensible, meaning that the sum will approach a specific value. You may notice that for ∣r∣>1, the series will diverge as there exists no L such that ∣L−xn∣ becomes arbitrarily small through increasing n, meaning that the sum will diverge. Using quite elementary techniques, we can work out this "specific value", as the given series is a geometric series, S, knowing the sum of a geometric series as S=1−ra1, where r is the common ratio, in this case 0.1
9k=1∑∞10k1=1−101(109)=109109=1.
Euler actually used this method to prove that 10=9.9.
If there are an infinite amount of numbers, then the number line they can be placed on must be infinite, meaning every number has its own space to preoccupy on this number line, which is why two cannot one space.
I'm not a mathematician, but I thought that infinity divided by infinity wasn't well-defined.
You are assuming that 1 and 0.9˙ are different numbers. We could do with a mathematician to answer that, but, as the difference is 0, I would say that they're the same number.
It has got nothing to do with rounding. They are different representations of the same number.
If you would like a more analytical proof, using slightly more rigorous mathematics, then I'll sketch one here. As there is an infinite number of "9"s, the number is equal to the infinite series below.
As 101<1, we know that the sum of the series will converge. In very very loose Layman's terms, this means that the difference between terms and a given limit L, ie. ∣L−xn∣ becomes arbitrarily small through increasing n. You may think of this as the value added on to the sum being increasingly small to the point at which the difference is almost incomprehensible. You may notice that for ∣r∣>1, the series will diverge as there exists no L such that ∣L−xn∣ becomes arbitrarily small through increasing n, meaning that the sum will diverge. Using quite elementary techniques, we can work out this "specific value", as the given series is a geometric series, L, knowing the sum of a geometric series as L=1−ra1, where r is the common ratio, in this case 0.1
9n=1∑∞10k1=1−101(109)=109109=1.
Euler actually used this method to prove that 10=9.9.
This is the summation to infinity, which is impossible. So theoretically yes it does = 1 but in reality, it never can.
I'm not a mathematician, but I thought that infinity divided by infinity wasn't well-defined.
You are assuming that 1 and 0.9˙ are different numbers. We could do with a mathematician to answer that, but, as the difference is 0, I would say that they're the same number.
Anything involving infinity isn't well defined. When you introduce infinity you move from absolute answers to theoretical absolutes.