I suspect that part of the problem in questions like this is that non-mathematicians assume that mathematics is describing the real world. It’s not – mathematics is about abstract systems, which may sometimes be set into an approximate relationship with the real world, but which almost always assume things that cannot happen in practice.
Two examples from this discussion:
(1) Decimal fractions are defined so that they can go on forever. So there is a book with the first million digits in the decimal expnsion of pi, but mathematicians know that there are still an infinite number of digits to go (which could be found if you had a computer with infinite memory and infinite time…). In the real world, when you are using a decimal fraction to describe a physical quality like length or time or mass, after a few digits you get to an accuracy which is not only beyond the capacity of our present measuring instruments, but which the uncertainty principle tells us will be beyond the capacity of any measuring instruments. So even the 100th digit of pi is meaningless in terms of of any measurements that you could make in the real world.
(2) Natural numbers are defined so that there is no greatest natural number. But as far as we can see, the universe is finite. A google may be about the right order of magnitude to so that no matter what you count in the universe, you will never get to a google. But then there’s a googleplex, and then you can have a googleplex raised to the power of a googleplex a googleplex number of times. For a mathematician, you’re just starting – there’s still an infinite number of natural numbers to go, in the theoretical world of mathematics – but you are way beyond anything that could possibly describe anything in the universe.
So in mathematics, you can’t appeal to what happens in the real world. You have to define things formally, and use logical arguments to work out the consequences of those formal definitions. Thus, mathematicians use formal definitions of infinite decimal expressions such as .999…, rather than informal ones based upon imagining a line drawn on a piece of paper, or pictures in the mind.
The sequence of formal definitions include:
(1) Defining the natural numbers, which have the property that you can always add 1 to get another natural number. (Because the system includes the natural numbers, it includes a concept of infinity, since the number of natural numbers is greater than any natural number. However, you don’t assume that infinity is a kind of number that you add, subtract, etc.)
(2) Defining fractions as the ratios of natural numbers.
(3) Defining finite decimal fractions as a special kind of fractions.
(4) Defining limits of series, where the definition of a series includes the concepts of functions and of natural numbers.
(5) Defining infinite decimal fractions as a limit of the finite decimal fractions that you get by truncating the infinite decimal fraction at 1, 2, 3, etc., places after the decimal point.
(Note that none of the above is a formaL definition – just an indication of what kind of formal definition you need.)
Then .999… would be defined as something like:
The limit, as n tends to infinity, of the sum (9/10 + 9/(10^2) + 9/(10^3) + … + 9/10^n)
And, given the formal framework that mathematicians normally use, it’s not too hard to prove that .999… = 1
