A question of numbers (one for the maths fans)

To multiply by 10, you shift the decimal point one position to the right. Where's the problem?

A s mall slip there. 10x-x is 9x, so that last line should be9x=3

Almost right. If there were a last digit, it would have to move over. But since there is no last digit, there isn't any problem. You still have an infinite string of '9' digits.

We don't lose any, because there is always another 9 to the right of it that can replace it.Here is a mathematician's secret. Don't tell anybody.Mathematics is fun precisely because we get to deal with the infinite. If it were not for that, then mathematics would be as boring as accounting balancing your checkbook.Edited by nwr, : reword the last sentence

From a mathematician's perspective, I have to disagree with that.Subtraction is something you do with numbers. The string of digits written on the paper is not a number. Rather, it is a physical representation of a number.The process you describe with "we subtract one column at a time" is an action on the representations. It is a mechanical procedure we carry out to find the representation of the result of subtraction. That the mechanical procedure is only defined for finite strings of digits, in no way proves that subtraction is impossible. At most, it shows that the mechanical procedure is impossible.Subtraction between numbers can be defined mathematically (or logically if you prefer that term) independent of the details of how we represent the number with ink and paper.

Then you missed the point entirely.Numbers do not have ends or beginnings. Those are not properties of numbers. Strings of digits may have ends or beginnings. But the numbers they represent do not.You need to make a distinction between numbers, and strings of digits.The string 0.999... does represent a number. And the reason it respresents a number, is that we have well established conventions that specify how strings of digits can represent numbers. These conventions extend to infinite decimal representations.

Neither of those actually make much sense. You cannot treat infinity as if it were an ordinary number.

That's the wrong question.We invent mathematical entities. Thus we can invent infinity. The mathematical sense of "exist" is different from the physical sense of "exist". If we can invent a suitable concept of infinity that does not cause any inconsistencies in our mathematics, then we can say that infinity exists in the mathematical sense.