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What is infinity minus one?

What is infinity minus one (or 10 or 100)? If it is still infinity, what do you have to subtract from infinity for it to be less than infinity? (In layman’s terms).
Judith Abbott from Essex (age 45-54)

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One Response

  1. Infinity isn’t really a number like 1, 2, 3, etc. and there are actually many ideas of infinity. I take it here that you are talking about counting things, starting with 1, 2, 3 …. If you do that forever, you reach infinity.

    If in that sense you’ve got infinitely many things, then removing finitely many things, like 1, 10, 100 any other number, you still have infinitely many things left. So it seems you have to remove infinitely many things to reach something less than infinity. However, if you do this, the answer is not clear. Say you’ve got a set of all positive integers, and remove all even numbers from it. You’ve got the odd numbers left. But there are infinitely many even numbers and infinitely many odd numbers. So that would mean infinity minus infinity is infinity. But you could also take all positive integers and remove all positive integers larger than, say, 1. Now you’ve only got one number, 1, left. So that would mean infinity minus infinity is 1 (or any other number depending precisely what you remove). If you allow negative numbers, etc. there will be even more possibilities, but not one single, consistent answer.

    So it really depends on what you count and remove, and in general it is best to say that subtracting something from infinity is not really defined at all. You are doing something illegal if you try it.

    Maybe it is best to think of infinity as the process of counting forever, starting from some number, like 1. Then infinity becomes the state you reach in the limit.

    For example if you’ve got the real number 0.999999… (infinitely many 9s), then you can count how many nines there are in this number and on your way you get the real numbers 0.9, 0.99, 0.999, 0.9999, etc. Obviously there are infinitely many of these. If you could count all of them, you’d reach infinity. So that is a process to describe infinity. But this process actually reaches the number 1 in the limit, because 1 – 0.9 = 0.1; 1-0.99 = 0.01, 1-0.999 = 0.001, etc. The distance between the number of nines you counted and 1 becomes smaller and smaller and so actually becomes 0 in the limit, which means 0.9999999…. = 1. Just the 0.99999…. is a process and 1 the state this process reaches. This is not too different from counting forever starting at 1 as the process and infinity itself as the state this process reaches.

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