In a mathematically rigorous sense, you can't add or subtract or multiply or divide infinity. Infinity isn't really a number; it's more of a concept. That said, if you want to try to assign some meaningful values to infinity, I'd use these:
Infinity plus infinity would be infinity. If you have two things that are without end, and you put them together, you're left with something without end.
Infinity times infinity should also be infinity, for a similar reason. (If you have so many objects in one bag that you'll never run out, and you also have so many bags that you'll never run out, you certainly have infinitely many objects in total.)
Infinity minus infinity is really tough to define! This is usually called an "indeterminate form" in mathematics. Let's see why.
Suppose that I have all the positive integers, {1, 2, 3, 4, ...}. If you take them all away, I'm left with zero. So maybe infinity minus infinity is zero. But suppose I have all the positive integers, {1, 2, 3, 4, ...}, and you just take away the ones that are bigger than 10; that is, you take {11, 12, 13, 14, ...}. Then you've taken infinitely many things from my original infinity, and I'm left with ten numbers. So maybe infinity minus infinity is 10.
Now, finally, suppose I have all the positive integers, {1, 2, 3, 4, ...}, and you take away just the odd ones, {1, 3, 5, 7, ...}. Then I'm left with all the even ones, {2, 4, 6, 8, ...}. You've taken infinitely many from me, but I STILL have infinitely many left!
So, for this reason, infinity minus infinity is usually called "indeterminate"--it could be anything, depending on the circumstances. Infinity divided by infinity is also indeterminate, for a similar reason. Ask yourself "if I have infinitely many objects, and I divide them into infinitely many equal-sized piles, how many are in each pile?"
The answer is "It depends how you do it." (This is never true for a normal number!) If I have all the positive integers {1, 2, 3, 4, ...}, I can divide them into infinitely many piles of one:
{1}, {2}, {3}, {4}, ...
So maybe infinity divided by infinity is one. But I can also divide them into infinitely many piles of three:
{1, 2, 3}, {4, 5, 6}, {7, 8, 9}, {10, 11, 12}, ...
So maybe infinity divided by infinity is three. If I get really clever, I can also divide infinity into infinitely many piles of infinity, as follows:
1. First put one object in the first pile.
2. Now put one object in the first pile, and one in the second pile.
3. Now put one object in the first pile, one in the second pile, and one in the third pile.
4. Now put one object in the first pile, one in the second pile, one in the third pile, and one in the fourth pile.
If I keep going (I'll have to keep going forever, because I have infinitely many objects), then when I'm done, all my piles will have infinitely many objects (I put one object in the first pile on each step; one object in the second pile on each step from the second step onward; one object in the third pile on each step from the third step onward; and one object in the millionth pile each step from the millionth step onward.) So maybe infinity divided by infinity is infinity!
So we say infinity divided by infinity is "indeterminate" as well. Good question though Natasha.
