I cottoned on to this when I was at primary school, when a teacher pointed out that there are an infinite number of radiuses to a given circle, and an infinite number of diameters, but only half as many diameters as radiuses.
Your teacher was wrong. A circle has exactly the same number of radiuses and diameters and it is the same as the number of real numbers.
A better example, perhps, is positive whole numbers and positive whole even numbers. I think the reason infinity doesn't behave itself is that, while the concept is useful to mathematicians, there's no such thing: even time and space are nowadays thought to be finite, and numbers don't exist either.
Again there is the same number of even numbers as whole numbers but the explanation is slightly easier to deal with than the circle diameter/radius case.
Mathematicians say two sets are the same size if their elements can be put into a one to one correspondence without leaving some elements from one set left out. So for the two sets { apple, tram, snake } and { Einstein, Kekulé, Newton } we can create a 1:1 correspondence (several actually). This means we can find a way to remove pairs with one item in the pair from each set and have nothing left over in either set at the end:
apple <-> Newton
tram <-> Eindstein
snake <-> Kekulé
so we know those two sets are the same size. But if the second set was { Einstein, Kekulé, Newton, Darwin } no matter what 1:1 correspondence we choose. There will always be something left over from the second set.
With infinite sets of numbers, we describe the correspondence by a mathematical relationship rather than listing all the pairs out which would take literally for ever. So, with the set of whole numbers and the set of even numbers we can pair each whole number with its double in the even numbers. This is a 1:1 relationship and by definition it mans the two sets are the same size.
A similar trick works for radiuses and diameters. Each radius crosses the circle at one point on its circumference. Each diameter crosses the circle at two points, one in the left half of the circle and one in the right half (the straight up and down diameter is a special case, we'll arbitrarily put the point at the top in the right half and the point at the bottom in the left half). So the number of radiuses is the same as the number of points on the circumference. The number of diameters is the same as the number of points in the right half of the circumference. So there are half as many diameters right? Wrong.
Imagine unwrapping the circle so that the circumference becomes a straight line. Imagine that the circumference was two units long. Now we have a line that is of length 2. All the points that were in the right half of the circle now lie between 0 and 1. All the points that were in the left half of the circle lie between 1 and 2 (technically speaking the line goes right up to but doesn't include the number 2, but this is a detail for mathematicians). So the number of diameters is the same as the number of points between 0 and 1 (not including 1 itself, see my comment about the up and down case) and the number of radiuses is the same as the number of points between 0 and 2 (not including 2 itself). Can they be put into a 1:1 correspondence. Yes they can. Just double the diameter point to get its equivalent radius point, or halve the radius point to get its corresponding diameter point.
For my next trick, I'll explain Cantor's diagonalisation proof that there are more real numbers than whole numbers, but not just now.
