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Yes. Both sets have an infinite number of elements, but some infinities are larger than others.

Natural numbers, whole numbers and integers are sets with the same number of elements. That number is the smallest of the infinities. Imagine having one computer for each set. The natural number computer starts counting "1, 2, 3 . . ." The whole number computer starts counting "0, 1, 2, 3 . . ." The integer computer starts counting "0, +1, -1, +2, -2, +3, -3" and it keeps counting. If you start these imaginary computers at the same time and let them run forever, each one will count out its entire set of numbers. This "thought experiment" lets us count to infinity.

Now, let's imagine a fraction computer. We want this computer to count every rational number between 0 and 1. So, it starts counting " 1/2, 1/3, 2/3, 1/4, 3/4, 1/5, 2/5, 3/5, 4/5, 1/6, 5/6 . . ." If we let that run forever, it will count out every rational number -- every distinct fraction -- between 0 and 1. Ah, but what about all the fractions between 1 and 2, or between 2 and 3?

Not only do you need to let this imaginary computer run forever, but you also need an infinite number of imaginary computers running forever.

In effect, infinity plus infinity equals the same infinity, but infinity times infinity equals a larger infinity.