A ratio of two integers. A rational number, when written in decimal notation, may have finitely many non-zero digits, or an infinite number which form a repeating sequence. The set of rational numbers is isomorphic to the set of integers.

Rational numbers have a cardinality of aleph null, as do the integers and natural numbers. this means that there must be a one-to-one correspondence (above: isomorphic) between any two of the three sets. Prove it to yourself with a very large sheet of graph paper:
1) write a 0 in the upper left corner square.
2) along the top (x-axis) and left side (y-axis), write the integers in increasing order.
3) in all the other squares, write the fraction made from the number directly up on the x-axis over the number directly over on the y-axis.
it's one quadrant of a graph, flipped upside-down. you just made an isomorphism. if you can do this with the real numbers, you are probably god.

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