#MathsInMinutes Day 26: Counting rational numbers

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Although not all infinite sets are countable, some very big sets are. These include the rational numbers – numbers made from a ratio of two integers a/b. We can prove this by looking at just the rationals between 0 and 1.

If the rationals between 0 and 1 are countable, then we should be able to put them in an order that creates a complete, if infinite, list. The natural ascending order of size is unhelpful here because between any two rational numbers one can always find another, so we could not write down even the first and second elements of such a list. But is there another way to list the numbers?

One solution is to order the numbers by their denominator, b, first, and then by the numerators a, as shown. There is some repetition in this approach, but each rational number between 0 and 1 will appear at least once in the list.

1/2

1/3, 2/3

1/4, 2/4, 3/4

1/5, 2/5, 3/5, 4/5

1/6, 2/6, 3/6, 4/6, 5/6

1/n, 2/n, 3/n, 4/n, … , (n-2)/n, (n-1)/n