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Cantor's infinity pairing proof. Diagram showing the pairing proof of the German mathematician Georg Cantor (1845-1918), which demonstrated that the infinite set of rational numbers is countable and has the same size (cardinality) as the natural numbers (1, 2, 3...). The rational numbers include the fractions formed from the natural numbers, but can be counted using the same method as that used to count the natural numbers. Here, the rational fractions are counted along the diagonals (red arrows) where the numbers forming the fraction add up to 1, 2, 3, 4, and so on. Repeated rational fractions are shown in blue. In contrast, the real numbers (infinite decimals) form an uncountable infinite set (see A900/140).
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