Countability of Rational Numbers: PWW

The rational numbers are naturally embedded into the set of integer pairs N² = {(n, m): n, m ∈ N}. The diagram below (of the 2004 vintage) makes a convincing argument that the latter can be stringed sequentially one after another, i.e., enumerated, making the set N² countable. Thus the same holds for the set Q of rational numbers.

a spiral that goes through all integer points in the plane

References

  1. Des MacHal, Mathematics Magazine v 77, n 1 (Feb 2004)

Countability of Rational Numbers

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