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Mathematics

The Completeness of the Real Numbers via Dedekind Cuts

Quick fact

Dedekind cuts, introduced by Richard Dedekind in 1872, prove that every gap on the number line is 'filled' by a real number, so the reals are complete. This construction is so elegant that it can be done using only sets of rational numbers.

Why this is interesting

Imagine the number line made of perfectly smooth, unbroken material. But mathematicians discovered that the rational numbers, despite being dense, have tiny holes that seem invisible—and filling them is the key to calculus.