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Mathematics

Dedekind Cuts: Constructing the Real Numbers from Rationals

Quick fact

A Dedekind cut is a single split of the rational numbers into two sets—one with all numbers less than the cut, the other with all numbers greater. Every possible cut corresponds to exactly one real number, and there are uncountably many such cuts, filling every gap in the rational line.

Why this is interesting

You know the number line is continuous—no gaps. But the rationals have holes everywhere. How can we fill those holes without inventing numbers out of thin air?