Dedekind cut

A Dedekind cut partitions the rational numbers into a lower set and an upper set, satisfying order conditions. Each cut represents a real number, including irrational numbers absent from the rationals.

Rational number: A number expressible as a ratio of two integers with a nonzero denominator. The cut divides this ordered number system into its lower and upper sets.
Real number: A number on the continuous number line, including rational and irrational values. Each Dedekind cut gives a construction of one real number.
Richard Dedekind: A German mathematician whose work helped establish the foundations of modern algebra and analysis. He introduced cuts as a way to define real numbers from rationals.
Square root of two: The positive real number whose square is two; it is irrational. Its cut consists of rationals whose squares are below two and those above it.
Ordered set: A set equipped with a relation that compares its elements consistently. Dedekind cuts rely on the rationals’ ordering to define a boundary.
Irrational number: A real number that cannot be expressed as a ratio of integers. A cut can represent an irrational value even when no rational equals its boundary.
Continuity and irrational numbers: Richard Dedekind’s 1872 essay on irrational numbers and the continuity of the number line. This work presented the cut construction as a foundation for arithmetic continuity.
Arithmetic of real numbers: The operations of addition, subtraction, multiplication, and division on real numbers. Cuts can define real-number operations directly from operations on rational numbers.
Lower set: A subset of an ordered set that contains every element below each of its members. The lower half of a cut must include every rational smaller than one of its elements.
Density of the rational numbers: The property that between any two distinct rational numbers lies another rational number. Density prevents a cut from leaving a gap between its lower and upper sets.

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