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In mathematics, the smallest positive number in a given number system is the first number greater than 0. For example, the smallest positive integer is 1. On the other hand, there is no smallest positive rational or real number.

Ordered fields
If x is a positive real or rational number, then x/2 is a smaller positive number. This proves that there is no smallest positive number in either number system, or indeed in any ordered field.

The proof can also be formulated as a proof by contradiction. Suppose that x is the smallest positive real number; then x/2 contradicts the assumption. This argument is sometimes given as a textbook example of the technique.

In particular, there is no smallest positive number in the ordered field of hyperreal numbers. Although the hyperreals do include positive infinitesimal numbers, which are smaller than any positive real number, there is no smallest such number.

History
Gottfried Leibniz developed his theory of calculus using infinitesimal numbers. He explicitly drew a distinction between infinitesimal numbers and the concept of a smallest positive number.