The AM-GM inequality, formally the inequality of arithmetic and geometric means, states that the arithmetic mean of a list of non-negative real numbers is always greater than or equal to the geometric mean of the same list, with the two means equal only when every number in the list is identical. The simplest non-trivial case covers two non-negative numbers.
Facts
StatementFor a list of non-negative real numbers, the arithmetic mean is greater than or equal to the geometric mean of the same list, with equality holding if and only if every number in the list is equal. 2 Proof YearDates the first published proof of the general n-variable case (Augustin-Louis Cauchy, Cours d'analyse, 1821); the simplest two-variable case was already known in antiquity. Classification
Statement FormCharacterization Theorem 1 Connections
Has Statement Form
Entity-backed identity for the statement-form enum value this theorem already carries, resolved to a mathematics concept by an explicit value-to-entity map (phase 3 bucket conversion, docs\design_entity_backed_browse_buckets_20260928.md). The statement-form fact itself stays on the theorem unchanged.
Sources
1. Wikipedia: AM-GM inequality
WikipediaLead section, statement-form referenceQuote, Lead section, statement-form reference
In mathematics, the inequality of arithmetic and geometric means, or more briefly the AM-GM inequality, states that the arithmetic mean of a list of non-negative real numbers is greater than or equal to the geometric mean of the same list; and further, that the two means are equal if and only if every number in the list is the same (in which case they are both that number).
2. AM-GM Inequality (Wikipedia)
Wikimedia Foundationopening paragraphQuote, opening paragraph
states that the arithmetic mean of a list of non-negative real numbers is greater than or equal to the geometric mean of the same list; and further, that the two means are equal if and only if every number in the list is the same
3. Cours d'Analyse (Wikipedia)
Wikimedia Foundationopening sentenceQuote, opening sentence
is a seminal textbook in infinitesimal calculus published by Augustin-Louis Cauchy in 1821.
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