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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.
The simplest non-trivial case – i.e., with more than one variable – for two non-negative numbers x and y, is the statement that
with equality if and only if x = y. This case can be seen from the fact that the square of a real number is always non-negative and from the elementary case = a ± 2ab + b of the binomial formula:
Hence ≥ 4xy, with equality precisely when = 0, i.e. x = y. The AM–GM inequality then follows from taking the positive square root of both sides and then dividing both sides by 2.