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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

{\displaystyle {\frac {x+y}{2}}\geq {\sqrt {xy}}}

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 (greater than or equal to zero) and from the elementary case (a ± b)2 = a2 ± 2ab + b2 of the binomial formula:

{\displaystyle {\BEGIN{aligned}0&\leq (x-y)^{2}\\&=x^{2}-2xy+y^{2}\\&=x^{2}+2xy+y^{2}-4xy\\&=(x+y)^{2}-4xy.\end{aligned}}}

Hence (x + y)2 ≥ 4xy, with equality precisely when (x − y)2 = 0, i.e. x = y. The AM-GM inequality then follows from taking the positive square root of both SIDES.

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