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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 and , is the statement that : with equality if and only if . 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 of the binomial formula: : In other words , with equality precisely when , i.e. . For a geometrical interpretation, consider a rectangle with sides of length and , hence it has perimeter and area . Similarly, a square with all sides of length has the perimeter and the same area as the rectangle. The simplest non-trivial case of the AM–GM inequality implies for the perimeters that and that only the square has the smallest perimeter amongst all rectangles of equal area. The general AM–GM inequality corresponds to the fact that the natural logarithm, which converts multiplication to addition, is a strictly concave function; using Jensen's inequality the general proof of the inequality follows. : Extensions of the AM–GM inequality are available to include weights or generalized means. == Background == The ''arithmetic mean'', or less precisely the ''average'', of a list of numbers is the sum of the numbers divided by : : The ''geometric mean'' is similar, except that it is only defined for a list of ''nonnegative'' real numbers, and uses multiplication and a root in place of addition and division: : If , this is equal to the exponential of the arithmetic mean of the natural logarithms of the numbers: : 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Inequality of arithmetic and geometric means」の詳細全文を読む スポンサード リンク
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