03.27.06

The arithmetic and geometric mean inequality

Posted in Mathematics at 9:47 pm by Haris

You all know the inequality between the arithmetic and geometric mean, right? In any case, it says that if x and y are two positive numbers, then their arithmetic mean is greater than or equal to their geometric mean, or in math notation:

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

It is of course not hard at all to prove this, but I just read in the Mathematics Magazine a cute “proof without words” for it, which to save the time of drawing the picture I will reproduce here with words. Consider a right angle triangle with the two sides being $2\sqrt{xy}$ and |x-y| . Then the hypotenuse is x+y .