Using Area to Prove the Arithmetic-Geometric Mean Inequality
If we have real numbers and
, we call
the arithmetic mean (AM) and
the geometric mean (GM) of
and
. In this post, we are going to examine the relationship of these two means.
To start off, let’s have a few examples. If and
, then
and
; if
and
, then
and
; if
and
, then
and
. What do you observe? Try a few example and see if your observations hold.
From a few examples above, and from your trials, you have probably observed that which means that
for some positive real numbers
and
. This is actually true for all positive real numbers
and
. In the following discussion, we are going to use the concept of area to prove that the statement is true.
To begin the proof, we construct a square with side length made up of four rectangles and a square at the center (technically, a square is also a rectangle). Clearly, the area of each of the four rectangles is
, and the square at the center has are
(Can you see why?). If we remove the square at the center, the remaining area is represented by the equation
. Note that
is the total area of the four rectangles. » Read more