The arithmetic and geometric means

You very well might remember the method for taking an average from your schooling, no matter how long ago that was.  If you have a list of numbers and we call the length of the list n, the average is the sum of the numbers divided by n.

The other name for average in math is arithmetic mean. When used as an adjective, it is pronounced a-rith-MET-ic instead of a-RITH-met-ic. In math, a "mean" is a number that lies somewhere in between the highest and lowest number on a list.

Let us look at the arithmetic mean in relation to another mean, the geometric mean. The simplest means to take are when we only have two numbers on our list. For the average, a and b can be any two numbers but the geometric mean is most useful when both the numbers are positive.

We already know that the average of a and b is (a + b)/2 or ½(a + b). The geometric mean of two numbers is the square root of their product, which I will write here as sqrt(ab). One way to think of the geometric mean is to have a rectangle with sides a and b. A square whose sides are the geometric mean has the same area as the rectangle.

Still dealing with positive numbers, if a = b, then the arithmetic mean is equal to the geometric mean. If they are not equal, then the arithmetic mean will be greater than the geometric mean. There are several ways to prove this. This diagram is one of them.

We first put two line segments of lengths a and b next to each other so the total length is a + b. The average is the midpoint of the long line segment. If we draw a half circle with the average as the radius from the midpoint and draw a perpendicular from the point where the a segment and b segment meet, we can make a right triangle where the hypotenuse length is the average (a + b)/2.

The horizontal leg of the right triangle is a - (a +  b)/2 = (a - b)/2. Since it's a right triangle we know the some of the squares of the legs equals the square of the hypotenuse, and with a little algebraic manipulation we get this.

[(a + b)/2]² =  [(a - b)/2]² + leg²     (write the squares out)

(a² + 2ab + b)²/4 = (a² - 2ab + b)²/4 + leg² (subtract the a² and b² out from both sides)

2ab/4 = - 2ab/4 + leg² (add 2ab/4 to both sides and simplify 4ab/4)

ab =  leg² (square root of both sides)

sqrt(ab) =  leg

The leg of a right triangle must be less than the hypotenuse and that completes our proof in the case where we have two numbers on our list. It is also true when there are more than two numbers on the list, but the proofs become less visual.

Tomorrow, an application of the geometric mean.