Means

What do we mean by mean?

Math isn't tough, but it can be mean.  The term "mean" in mathematics simply reflects a specific relationship of one number as the middle point of two extremes.

The arithmetic mean of 2 and 6 is 4, as 4 is equally distant between the two in addition:

2 + 2 = 4
and
4 + 2 = 6

For the arithmetic mean (b) of two numbers (a) and (c):

b = ( a + c ) / 2

4 = ( 2 + 6 ) / 2

The arithmetic mean is thus the simple average between two numbers. 

Geometric means

The geometric mean is similar, but based on a common multiplier that relates the mean to the other two numbers. As an example, the geometric mean of 2 and 8 is 4, as 4 is equally distant between the two in multiplication:

2 * 2 = 4
and
4 * 2 = 8

So 2 is to 4 as 4 is to 8.

The Golden Mean

The Golden Mean is a very specific geometric mean.  In the geometric mean above, we see the following lengths of line segments on the number line:

Here, 2 x 2 = 4 and 4 x 2 = 8, but 2 + 4 = 6, not 8.  The Golden Mean imposes the additional requirement that the two segments that define the mean also add to the length of the entire line segment:

This occurs only at one point, which as you can see above is just a little less than 5/8ths, or 0.625.  The actual point of the Golden Mean is at 0.6180339887..., where:

A is to B as B is to C
AND
B + C = A