## 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.

## Arithmetic means

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.

b is the square root of a times c.

b = √( a * c )

4 = √( 2 * 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:

Yellow line = 2

Blue line = 4

White line = 8

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

this gives Phi its unusual properties:

B = √ ( A * C ) AND B + C = A

1 = √ ( Phi * 1/Phi ) AND 1 + 1/Phi = Phi

1 = √ ( 1.618 … * 1/1.618 … )

AND

1 + 1 / 1.618 … = 1.618 …

Note also that:

1 / 1.618 … = 0.618… = 1.618 … – 1

1 / Phi = 0.618… = Phi – 1

Heidi says

How can the golden ratio be expressed as the geometric mean?