## 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?

David Mann says

Find geometric mean of the numbers 1.618 and 0.618.

That’s easy when you run it because 1.618*0.618 = 1. Square root of 1 is 1. I’ve never understood phi geometrically but this is quite an interesting little property.

Zack says

Fascinating, and I don’t get it .. but … but I am wondering if the Golden mean is the invention of the Geometric mean, almost like an Echo, or second rainbow of each other