Math

Mathsy Shortcuts

I have written this page to add small shortcuts and insights in reasoning that can be useful.

Geometric mean is the (arithmetic) mean of magnitudes

For a set of positive numbers $ A = \{ x_1, x_2, … ,x_n \} $ the arithmetic mean is defined as $\frac{1}{n} \sum {x_i} $ whereas the geometric mean is defined as $ \sqrt[n]{ \prod x_i} $, resulting in:

$ A $ Arithmetic Mean Geometric Mean
$ \{ 4,9 \} $ $6.5$ $ 6 $
$ \{ 1,2,4,8,16 \} $ $6.2$ $ 4 $

The Geometric Mean becomes obvious if we write all numbers as magnitudes:

$ A $ Arithmetic Mean Geometric Mean
$ \{ 2^2,2^{3.1699} \} $ $ 2^{2.7004} $ $ 2^{2.58496} $
$ \{ 2^0,2^1,2^2,2^3,2^4 \} $ $ 2^{2.632268} $ $ 2^2 $

Note that $2.58496$ is just the mean of $ \{ 2 , 3.1699 \}$, and for the second line $2$ is the mean of $ \{ 0,1,2,3,4 \}$. I have written them base $2$ but any magnitude would suffice.