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.