# How do I find the order of magnitude of a number?

Oct 2, 2015

The mathematical explanation is to take the base ten log of a number and floor it, however, that's just a fancy math way of saying count the digits before the decimal marker then take 1.

Oct 4, 2015

Orders of magnitude are approximately common logarithms of ratios of numbers, typically rounded to the nearest integer below.

If your number already represents a ratio, then take its common log.

#### Explanation:

If one number is $10$ times another, then we can say that it is an order of magnitude larger.

Orders of magnitude represent a comparison of scale, for example, $1 \text{kg}$ is three orders of magnitude larger than $1 \text{g}$

The "order of magnitude of $500$" is meaningless except in comparison to another number. The order of magnitude of the ratio $500 : 1$ is meaningful, typically by saying that one thing is $2$ orders of magnitude larger than another thing.

Note that orders of magnitude try not to exaggerate the scale, choosing to express that $\frac{500}{1} > {10}^{2}$ rather than the fact that $\log \left(500\right) \approx 2.69897$ so $\frac{500}{1} \approx {10}^{2.69897}$

If we are faced with a ratio like $900 : 1$, then we would probably say that something was "roughly three orders of magnitude larger than" another thing.