# Orders of Magnitude

## Key Questions

• 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.

It depends on what you mean by "order of magnitude".

#### Explanation:

In one meaning, it is a number's nearest power of 10.

For example, 9.9 × 10^2 is much closer to ${10}^{3}$ than to ${10}^{2}$, so we say that the order of magnitude is $3$.

By the same argument, 9.9 × 10^-3 is closer to ${10}^{-} 2$ than to ${10}^{-} 3$, so we say that the order is $- 3$.

In another meaning, it is a difference of $1$ unit in the exponent of $10$ in the value of a quantity.

Thus, $2315$ is one order of magnitude larger than $231.5$, and $23.15$ is one order of magnitude less than $231.5$.

If the value of a number decreased from $2315$ to $23.15$, we would say that it decreased by two orders of magnitude or that the values differed by two orders of magnitude.

In that sense, a difference in orders of magnitude is always expressed as a positive number.