Orders of magnitude

An order of magnitude is an approximate measure of the size of a number, equal to the logarithm (base 10) rounded to a whole number. For example, the order of magnitude of 1500 is 3, because 1500 = 1.5 × 10³.

Differences in order of magnitude can be measured on a base-10 logarithmic scale in “decades” (i.e., factors of ten). Examples of numbers of different magnitudes can be found at Orders of magnitude (numbers).

Generally, the order of magnitude of a number is the smallest power of 10 required to represent that number. To work out the order of magnitude of a number ${\displaystyle N}$, the number is first expressed in the following form:

where ${\displaystyle 0.5<a\leq 5}$. Then, ${\displaystyle b}$ represents the order of magnitude of the number. The order of magnitude can be a positive integer, zero, or a negative integer. The table below enumerates the order of magnitude of some numbers in light of this definition:

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