What is an order of magnitude?

Definition

The order of magnitude of a number is the power of 10 closest to that number. It gives a quick sense of the size of a quantity, without worrying about the precise digits after the decimal point.

How to determine it

1. Write the number in scientific notation: a x 10^n.
2. If a < 5, the order of magnitude is 10^n.
3. If a >= 5, the order of magnitude is 10^(n+1).

This simplified rule (threshold at 5) is the one used in middle school.

Examples

• 420 = 4.2 x 10^2 -> a = 4.2 < 5 -> order of magnitude = 10^2 = 100
• 780 = 7.8 x 10^2 -> a = 7.8 >= 5 -> order of magnitude = 10^3 = 1000
• 0.03 = 3 x 10^-2 -> order of magnitude = 10^-2

What it's used for

Estimating an order of magnitude allows you to quickly check whether a calculation result is plausible, without doing a precise calculation. It's a useful reflex in physics for spotting an error, for example a factor of 1000 forgotten in a unit conversion.

Classic pitfall

The order of magnitude is not the first digit of the number, it is a power of 10. Saying that the order of magnitude of 780 is 7 is wrong: it is 1000 (10^3), not 7 or 7.8.