Order-of-magnitude estimation

An order of magnitude describes the approximate scale of a quantity using a power of ten. For example, a length of about $10^3,\text{m}$ is on the scale of a thousand metres. Each step from one power of ten to the next represents a factor of ten, so quantities whose orders of magnitude differ by one are roughly ten times different in scale.

An order-of-magnitude estimate identifies a suitable power of ten rather than trying to give a precise value. The units must be stated: $10^3,\text{m}$ and $10^3,\text{km}$ describe different lengths.

Choosing an order of magnitude

First judge a plausible value from the available information, including its units. Then express it using a power of ten and select the nearest order of magnitude. Powers of ten are compared on a multiplicative, or logarithmic, scale. For example, $2 \times 10^3,\text{m}$ is closer in scale to $10^3,\text{m}$ , while $7 \times 10^3,\text{m}$ is closer in scale to $10^4,\text{m}$ .

Combining estimates

A physical relationship can be used to estimate a quantity from other estimated values. When multiplying powers of ten, add their exponents; when dividing them, subtract the exponent in the denominator from the exponent in the numerator.

10^a \times 10^b = 10^{a+b}, \qquad \frac{10^a}{10^b} = 10^{a-b}

For example, suppose a journey is estimated to cover $3 \times 10^3,\text{m}$ in $2 \times 10^2,\text{s}$ . Using average speed as distance divided by time gives:

\text{average speed} \approx \frac{3 \times 10^3\,\text{m}}{2 \times 10^2\,\text{s}} = 1.5 \times 10^1\,\text{m/s}

The estimated speed is therefore of order $10^1,\text{m/s}$ . The result is approximate because the distance and time were themselves estimates. Different plausible estimates or assumptions could give a different result.