Estimation and Orders of Magnitude

The $\text{order of magnitude}$ of a quantity is the power of ten closest to it. A 70 kg person has mass of order $10^2$ kg; the Earth–Sun distance, $1.5\times 10^{11}$ m, is of order $10^{11}$ m.

A value such as $4\times 10^5$ is usually rounded to $10^5$ or $10^6$ by comparing it with $\sqrt{10}\approx 3.2$: below that, round down; above it, round up.

Two quantities are "three orders of magnitude apart" if their ratio is about $10^3$.

$\text{Masses}$: apple $0.1$ kg; adult $70$ kg; car $1000$ kg; bus $10^4$ kg.

$\text{Lengths}$: atom $10^{-10}$ m; human hair width $10^{-4}$ m; height of a person $1.7$ m; height of a room $2.5$ m; Earth’s radius $6\times 10^6$ m.

$\text{Speeds}$: walking $1.5\mathrm{m}{\mathrm{s}}^{-1}$; sprinter $10\mathrm{m}{\mathrm{s}}^{-1}$; motorway car $30\mathrm{m}{\mathrm{s}}^{-1}$; sound in air $340\mathrm{m}{\mathrm{s}}^{-1}$; light $3\times 10^8\mathrm{m}{\mathrm{s}}^{-1}$.

$\text{Energy and power}$: kettle $2$ kW; light bulb $10$–$60$ W; human resting power $100$ W; food energy per day $10^7$ J.

Break the problem into quantities you can estimate, and write down each assumption — this is where the marks are.

Use rounded values, often to one significant figure, and keep the arithmetic simple.

Give the final answer to one significant figure or as an order of magnitude. Quoting three significant figures from rough inputs implies precision you do not have.

Before trusting a calculated answer, estimate what it should be. A factor of $10^3$ discrepancy usually means a missed prefix, such as grams for kilograms or millimetres for metres.

Watch the powers of ten in particular: a missed $10^{-6}$ from $\mathrm{m}{\mathrm{m}}^2$ or $\mu$F turns a correct method into an answer that is wildly off.

In practical work, estimate expected values before measuring, so you notice immediately if equipment is set up wrongly.