Significant Figures and Order of Magnitude

What the digits in a measurement tell us

A calculator can display many digits, but that does not mean a measurement supports all of them. Reporting an appropriate number of significant figures helps us communicate a value without suggesting more detail than the measurements justify.

Significant figures are counted from the first non-zero digit, reading from left to right. For example, 346, 0.346 and 0.0346 each have three significant figures. The zeros before the 3 in the decimal values locate the digits; they do not count as significant figures.

Zeros between non-zero digits do count. In 3097, the zero is the second significant figure, so the number has four significant figures. Zeros at the end of a decimal also count: 4.00 has three significant figures, whereas 4 has one. These values are numerically equal, but they communicate different levels of detail.

Significant figures are not the same as decimal places. Decimal places count every position after the decimal point. For example, 0.0346 has four decimal places but only three significant figures.

Rounding to significant figures

First locate the first non-zero digit. Count along to the last significant figure you need to keep, then inspect the next digit. If that next digit is 0–4, leave the retained digit unchanged. If it is 5–9, increase the retained digit by one.

For example, to round 0.002956314 to three significant figures, retain the digits 2, 9 and 5. The next digit is 6, so round upwards to 0.00296. The zeros before the 2 remain because they preserve the size of the number.

For large whole numbers, replace discarded digits with zeros to preserve place value. Thus 34 568 becomes 35 000 to two significant figures, not 35. Standard form can make the intended number of significant figures clearer: $3.5 \times 10^4$ explicitly shows two.

Rounding can carry into another column. To three significant figures, 3.997 rounds to 4.00. Both zeros must remain to show the requested three significant figures.

Choosing how many figures to report

If a question specifies a number of significant figures, use it. Otherwise, consider the detail in the measurements and the purpose of the answer. A result calculated from approximate measurements should not be reported with a long string of calculator digits as though all were meaningful.

Keep the full calculator value during intermediate steps and round once, at the end. Rounding each intermediate result can introduce extra rounding error, especially when that result is used in further calculations. An appropriate final presentation does not require throwing away useful digits while you are still working.

Order of magnitude describes the scale

An order of magnitude describes the approximate size of a quantity using a power of ten. For example, 700 J has an order of magnitude of $10^3$ J: its scale is hundreds to around a thousand joules, rather than millions of joules.

A factor of ten is one order of magnitude; a factor of $10^3$, or 1000, is three orders of magnitude. This lets us compare very different sizes without writing out many zeros. The units matter: convert quantities to the same units before comparing their orders of magnitude.

Making an order-of-magnitude calculation

Start with sensible approximate values, state your assumptions and combine them using the relevant equation. The aim is to establish the scale of the answer, not a highly detailed result.

For example, estimate the energy transferred when a person climbs a flight of stairs. Take the person's mass as about 70 kg, gravitational field strength as about 10 N/kg and the vertical rise as about 3 m. Their weight is approximately $70 \times 10 = 700$ N. Using energy transferred = force × distance gives:

$$E \approx 700 \times 3 = 2100\text{ J}$$

This is roughly 2000 J, with an order of magnitude of $10^3$ J. The estimate depends on the person and the staircase, but it establishes a useful scale.

If a more detailed calculation gave $10^6$ J, it would be about three orders of magnitude larger than this estimate. That is a reason to check the inputs, unit conversions and powers of ten. An estimate is a check on plausibility, not proof that every step of a calculation is correct.

Significant figures

• Count from the first non-zero digit.
• Leading zeros do not count; zeros between significant digits do.
• Trailing decimal zeros count.
• Inspect the next digit: 0–4 rounds down; 5–9 rounds up.
• Preserve place value and the requested number of figures.
• Follow any stated rounding instruction; otherwise use the measurement context. Round only the final result.

Order of magnitude

Express the approximate scale as a power of ten, with a unit.

• One order of magnitude: factor of $10$.
• Three orders of magnitude: factor of $$.