Significant Figures

Writing more digits does not make a measurement more accurate: accuracy means closeness to the true value. The aim is to report the answer at an appropriate level of detail, rather than simply copying every digit from the calculator.

Counting significant figures

Start at the first non-zero digit and count from there. The decimal point itself does not count.

The numbers $346$, $0.346$ and $0.0346$ each have three significant figures: the digits 3, 4 and 6. The zeros before the 3 only locate the number's size; they do not count as significant figures.

Zeros need particular care:

• Leading zeros do not count: $0.0075$ has two significant figures.
• Zeros between non-zero digits count: $1.045$ has four significant figures.
• Final zeros after a decimal point count: $5.00$ has three significant figures. Those zeros show that the value is stated to the hundredths place.

Final zeros in whole numbers can be ambiguous. For example, $500$ alone does not clearly show whether its zeros indicate precision or simply place value. Standard form makes the intended precision clear: $5.0\times10^2$ has two significant figures, while $5.00\times10^2$ has three.

Significant figures are not decimal places

Decimal places count all digit positions after the decimal point. Significant figures count from the first non-zero digit instead.

For example, $0.0346$ has four decimal places, but only three significant figures. This distinction matters when a question tells you how to round an answer.

Rounding to a specified number of significant figures

First find the first non-zero digit. Count along to the last digit you need to keep, then examine the next digit:

• If it is 0–4, leave the last retained digit unchanged.
• If it is 5–9, increase the last retained digit by one.

Remove the remaining digits after the decimal point. For a whole number, replace discarded digits with zeros where needed to preserve its size.

For example, $15.22$ to three significant figures is $15.2$: the next digit is 2, so the retained digits stay unchanged. By contrast, $121.45$ to four significant figures is $121.5$, because the next digit is 5.

Rounding can carry into an earlier digit. If that happens, check that the final written answer still shows the requested number of significant figures.

Choosing precision in calculations

Follow any rounding instruction in the question. If it asks for three significant figures, count three significant digits in your final answer, even if the calculator displays many more.

For multiplication and division, when no different instruction is given, a useful rule is to report the result to the same number of significant figures as the measured input with the fewest significant figures. Exact counts and defined conversion factors do not limit the precision in this way.

For addition and subtraction, the corresponding rule concerns decimal places: report the result to the least detailed decimal place among the inputs.

Throughout a multi-step calculation, keep the calculator's unrounded values. Round once, at the end. For instance, a displayed result of $0.34564524$ becomes $0.346$ to three significant figures. Replacing an intermediate value with $0.346$ too early can change the final result.