Rounding to Significant Figures (GCSE & IGCSE): Rules, Method and Worked Examples
In GCSE and IGCSE Maths, significant figures are used to show how precise a number is. This introduction explains how to identify the first significant digit, locate the correct rounding position, and decide whether to round up or down. You will also see how the rules apply differently to whole numbers, decimals, and values with leading or trailing zeros. The examples below follow the method examiners expect and highlight the common errors that cost marks.
Significant figures are the digits in a number that carry information about its precision. Not every digit counts. Here are the four rules with real examples.
All non-zero digits are always significant.
3 s.f.Zeros between non-zero digits are significant.
3 s.f.Leading zeros are not significant.
2 s.f.Trailing zeros after a decimal point are significant.
3 s.f.How to Round to Significant Figures: Step by Step
Follow these four steps to round any number to a given number of significant figures.
- Find the first significant digit. This is the first non-zero digit from the left. In \( 0.00958 \) the first significant digit is 9.
- Count to the n-th significant digit. For 3 s.f., move across to the 3rd significant digit. Remember: zeros between non-zero digits do count.
- Look at the next digit. If it is 5 or more, round up. If it is 4 or less, keep the rounding digit the same.
- Rewrite the number. For whole numbers, replace later digits with zero placeholders so the number stays the right size. For decimals, simply trim the extra digits.
Quick example: to round \( 0.007865 \) to 2 s.f., the first significant digit is 7, the second is 8, and the next digit is 6 (5 or more), so round up to get \( 0.0079 \).
Worked Examples
💡 Example 1: Decimals Less Than 1
Round \( 0.007865 \) to 2 significant figures.
\( 0.00\textcolor{#b91c1c}{7}865 \)
\( 0.00\textcolor{#b91c1c}{78}65 \)
What's happening?
Find the first significant digit (\(7\)), then count to the 2nd (\(8\)). Leading zeros are ignored.
\( 0.0078\textcolor{#b91c1c}{6}5 \)
Next digit is \(\textcolor{#b91c1c}{6} \ge 5\), so round up.
\( 0.007865 \approx \textcolor{#b91c1c}{0.0079} \)
What's happening?
The digit after the 8 is 6, which is 5 or more, so round the 8 up to 9.
💡 Example 2: Whole Numbers
Round \( 12{,}784 \) to 3 significant figures.
\( \textcolor{#b91c1c}{1}2{,}784 \)
\( \textcolor{#b91c1c}{12}{,}784 \)
\( \textcolor{#b91c1c}{12{,}7}84 \)
What's happening?
Count the first three significant digits: 1, 2, 7.
\( 12{,}7\textcolor{#b91c1c}{8}4 \)
Next digit is \(\textcolor{#b91c1c}{8} \ge 5\), so round up.
\( 12{,}784 \approx \textcolor{#b91c1c}{12{,}800} \)
What's happening?
The 8 pushes the 7 up to 8. The last two digits become zero placeholders to keep the size of the number right.
💡 Example 3: Zeros in the Middle
Round \( 3.4061 \) to 3 significant figures.
\( \textcolor{#b91c1c}{3}.4061 \)
\( \textcolor{#b91c1c}{3.4}061 \)
\( \textcolor{#b91c1c}{3.40}61 \)
What's happening?
The zero between 4 and 6 is a significant digit, so the first three are 3, 4, 0.
\( 3.40\textcolor{#b91c1c}{6}1 \)
Next digit is \(\textcolor{#b91c1c}{6} \ge 5\), so round up.
\( 3.4061 \approx \textcolor{#b91c1c}{3.41} \)
What's happening?
The 6 bumps the 0 up to 1.
💡 Example 4: When to Keep the Digit
Round \( 0.05243 \) to 3 significant figures.
\( 0.0\textcolor{#b91c1c}{5}243 \)
\( 0.0\textcolor{#b91c1c}{52}43 \)
\( 0.0\textcolor{#b91c1c}{524}3 \)
What's happening?
Leading zeros are ignored. The first three significant digits are 5, 2, 4.
\( 0.0524\textcolor{#b91c1c}{3} \)
Next digit is \(\textcolor{#b91c1c}{3} < 5\), so keep the 4.
\( 0.05243 \approx \textcolor{#b91c1c}{0.0524} \)
What's happening?
Because the next digit is less than 5, the rounding digit stays the same.
🔑 Key Points
- Start counting from the first non-zero digit, not from the decimal point.
- Zeros between non-zero digits are significant.
- Leading zeros are never significant. Trailing zeros after a decimal point are.
- Check the digit immediately after your rounding digit: 5 or more rounds up, 4 or less keeps it the same.
- Whole numbers need zero placeholders: \( 12{,}784 \) to 3 s.f. is \( 12{,}800 \), not \( 128 \).
⚠️ Common Pitfalls
- Starting from the decimal point instead of the first non-zero digit. In \( 0.00485 \), the first significant digit is \(4\), not the zero after the point.
- Dropping necessary zeros. \( 3.40 \) (three s.f.) is not the same as \( 3.4 \) (two s.f.).
- Confusing significant figures with decimal places. \( 0.004 \) has one s.f. but three decimal places.
- Forgetting placeholders on whole numbers. Rounding \( 12{,}784 \) to 3 s.f. gives \( 12{,}800 \), not \( 127 \).
- Rounding twice. Always round directly from the original number, not from an already-rounded version.