Significant figures: how to round and estimate with them

What are significant figures?

Significant figures are the digits in a number that tell you how big it is, starting from the first digit that isn't zero. In 4,872, the 4 is the first significant figure, the 8 is the second, the 7 the third and the 2 the fourth.

Three rules decide which digits count:

• The first significant figure is the first digit that isn't zero. In 0.04567 it's the 4. The zeros in front only show where the decimal point is.
• Zeros between other digits count. In 3,047 the 0 is the second significant figure.
• Zeros at the end of a decimal count if they're written. 2.00 has three significant figures, because someone chose to show those zeros.

Number	1st significant figure	2nd significant figure
4,872	4 (worth 4,000)	8 (worth 800)
3,047	3 (worth 3,000)	0
0.04567	4 (worth 0.04)	5 (worth 0.005)
0.849	8 (worth 0.8)	4 (worth 0.04)

Significant figures are different from decimal places. Decimal places always count from the decimal point. Significant figures count from the first non-zero digit, wherever it is, so they work for big numbers and small ones alike.

How to round to significant figures

To round to significant figures, use the same rule as any other rounding, starting from the first non-zero digit:

1. Find the first significant figure, then count along to the number of significant figures you need.
2. Look at the next digit along. If it's 5 or more, round up. If it's 4 or less, keep the last digit as it is.
3. Fill the places up to the decimal point with zeros, so the number stays the same size. Don't write zeros after the last significant figure in a decimal.

This is ordinary rounding, which children learn at primary school. Rounding 4,872 to 1 significant figure is the same as rounding it to the nearest 1,000. The new part is working out which place to round to. Our guide to rounding covers the basics, including what to do with a 5.

How to round to 1 significant figure

To round to 1 significant figure, keep only the first non-zero digit, rounded using the digit after it.

Worked example: round 4,872 to 1 significant figure. The first significant figure is the 4, in the thousands. The next digit is 8, so round up. The answer is 5,000.

More examples:

• 827 to 1 significant figure is 800 (the next digit is 2).
• 38,562.7 to 1 significant figure is 40,000 (the next digit is 8, so 3 becomes 4).
• 0.849 to 1 significant figure is 0.8 (the next digit is 4).
• 0.0372 to 1 significant figure is 0.04 (the next digit is 7).

How to round to 2 significant figures

To round to 2 significant figures, keep the first two digits from the first non-zero digit, and use the third to decide which way to round.

Worked example: round 7,982 to 2 significant figures. The first two significant figures are 7 and 9, so the number is between 7,900 and 8,000. The next digit is 8, so round up. The answer is 8,000.

8,000 looks as if it has only 1 significant figure, but it's the correct answer to 2 significant figures. The 9 rounded up and carried into the thousands.

Worked example: round 0.04567 to 2 significant figures. The first significant figure is the 4, so the first two are 4 and 5, in the hundredths and thousandths. The next digit is 6, so round up. The answer is 0.046.

Two more to watch: 3,047 to 2 significant figures is 3,000, and 0.9968 to 2 significant figures is 1.0. Keep the zero after the point in 1.0, because it's the second significant figure.

How to round to 3 significant figures

To round to 3 significant figures, keep three digits from the first non-zero digit, and look at the fourth.

• 4,872 to 3 significant figures is 4,870 (the next digit is 2).
• 38,562.7 to 3 significant figures is 38,600 (the next digit is 6).
• 3,047 to 3 significant figures is 3,050 (the next digit is 7, so 304 becomes 305).
• 0.003049 to 3 significant figures is 0.00305.
• 2.0049 to 3 significant figures is 2.00. Both zeros stay, because they're significant.

Estimating by rounding to 1 significant figure

To estimate an answer, round every number to 1 significant figure and work out the easier sum in your head. The national curriculum asks secondary pupils to estimate answers by rounding, and to check calculations, including calculator answers, this way.

Worked example: estimate 4.87 × 31.2. Round to 1 significant figure: 4.87 is about 5, and 31.2 is about 30. Then 5 × 30 = 150. The calculator answer is 151.944, so it looks right.

Worked example: estimate 612 ÷ 0.48. Round to 600 and 0.5. Then 600 ÷ 0.5 = 1,200. Dividing by a number less than 1 makes the answer bigger, and the exact answer is 1,275.

Write an estimate with "is about" or the ≈ sign, not an equals sign, because it isn't exact. Dividing decimals and multiplying decimals help with the steps once the numbers are rounded.

Error intervals

An error interval gives the range of values a rounded number could have come from. Key stage 3 introduces this with inequality notation, and the GCSE content includes error intervals due to rounding and truncation for all students, so they're on both tiers.

If a number rounded to 2 significant figures is 4,900, it could have been anything from 4,850 up to, but not including, 4,950:

4,850 ≤ x < 4,950

The lower end is included, because 4,850 rounds up to 4,900. The upper end isn't, because 4,950 would round up to 5,000. If 4,800 came from truncating to 2 significant figures (cutting the other digits off rather than rounding), the interval is 4,800 ≤ x < 4,900. Using these limits in calculations, called upper and lower bounds, is on the higher tier only.

Common mistakes with significant figures

• Counting zeros at the front. 0.0372 to 1 significant figure is 0.04, not 0.
• Dropping the zeros in whole numbers. 4,872 to 1 significant figure is 5,000, not 5. The zeros keep it the right size.
• Adding extra zeros to decimals. 0.04567 to 2 significant figures is 0.046, not 0.04600.
• Mixing up significant figures and decimal places. 38,562.7 to 1 decimal place is 38,562.7, but to 1 significant figure it's 40,000.
• Losing a significant zero. 0.9968 to 2 significant figures is 1.0, and 2.0049 to 3 significant figures is 2.00.
• Rounding twice. Round from the original number in one go, not 4,872 to 4,870, then to 4,900, then to 5,000.

How to help at home

• Round real numbers. Take a football crowd, a house price or a town's population and ask your child to round it to 1, 2 and 3 significant figures.
• Estimate before using a calculator. For a shopping bill or a recipe, round to 1 significant figure first, then compare with the exact answer.
• Point to the first significant figure. For any number, ask "which digit is the first one that counts, and what is it worth?" Place value and decimals are the foundations here.

When do children learn significant figures?

Significant figures are taught at secondary school. The national curriculum says that in Year 6 children round answers to the nearest 10, 20 or 50, but not to a number of significant figures (see Year 6 maths). Rounding to significant figures and estimating with them are in the key stage 3 programme of study, which covers Years 7 to 9 as one key stage: schools decide when each topic is taught, so ask the school for its scheme of work. They come up again at GCSE, on both tiers, along with error intervals. Significant figures are also used in standard form. The KS3 maths guide covers the rest of the key stage.

Significant figures questions to try

1. Round 6,381 to 1 significant figure.
2. Round 0.0729 to 1 significant figure.
3. Round 25,446 to 2 significant figures.
4. Round 0.684 to 2 significant figures.
5. Round 3,961 to 3 significant figures.
6. Estimate 8.9 × 52.3 by rounding each number to 1 significant figure.

Answers

1. 6,000
2. 0.07
3. 25,000
4. 0.68
5. 3,960
6. About 450 (9 × 50 = 450; the exact answer is 465.47)