Rounding to Significant Figures

Introduction

A calculator shows 0.043 857 142 9 for the concentration of a solution, but your measurements justify only three significant figures. You need to shorten the number without changing its value more than necessary. That is rounding . Rounding looks simple, but it has traps: dropping place-holding zeros changes the size of the number, and rounding too early in a calculation can shift the final answer. This page sets out a reliable method.

Core explanation

The basic method. To round to n significant figures:

1. Count n significant figures from the first non-zero digit. 2. Look at the next digit — the deciding digit . 3. If the deciding digit is 5 or more, increase the last kept digit by one. If it is 4 or less, leave the last kept digit unchanged. 4. Drop the remaining digits, replacing them with zeros where needed to hold the place value.

For example, 0.043 857 to 3 s.f.: the first three significant figures are 4, 3 and 8; the deciding digit is 5, so 8 becomes 9, giving 0.0439.

Keep the size of the number. Rounding must not change the magnitude. Rounding 18 460 to 2 s.f. gives 18 000, not 18. The zeros are placeholders here. To show clearly that only two figures are significant, write 1.8 × 10⁴.

Carrying a 9. When the last kept digit is 9 and must be rounded up, the carry moves left. 2.996 to 3 s.f. becomes 3.00: the 9s roll over, and the trailing zeros must be written to show three significant figures. Writing just 3 would wrongly suggest one significant figure.

Round only at the end. In a calculation with several steps, keep at least one or two extra digits in intermediate values (or keep them in the calculator's memory) and round only the final answer. Rounding each step introduces small errors that can add up and change the last significant figure of the result.

Avoid false precision. Rounding is not about losing information; it is about not inventing information. A result written with too many digits claims a precision that the measurements do not have. Rounding to a justified number of significant figures makes the answer honest.

How many figures to keep? At this level, the usual guide is to give answers to the same number of significant figures as the least precise data used, which is explored further for multiplication and division.

Step-by-step reasoning

1. Identify how many significant figures the answer needs. 2. Mark the last digit to keep and look at the digit after it. 3. Round up for 5 or more; leave alone for 4 or less. 4. Replace dropped digits before the decimal point with zeros, or use scientific notation. 5. Check that the rounded number is close to the original in size.

Visual explanation

Picture a number line from 2.3 to 2.4 with 2.35 marked in the middle. Any value to the right of the middle, such as 2.37, lies closer to 2.4 and rounds up. Any value to the left, such as 2.33, lies closer to 2.3 and rounds down. The halfway point is conventionally rounded up.

Real-world analogy

Rounding is like telling someone the time. If it is 10:58, you might say "about eleven o'clock", not "about ten o'clock". You pick the nearest sensible value, and you do not pretend to know the time to the second when you only glanced at a clock.

Real-world example

Nutrition labels round energy values. A cereal portion might contain 146.7 kcal by calculation, but the label says 147 kcal, or sometimes 150 kcal, because the natural variation between portions makes further digits meaningless.

Why?

Why does rounding too early cause errors? Each rounding step changes the value slightly. When rounded values are multiplied or divided in later steps, those small changes are carried forward and can combine, so the final answer may differ from the correctly calculated one in its last significant figure.

Common misconception

"Rounding 7 650 to 2 s.f. gives 77." This drops the place value and makes the number a hundred times smaller. The correct answer is 7 700 (or 7.7 × 10³). Rounding changes the number of significant figures, never the magnitude.

Worked example

Question: A student calculates the mass of a product as 0.4995 g. Round this to (a) 3 s.f. and (b) 2 s.f.

Reasoning: (a) The first three significant figures are 4, 9, 9; the deciding digit is 5, so the last 9 rounds up. The carry turns 0.499 into 0.500. (b) The first two significant figures are 4 and 9; the deciding digit is 9, so 0.49 rounds up to 0.50.

Answer: (a) 0.500 g, (b) 0.50 g.

Quick check

1. Round 58 362 to 2 significant figures. Answer: 58 000, or 5.8 × 10⁴.

Exam focus

Examiners often ask for answers "to 3 significant figures". Do not lose marks by dropping trailing zeros that are significant, such as writing 0.5 instead of 0.500. Keep full calculator values in the middle of a calculation and round only the final answer.

Advanced insight

The "round half up" rule introduces a slight upward bias when many values ending in exactly 5 are rounded. Scientific software and financial systems often use "round half to even" (banker's rounding): 2.45 rounds to 2.4, and 2.55 rounds to 2.6. Over large data sets, this removes the bias. Computers also store decimals in binary, so a value such as 2.675 may be held as slightly less than 2.675 and round down unexpectedly.

Summary

To round to n significant figures, look at the digit after the nth figure: round up for 5 or more, leave alone for 4 or less. Keep place-holding zeros so the size is unchanged, or use scientific notation. Write significant trailing zeros produced by rounding, round only at the end of a calculation, and avoid false precision.

Practice questions

1. Round 3.14159 to 3 significant figures. Answer: 3.14. 2. Round 0.006 748 to 2 significant figures. Answer: 0.0067, because the deciding digit is 4. 3. Round 9.97 to 2 significant figures and explain the result. Answer: 1.0 × 10¹; the deciding digit 7 rounds 9.9 up, the carry moves into the tens place, and scientific notation shows that the result still has two significant figures. 4. Round 264 500 to 3 significant figures in scientific notation. Answer: 2.65 × 10⁵. 5. Why should intermediate values in a calculation not be rounded? Answer: Early rounding introduces small errors that are carried forward and can change the last significant figure of the final answer.