Number · Number and Numeration Systems · Lesson 5
Significant Figures and Decimal Places
A tape measure, a kitchen scale at Kejetia market and a thermometer at the clinic all give readings that are only accurate to a certain number of digits. Scientists, engineers and BECE examiners describe this accuracy using decimal places and significant figures. In this lesson you will learn which digits are significant (and when a zero counts), round decimals to a given number of decimal places or significant figures, and work out what an actual measurement could have been.
By the end you will be able to
- ✔Express decimals correct to a given number of decimal places.
- ✔Identify the significant figures in a decimal and explain when zero is significant.
- ✔Express decimals correct to a given number of significant figures.
- ✔Find the range of possible actual values of a measurement given correct to a number of decimal places.
Key words
- Decimal places (d.p.)
- The number of digits after the decimal point. 745.97 has 2 decimal places.
- Significant figures (s.f.)
- The digits that show how accurate a number is, counted from the first non-zero digit. 0.360 has 3 significant figures.
- Leading zeros
- Zeros before the first non-zero digit, e.g. the zeros in 0.00234. They are never significant.
- Trailing zeros
- Zeros at the end of a decimal, after the point, e.g. the last 0 in 0.360. They are significant.
- Lower and upper bound
- The smallest value and the value just above the largest that a rounded measurement could have come from.
Curriculum reference
Content standard: B7.1.1.1 Demonstrate understanding and the use of place value for expressing quantities recorded as base ten numerals as well as rounding these to given decimal places and significant figures.
Indicator: Express decimal numerals to given significant and decimal places.
Starter
Musa's table
Musa measured his teacher's table with a metre rule and wrote "0.76 m, correct to 2 decimal places". His friend Fatima measured the same table and got 0.758 m. Kwame got 0.762 m. Did someone make a mistake? Not necessarily! Every one of these readings becomes 0.76 m when rounded to 2 decimal places. By the end of this lesson you will be able to say exactly which readings Musa might really have obtained.
1. Correcting to decimal places
Decimal places are counted from the decimal point to the right. "Correct to 2 decimal places" means the same as "to the nearest hundredth", and "correct to 1 decimal place" means "to the nearest tenth". You use the same deciding-digit rule as in the last lesson: look at the next digit; 0–4 keep, 5–9 add 1.
| Correct to… | Keep up to | Deciding digit | Answer |
|---|---|---|---|
| 3 d.p. | 745.967 | 4 → keep | 745.967 |
| 2 d.p. | 745.96 | 7 → add 1 | 745.97 |
| 1 d.p. | 745.9 | 6 → add 1 (carry) | 746.0 |
Look at the last row. Adding 1 to the tenths digit 9 makes 10, so we carry 1 into the ones: 745 becomes 746. Because the question asks for 1 decimal place, we must still write one digit after the point: 746.0, not 746.
2. Which digits are significant?
The number of decimal places depends on where the decimal point is. Significant figures are different: they count the digits that carry real information about the size and accuracy of a number, wherever the point is. Learn these four rules — they explain exactly when zero is significant.
- All non-zero digits are significant. 4.76 has 3 s.f.
- Zeros between non-zero digits are significant. In 7.021 the 0 is sandwiched between 7 and 2, so 7.021 has 4 s.f.
- Leading zeros are NOT significant. In 0.00234 the zeros before the 2 only show where the decimal point is. 0.00234 has 3 s.f.
- Trailing zeros after the decimal point ARE significant. In 0.360 the final 0 tells us the measurement is accurate to the thousandths place, so 0.360 has 3 s.f. (the leading 0 before the point does not count).
| Number | Significant digits | Number of s.f. | Rule used |
|---|---|---|---|
| 0.360 | 3, 6, 0 | 3 | trailing zero after the point counts |
| 7.021 | 7, 0, 2, 1 | 4 | sandwiched zero counts |
| 0.0050 | 5, 0 | 2 | leading zeros do not count; trailing zero does |
| 30.04 | 3, 0, 0, 4 | 4 | sandwiched zeros count |
| 0.00234567 | 2, 3, 4, 5, 6, 7 | 6 | leading zeros do not count |
▶ Notice that counting starts at the first non-zero digit; the leading zeros only hold the place of the decimal point.
3. Correcting to significant figures
- 1Find the first non-zero digit. This is the 1st significant figure.
- 2Count along to the required number of significant figures (zeros after the first non-zero digit now count).
- 3Look at the next digit (the deciding digit): 0–4 keep, 5–9 add 1 to the last kept digit.
- 4Drop the digits after it (for decimals) — but keep any leading zeros, and keep trailing zeros that are part of the required s.f.
| Number | 3 s.f. | 4 s.f. | 6 s.f. |
|---|---|---|---|
| 0.00234567 | 0.00235 | 0.002346 | 0.00234567 |
| 84.40995000 | 84.4 | 84.41 | 84.4100 |
For 84.40995000 to 6 s.f., the digits kept are 8, 4, 4, 0, 9, 9 and the deciding digit is 5. Adding 1 to the last 9 makes it 10, which carries into the next 9, which carries into the 0: 84.4099 becomes 84.4100. Both final zeros must be written, because the question asked for six significant figures.
4. Measurements: what could the real value be?
When a measurement is given correct to some decimal places, the real value could be anything that rounds to it. Musa's answer is 0.76 m correct to 2 d.p. Which numbers round to 0.76? Those from halfway below (0.755) up to — but not including — halfway above (0.765). We write 0.755 ≤ length < 0.765.
▶ The green band shows every actual length that would be written as 0.76 m to 2 decimal places.
If Musa's metre rule could read to the nearest millimetre (3 d.p. in metres), his actual reading must have been one of 0.755, 0.756, 0.757, 0.758, 0.759, 0.760, 0.761, 0.762, 0.763 or 0.764 m. So Fatima's 0.758 m and Kwame's 0.762 m are both possible.
Worked examples
Try each one yourself first, then open the solution and compare every step.
Example 1
When is zero significant?
Show the solution
- Step 1(a) The 0 before the point is a leading zero (not significant). 3 and 6 are significant, and the trailing 0 after the point is significant: 3 s.f.
- Step 2(b) 7, 2 and 1 are significant, and the 0 lies between non-zero digits: 4 s.f.
- Step 3(c) The zeros before the 5 are leading zeros (not significant). The 5 and the trailing 0 count: 2 s.f.
Answer: (a) 3 s.f. (b) 4 s.f. (c) 2 s.f.
Example 2
Decimal places
Show the solution
- Step 1(a) Keep 745.967; deciding digit 4 → keep: 745.967.
- Step 2(b) Keep 745.96; deciding digit 7 → 6 becomes 7: 745.97.
- Step 3(c) Keep 745.9; deciding digit 6 → 9 becomes 10, carry 1 to the ones: 746.0.
Answer: (a) 745.967 (b) 745.97 (c) 746.0
Example 3
Significant figures of a small decimal
Show the solution
- Step 1The first significant figure is 2 (the zeros before it do not count).
- Step 2(a) Keep 2, 3, 4; deciding digit 5 → 4 becomes 5: 0.00235.
- Step 3(b) Keep 2, 3, 4, 5; deciding digit 6 → 5 becomes 6: 0.002346.
- Step 4(c) The number has exactly 6 significant figures already: 0.00234567.
Answer: (a) 0.00235 (b) 0.002346 (c) 0.00234567
Example 4
Carrying and keeping zeros
Show the solution
- Step 1(a) Keep 8, 4, 4; deciding digit 0 → keep: 84.4.
- Step 2(b) Keep 8, 4, 4, 0; deciding digit 9 → 0 becomes 1: 84.41.
- Step 3(c) Keep 8, 4, 4, 0, 9, 9; deciding digit 5 → add 1 to the last 9. 9 + 1 = 10, carry; 9 + 1 = 10, carry; 0 + 1 = 1.
- Step 4So 84.4099 becomes 84.4100 — keep both zeros to show 6 s.f.
Answer: (a) 84.4 (b) 84.41 (c) 84.4100
Example 5
Musa's measurement
Show the solution
- Step 1Half of the last place (0.01) is 0.005.
- Step 2Lower bound: 0.76 − 0.005 = 0.755 (this rounds up to 0.76, so it is included).
- Step 3Upper bound: 0.76 + 0.005 = 0.765 (this would round to 0.77, so it is NOT included).
- Step 4So 0.755 m ≤ actual length < 0.765 m. To 3 d.p. the possible readings are 0.755, 0.756, …, 0.764 m.
Answer: Any reading from 0.755 m up to (but not including) 0.765 m, e.g. 0.755, 0.758, 0.762 or 0.764 m.
Summary
- Decimal places are counted from the decimal point; '2 d.p.' means the nearest hundredth.
- Significant figures are counted from the first non-zero digit.
- Zero is significant when it lies between non-zero digits (7.021) or is a trailing zero after the decimal point (0.360); leading zeros (0.00234) are never significant.
- Use the deciding digit to round (0–4 keep, 5–9 add 1) and keep any zeros needed to show the required accuracy (746.0, 84.4100).
- A measurement such as 0.76 m (2 d.p.) could be any value from 0.755 m up to, but not including, 0.765 m.
Practice: 15 questions
5 easy, 5 medium, 5 hard. Each one is marked straight away, with a full worked solution.
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