3.2

Rounding Off Numbers to Specified Significant Figures

Activity 1

Objective: To determine the appropriate degree of accuracy of numbers.

1. The exact thickness of a piece of glass is given as 0.004 503 m.

(a) Round off the thickness of the glass to 2 decimal places.

= m

(b) Would the rounded value in (a) make sense to a handyman? Why?

自评：

m （或 mm 单位）

2. The population of Singapore was 5 791 901 at a particular point in time in 2018.

(a) Give a meaningful estimation:

(b) Why is it a meaningful estimation?

自评：

From Activity 1, a number should be rounded off appropriately so that it is more practical. The desired degree of accuracy is not simply a matter of decimal places. We must recognise the important digits — the significant figures (s.f.).

In a number, significant figures (s.f.) are the important digits used to express the degree of accuracy that is appropriate. When a number is given to more significant figures, it is more accurate.

In a number, the most significant digit is the first non-zero digit when reading from left to right. This digit is the first significant figure. The digit to its right is the second s.f., and so on.

The 5 Rules to Identify Significant Figures

RULE 1

All non-zero digits are significant.

54 332 → 5 s.f. 0.16 → 2 s.f.

RULE 2

All zeros between non-zero digits are significant.

100 009 → 6 s.f. 0.040 05 → 4 s.f.

RULE 3

For any integer, the zeros at the end may or may not be significant. This depends on how the number is approximated.

500 → 1 s.f. (correct to nearest 100)
500 → 2 s.f. (correct to nearest 10)

RULE 4

For any decimal, all zeros before the first non-zero digit are NOT significant.

0.0008 → 1 s.f. 0.011 → 2 s.f.

RULE 5

For any decimal, zeros after a non-zero digit ARE significant.

1.60 → 3 s.f. 0.070 00 → 4 s.f.

📖 WORKED EXAMPLE 6

State the number of significant figures in each number.

(a)(i) 72 301 (ii) 5.038 (iii) 0.004 030 0 (b) 49 000

Solution

(a)(i) 72 301 → Rule 1 + Rule 2 (zero between) → 5 s.f.

(a)(ii) 5.038 → Rule 1 + Rule 2 → 4 s.f.

(a)(iii) 0.004 030 0 → Rules 1, 2, 5 (trailing zeros after non-zero are significant) → 5 s.f.

(b) 49 000 — the zeros may or may not be significant:

49 372 = 49 000 (nearest 1000) → 2 s.f.
49 016 = 49 000 (nearest 100) → 3 s.f.
48 996 = 49 000 (nearest 10) → 4 s.f.
49 000.2 = 49 000 (nearest integer) → 5 s.f.

So 49 000 can have 2, 3, 4 or 5 s.f. depending on how it was approximated.

✏️ TRY IT YOURSELF 6

State the number of significant figures.

(a)(i) 3847 = s.f. (ii) 3.120 = s.f. (iii) 0.050 060 = s.f.

(b)(i) 130 000 = s.f. (ii) 7900 = s.f.

When we round off numbers to a specified number of significant figures,

look for the specified significant figure starting from the left,
then consider the next digit to its right. If 5 or more, round up; else round down.

📖 WORKED EXAMPLE 7

Round off

(a) 60 220 to 2 s.f., (b) 0.008 101 to 3 s.f., (c) 89.950 to 3 s.f.

Solution

(a) 2nd s.f. is 0; digit to right is 2 (<5) → round down. 60 220 = 60 000 (correct to 2 s.f.).

(b) 3rd s.f. is 1; digit to right is 0 (<5) → round down. 0.008 101 = 0.008 10 (correct to 3 s.f.).

✏️ TRY IT YOURSELF 7

Round off:

(a) 70 049 to 3 s.f. = (b) 0.070 185 to 3 s.f. =

📖 WORKED EXAMPLE 8

Mr Singh bought a car for $128 175. Round off the price of the car to (a) 3 s.f., (b) 4 s.f.

Solution

(a) 3rd s.f. = 8 (in "128"); digit to right is 1 (<5) → round down. = $128 000 (correct to 3 s.f.).

(b) 4th s.f. = 1 (in "1281"); digit to right is 7 (≥5) → round up. = $128 200 (correct to 4 s.f.).

NOTE: We can say that the price of the car is about $128 000.

✏️ TRY IT YOURSELF 8

Mrs Tan spent $17 345 on a watch.

(a) 3 s.f. = $ (b) 4 s.f. = $

📖 WORKED EXAMPLE 9

The total number of passengers who arrived in Singapore by air in September 2017 was 2 472 650. Find the average number per day, correct to 3 s.f.

Solution

Avg = 2 472 650 ÷ 3030 days in September.
= 82 421.666 …
= 82 400correct to 3 s.f.

NOTE: The average number of passengers per day in September 2017 was about 82 400.

✏️ TRY IT YOURSELF 9

Population density = people per km². In June 2017, Singapore population ≈ 5.61 million; land area ≈ 721.50 km². Find population density correct to 2 s.f.

Density ≈ people/km²

📝 PRACTICE EXERCISE 3.2

BASIC MASTERY

State the number of significant figures in each.

(a) 38 = s.f. (b) 30 001 = s.f. (c) 15.340 = s.f.

(d) 0.024 908 0 = s.f. (e) 6800 (to nearest integer) = s.f. (f) 301 000 (to nearest 100) = s.f.
Round off to the number of s.f. given in brackets.

(a) 13.67 (1 s.f.) = (b) 0.0392 (1 s.f.) = (c) 69 352 (3 s.f.) =

(d) 13.047 (3 s.f.) = (e) 89.9999 (4 s.f.) = (f) 8.004 036 25 (5 s.f.) =

INTERMEDIATE

(a) Convert 6/7 to a decimal.

=

(b) Express the answer in (a) correct to 4 s.f.:

=
Evaluate −4.937 + (−23.025) correct to 3 s.f.

=
(a) Evaluate 1 1/4 × 2/9 + (−2 1/2) × (−1/6) (exact value).

=

(b) Express as a decimal correct to 4 s.f.

=

(c) Find the exact value of the square root of the answer in (a).

=
Evaluate 0.049 23 / (∛23.56 − √13.67) correct to 2 s.f.

=
In 2018, the number of people employed in Singapore was 3 715 800.

(a) How many significant figures? s.f.

(b) Write this number to 3 s.f.:
In 2017, the total land area of Hong Kong was 1106 km² and the population was about 7 392 000. Find the average number of people who lived on 1 km² of the land. Give your answer correct to 3 s.f.

=
The numbers of private cars in Singapore in 2016 and 2017 were 504 160 and 502 187 respectively. The number of resident households in 2017 was 1 289 900. Give answers correct to 3 s.f.

(a) Decrease in private cars in 2017 =

(b) Average cars per household in 2017 =

ADVANCED

Round off 0.093 800 to all possible numbers of significant figures. What is the largest value of all the possible approximations?

Largest =

💡 思考后展开详细 breakdown

1 s.f. → 0.09
2 s.f. → 0.094
3 s.f. → 0.0938
4 s.f. → 0.093 80
5 s.f. → 0.093 80（即原数）
Largest = 0.094。
John's height is measured as 162.80 cm.

(a) How many significant figures are there in this number? s.f.

(b)(i) Convert his height to metres: m (ii) s.f. in metres: s.f.

(b)(iii) Round off the metres answer to 3 s.f.: m

(c) When changing units from cm to m, what happens to the degree of accuracy? Why?

ENDegree of accuracy does NOT change. Changing units (cm ↔ m) just rescales the number — the same physical measurement is preserved, and the same digits are significant. 162.80 cm and 1.6280 m both have 5 s.f.

⏱ 15 秒后查看中文版

中文精度不变。单位变化（cm ↔ m）只是数值的缩放，物理测量本身没有变。162.80 cm 和 1.6280 m 都是 5 位有效数字。

自评：
[OPEN] A number is rounded off to 0.0506, correct to 3 s.f. Give 3 possible actual values.

ENRange: [0.050 55, 0.050 65). Examples: 0.050 55, 0.050 60, 0.050 64. Any value in this range rounds to 0.0506 at 3 s.f.

⏱ 15 秒后查看中文版

中文范围：[0.050 55, 0.050 65)。例：0.050 55、0.050 60、0.050 64。任何在这个区间内的数 3 s.f. 都是 0.0506。

自评：
The population of Singapore in 2019 was 5 900 000, correct to 3 s.f.

(a) Largest possible population =

(b) Smallest possible population =
The dimensions of a room floor are 6.7 m by 3.8 m, correct to 2 s.f.

Is it possible that the floor area is 27 m² (to 2 s.f.)?

How about 25 m²?

Explain your answer:

ENTrue length L ∈ [6.65, 6.75); true width W ∈ [3.75, 3.85). So area ∈ [6.65 × 3.75, 6.75 × 3.85) = [24.94, 25.99) m².
To 2 s.f., possible area values are 25 (covers [24.5, 25.5)) or 26 (covers [25.5, 26.5)).
• 27 m² is NOT possible — area max 25.99 < 26.5.
• 25 m² IS possible — most actual (L,W) give area in [24.94, 25.5).

⏱ 30 秒后查看中文版

中文真长 L ∈ [6.65, 6.75)；真宽 W ∈ [3.75, 3.85)。面积 ∈ [6.65×3.75, 6.75×3.85) = [24.94, 25.99) m²。
2 s.f. 可能取值：25（[24.5, 25.5)）或 26（[25.5, 26.5)）。
• 27 m² 不可能，最大 25.99 < 26.5。
• 25 m² 可以，大多数 (L,W) 组合面积都在 [24.94, 25.5)。

自评：

章末概念检查 · Concept Checkpoints

5 道封闭题。关键：5 条 s.f. 规则要烂熟于心。