Basic 8 · Mathematics · Significant Figures

Significant Figures, Decimal Places and Standard Form for Small Numbers

In this topic you will express numbers to a given number of significant figures and decimal places, and write small numbers in standard form using negative indices.

By the end of these notes you should be able to:

Express integers to a given number of significant figures
Express numbers to a given number of decimal places
Write small numbers in standard form

What you already know

Standard form for large numbers
Rounding and significant figures (Basic 7)

Copy everything that has a blue line on its left into your notebook. The other parts are for reading.

Lesson 1

Strand: Number
Sub-strand: Significant Figures
Indicator: B8.1.1.1.5 – Express integers in a given number of significant and decimal places

Introduction

A newspaper will not report a crowd as 41,387 people. It says "about 41,000". This is a number given to two significant figures.

1. Significant Figures in Integers

For whole numbers, the first non-zero digit from the left is the first significant figure. To correct to a given number of significant figures, keep that many digits, round using the next digit, and replace the rest with zeros so that the place value does not change.

Rounding 7,582,916

Significant figures	Answer
1 s.f.	8,000,000
2 s.f.	7,600,000
3 s.f.	7,580,000
4 s.f.	7,583,000

2. Worked Example

Correct to 3 s.f.

Correct 2,046,700 to 3 significant figures.

First three significant figures: 2, 0, 4. (The zero is between non-zero digits, so it counts.)
Next digit: 6, so round up the 4 to 5.
Replace the remaining digits with zeros.

Answer: 2,050,000

3. Why Use Significant Figures?

Numbers measured in real life are never exact. It makes little sense to report a town's population as 83,417 when people move every day. Saying "83,000" (2 s.f.) gives a useful and honest picture. Scientists use significant figures to show how accurate a measurement is.

A crowd of football fans cheering

Figure 1: Counting a crowd Round the attendance figures of three events to 2 s.f.

4. Significant Figures and Estimation

Rounding each number to 1 significant figure gives a quick estimate. To estimate 4,870 × 62, use 5,000 × 60 = 300,000. To estimate 7,912 ÷ 38, use 8,000 ÷ 40 = 200. Use this method to check calculator answers.

New words

significant figure: A digit that carries meaning in a number.
round: To replace a number with a simpler one close to it.
accurate: Close to the true value.
estimate: An approximate answer.

Summary

In integers, significant figures start from the first non-zero digit.
Replace the remaining digits with zeros to keep the place value.
Rounding to 1 s.f. gives quick estimates.

Class activity: Rounding Attendance

List the attendance figures of five sporting events.
Round each to 1, 2 and 3 significant figures.
Discuss which is most useful for a news report.

You need: Printed power-of-ten cards, Decimal place-value chart, Graph paper

Exercises

Answer in your exercise book first, then check your answers.

1. 3,780 to 2 s.f. is…
- A. 3,800
- B. 3,700
- C. 38
- D. 4,000
Show answer
Answer: 3,800Next digit 8.
2. 60,548 to 3 s.f. is…
- A. 60,500
- B. 60,600
- C. 605
- D. 61,000
Show answer
Answer: 60,500Next digit 4.
3. Zeros between non-zero digits are significant.
- A. True
- B. False
Show answer
Answer: True2,046.
4. 7,582,916 to 1 s.f. is ______.
Show answer
Answer: 8,000,000First digit.
5. Correct 495,300 to 2 s.f.
Show answer
Answer: 500,00049 becomes 50.
6. Estimate 6,120 × 48 by rounding to 1 s.f.
Show answer
Answer: 6,000 × 50 = 300,000Estimation.

Lesson 2

Strand: Number
Sub-strand: Significant Figures
Indicator: B8.1.1.1.5 – Express integers in a given number of significant and decimal places

Introduction

When you divide GH₵100 among 3 people, the calculator shows 33.3333333. How should the answer be written? Decimal places tell us how exact to be.

1. Review of Decimal Places

Decimal places count the digits after the decimal point. To correct to n decimal places, keep n digits after the point, and look at the next digit. If it is 5 or more, add 1 to the last digit kept.

Rounding 12.34567

Decimal places	Answer
1 d.p.	12.3
2 d.p.	12.35
3 d.p.	12.346
4 d.p.	12.3457

2. Calculations with Decimal Places

Calculate and correct

Share GH₵100 among 3 people. Give the answer to 2 d.p.

100 ÷ 3 = 33.3333...
Keep two decimal places: 33.33.
Next digit: 3, so round down.

Answer: GH₵33.33

Notice that 3 × 33.33 = 99.99, so one pesewa is left over. Rounded answers are close but may not be exact.

A calculator showing a long decimal number

Figure 2: Calculator display Correct three calculator results to 2 d.p.

3. Decimal Places or Significant Figures?

Comparing d.p. and s.f.

Number	2 d.p.	2 s.f.
45.678	45.68	46
0.03456	0.03	0.035
1,234.567	1,234.57	1,200

Read questions carefully, because the answers are different. Use decimal places for money and measurements with fixed units, and significant figures for estimates and large values.

4. Rounding in the Middle of a Calculation

Do not round too early. If a problem has several steps, keep extra digits until the end, then round the final answer. Rounding at each step can make the final answer less accurate.

New words

decimal place: A digit position after the decimal point.
recurring decimal: A decimal with repeating digits.
round down: To keep the last digit unchanged.
accuracy: How close a value is to the true value.

Summary

Keep the required digits after the point and use the next digit to round.
Decimal places and significant figures give different answers.
Round only at the end of a multi-step calculation.

Class activity: Sharing Costs

Share given amounts of money equally among 3, 6 and 7 people.
Give each answer to 2 d.p.
Multiply back and find how much is left over.

You need: Printed power-of-ten cards, Decimal place-value chart, Graph paper

Exercises

Answer in your exercise book first, then check your answers.

1. 8.256 to 1 d.p. is…
- A. 8.3
- B. 8.2
- C. 8.26
- D. 8.0
Show answer
Answer: 8.3Next digit 5.
2. 0.04567 to 2 s.f. is…
- A. 0.046
- B. 0.05
- C. 0.04
- D. 0.045
Show answer
Answer: 0.046Next digit 6.
3. Money is usually given to 2 d.p.
- A. True
- B. False
Show answer
Answer: TruePesewas.
4. 22 ÷ 7 to 2 d.p. is ______.
Show answer
Answer: 3.143.1428...
5. Correct 19.9962 to 2 d.p.
Show answer
Answer: 20.00Round up and carry.
6. Why should you round only at the end of a calculation?
Show answer
Answer: Rounding at each step adds small errors that can make the final answer inaccurate.Accuracy.

Lesson 3

Strand: Number
Sub-strand: Standard Form
Indicator: B8.1.1.1.4 – Express integers of any size into standard form

Introduction

Standard form also works for tiny numbers. The width of a human hair is about 0.00008 metres, or 8 × 10⁻⁵ m.

1. Negative Indices

Dividing by ten gives a negative power: 10⁻¹ = 1/10 = 0.1; 10⁻² = 1/100 = 0.01; 10⁻³ = 1/1,000 = 0.001. A negative index shows how many places the digits move to the right.

Powers of ten

Power	Value
10²	100
10¹	10
10⁰	1
10⁻¹	0.1
10⁻²	0.01
10⁻³	0.001

2. Writing Small Numbers

Write in standard form

Write 0.00056 in standard form.

Move the point after the first non-zero digit: 5.6.
It moved four places to the right, so the index is −4.

Answer: 5.6 × 10⁻⁴

3. Converting Back

Change to ordinary form

Write 3.2 × 10⁻³ as an ordinary number.

The index −3 means dividing by 1,000.
Move the digits three places to the right.

Answer: 0.0032

Tiny organisms seen through a microscope

Figure 3: Tiny things Write the sizes of three tiny things in standard form.

4. Comparing in Standard Form

To compare numbers in standard form, look at the index first. A larger index means a larger number: 3 × 10⁵ > 9 × 10⁴. If the indices are equal, compare the first numbers: 4.2 × 10³ > 3.9 × 10³. With negative indices, −3 is greater than −5, so 2 × 10⁻³ > 7 × 10⁻⁵.

5. Why Scientists Use Standard Form

Writing 0.000000001 m is slow and easy to get wrong, but 1 × 10⁻⁹ m is short and clear. Scientists use it for atoms, germs and the very small parts in computer chips, and also for huge distances in space.

New words

negative index: A power showing division by ten.
ordinary form: A number written without powers of ten.
micro-organism: A very tiny living thing.
compare: To decide which is greater.

Summary

Small numbers in standard form have negative indices.
A negative index shows the digits moving to the right.
Compare standard form numbers by the index first.

Class activity: Tiny Measures

Look up the sizes of a bacterium, a grain of sand and a hair.
Write each in standard form.
Order them from smallest to largest.

You need: Printed power-of-ten cards, Decimal place-value chart, Graph paper

Exercises

Answer in your exercise book first, then check your answers.

1. 10⁻² =
- A. 0.01
- B. 0.1
- C. 100
- D. 0.001
Show answer
Answer: 0.011/100.
2. 0.0047 in standard form is…
- A. 4.7 × 10⁻³
- B. 47 × 10⁻⁴
- C. 4.7 × 10³
- D. 0.47 × 10⁻²
Show answer
Answer: 4.7 × 10⁻³Three places.
3. 3 × 10⁵ is greater than 9 × 10⁴.
- A. True
- B. False
Show answer
Answer: TrueLarger index.
4. 6 × 10⁻² = ______.
Show answer
Answer: 0.06Two places right.
5. Write 8.1 × 10⁻⁴ as an ordinary number.
Show answer
Answer: 0.00081Four places.
6. Which is greater: 2 × 10⁻³ or 7 × 10⁻⁵? Explain.
Show answer
Answer: 2 × 10⁻³, because −3 > −5 and 0.002 > 0.00007.Compare.

Homework

Correct ten numbers to 2 s.f. and 2 d.p.
Write five small measurements in standard form.