Download National 5 Mathematics Course Materials Topic 1: Rounding and

SCHOLAR Study Guide
National 5 Mathematics
Course Materials
Topic 1: Rounding and scientific
notation
Authored by:
Margaret Ferguson
Reviewed by:
Jillian Hornby
Previously authored by:
Eddie Mullan
Heriot-Watt University
Edinburgh EH14 4AS, United Kingdom.
First published 2014 by Heriot-Watt University.
This edition published in 2016 by Heriot-Watt University SCHOLAR.
Copyright © 2016 SCHOLAR Forum.
Members of the SCHOLAR Forum may reproduce this publication in whole or in part for
educational purposes within their establishment providing that no profit accrues at any stage,
Any other use of the materials is governed by the general copyright statement that follows.
All rights reserved. No part of this publication may be reproduced, stored in a retrieval system
or transmitted in any form or by any means, without written permission from the publisher.
Heriot-Watt University accepts no responsibility or liability whatsoever with regard to the
information contained in this study guide.
Distributed by the SCHOLAR Forum.
SCHOLAR Study Guide Course Materials Topic 1: National 5 Mathematics
1. National 5 Mathematics Course Code: C747 75
Acknowledgements
Thanks are due to the members of Heriot-Watt University's SCHOLAR team who planned and
created these materials, and to the many colleagues who reviewed the content.
We would like to acknowledge the assistance of the education authorities, colleges, teachers
and students who contributed to the SCHOLAR programme and who evaluated these materials.
Grateful acknowledgement is made for permission to use the following material in the
SCHOLAR programme:
The Scottish Qualifications Authority for permission to use Past Papers assessments.
The Scottish Government for financial support.
The content of this Study Guide is aligned to the Scottish Qualifications Authority (SQA)
curriculum.
All brand names, product names, logos and related devices are used for identification purposes
only and are trademarks, registered trademarks or service marks of their respective holders.
1
Topic 1
Rounding and scientific notation
Contents
1.1
Rounding to a given number of significant figures . . . . . . . . . . . . . . . .
3
1.2
1.3
Scientific Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Learning Points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6
10
1.4
End of topic test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
11
2
TOPIC 1. ROUNDING AND SCIENTIFIC NOTATION
Learning objectives
By the end of this topic, you should be able to:
•
round to a given number of significant figures;
•
perform calculations using scientific notation.
© H ERIOT-WATT U NIVERSITY
TOPIC 1. ROUNDING AND SCIENTIFIC NOTATION
1.1
3
Rounding to a given number of significant figures
Learning objective
By the end of this topic you should be able to round calculations to a required number
of significant figures.
Rounding to decimal places
When £28 is divided by 13 the calculator gives the result 2·153846153.
Since the result is an amount of money we only want to give the answer to two decimal
places. The digits in the other decimal places have no practical meaning.
2·15[3846153]
You have to decide between £2·15 and £2·16. To decide. . .
•
think of what we want to lose;
•
the first of these digits is 3.
If this digit is bigger than 4 then we choose the higher answer. . . £2·16.
If not we choose the lower. . . £2·15.
Since 3 is less than 4 we choose £2·15 as the answer and write £28 ÷ 13 = £2 · 15
(to 2 d.p.)
Now try some examples.
Q1: £85 divided by 13 giving your answer to 2 decimal places.
..........................................
Q2: £25 divided by 7 giving your answer to 2 decimal places.
..........................................
Example : Rounding to decimal places
Problem:
A calculator displays 25·0425678. This can be recorded in a variety of ways.
Solution:
1. 25·0426 (to 4 decimal places)
2. 25·043 (to 3 decimal places)
3. 25·04 (to 2 decimal places)
4. 25·0 (to 1 decimal places)
5. 25 (to the nearest whole number)
6. 30 (to the nearest ten)
© H ERIOT-WATT U NIVERSITY
Go online
4
TOPIC 1. ROUNDING AND SCIENTIFIC NOTATION
..........................................
Q3:
A calculator reads 74·20345. Round this to 1 decimal place.
..........................................
Rounding to significant figures
Go online
In mathematics a digit is significant if it adds value to the number or adds accuracy
to the number.
In mathematics, a digit is significant if it adds value or accuracy to a number. For
example, look at the number 0·00001325800
The first 5 digits don’t add value or accuracy to the number. They just help to position
the decimal point.
The 1 is significant. . . it’s the first and most significant figure. . . it adds value to the
number.
After the most significant figure, all other digits are significant. . . 0·00001325800 has 7
significant figures.
Trailing zeros are significant because they make the number more precise. . . 11 decimal
places rather than 9.
..........................................
Example : Rounding to significant figures
Problem:
Round the numbers 0·040203 and 3·02 to a variety of significant figures.
Solution:
0 · 040203
0 · 040203
0 · 040203
0 · 040203
=
=
=
=
0 · 04020 (to 4 significant figures)
0 · 0402 (to 3 significant figures)
0 · 040 (to 2 significant figures)
0 · 04 (to 1 significant figure)
3 · 02 = 3 · 0 (to 2 significant figures)
3 · 02 = 3 (to 1 significant figure)
..........................................
Q4:
A calculator reads 3·00345. Round this to 3 significant figures.
..........................................
© H ERIOT-WATT U NIVERSITY
TOPIC 1. ROUNDING AND SCIENTIFIC NOTATION
Significant figures and whole numbers
With whole numbers it’s more difficult to decide when trailing zeros are significant. It
will depend on the story-line.
To the nearest 10 there are 50 weeks in a year . . . this number has only 1 significant
figure . . . the 5. The zero just makes sure the 5 is in the 10s column.
This coin is worth 50p exactly . . . there are 2 significant figures.
The zero counts . . . its not worth 49p or 51p but 50p exactly.
There are roughly 360 days in a year . . . this number has only 2 significant figures . . .
the 3 and 6. The zero just makes sure the 3 and 6 are in the right columns.
There are 360 ◦ in a revolution.
The zero counts . . . its not 359 ◦ or 361◦ but
360◦ exactly.
In this situation 360 has 3 significant figures.
Example : Rounding whole numbers to significant figures
Problem:
Round 354, 729 and 3942 to a variety of significant figures.
Solution:
354 = 350 (to 2 significant figures)
354 = 400 (to 1 significant figure)
729 = 730 (to 2 significant figures)
729 = 700 (to 1 significant figure)
3942 = 3940 (to 3 significant figures)
3942 = 3900 (to 2 significant figures)
3942 = 4000 (to 1 significant figures)
© H ERIOT-WATT U NIVERSITY
5
6
TOPIC 1. ROUNDING AND SCIENTIFIC NOTATION
..........................................
Rounding calculations exercise
Q5:
Go online
a) Round 85 · 88235294 to 1 decimal place.
b) Round 85 · 88235294 to 2 decimal places.
c) Round 85 · 88235294 to 3 decimal places.
d) Round 85 · 88235294 to the nearest whole number.
e) Round 85 · 88235294 to the nearest ten.
..........................................
Q6:
a) Round 5 · 54 to 2 significant figures.
b) Round 7922 to 3 significant figures.
c) Round 0 · 544056 to 4 significant figures.
d) Round 23 · 98421 to 3 significant figures.
e) Round 5426859 to 3 significant figures.
..........................................
1.2
Scientific Notation
Scientific notation or standard form as it is also known, is a way of expressing very
large and very small numbers. You should already be able to convert and interpret
numbers in scientific notation but here is a reminder.
Expressing large and small numbers
..........................................
Go online
Carrying out calculations involving standard form requires you to be able to enter
numbers in scientific notation into your calculator. They all have a slightly different
method. Check out how your calculator works.
© H ERIOT-WATT U NIVERSITY
TOPIC 1. ROUNDING AND SCIENTIFIC NOTATION
Try entering: 9·3 × 10x 7
Try entering:
9 · 3 exp 7
Examples
1.
Problem:
Light travels at a speed of 3 × 1010 cm per second. How long will it take to travel from
the TV to you which is a distance of 400 cm? Give your answer in scientific notation
correct to 3 significant figures.
Solution:
400
From your prior knowledge of Speed, Distance and Time we get T = D
S = 3×1010 =
1 · 333333333 × 10−8 = 1 · 33 × 10−8 seconds to 3 significant figures. Notice only the
decimal part of the number is rounded.
Most modern scientific calculators set in MATH mode will give 3 possible answers
1
750000000 ,
•
0·00000001 3 (note the dot over the three which means it is a recurring decimal)
or 1 · 33 × 10−8 .
..........................................
© H ERIOT-WATT U NIVERSITY
7
8
TOPIC 1. ROUNDING AND SCIENTIFIC NOTATION
2.
Problem:
The mass of an electron is 9 · 11 × 10-31 kg.
Calculate the mass of 5 × 1036 electrons.
Give your answer correct to 2 significant figures.
Solution:
9 · 11 × 10-31 × 5 × 1036 = 4555000
4555000 = 4 · 555 × 106 = 4 · 6 × 106 to 2 significant figures.
..........................................
3. Rounding to significant figures in scientific notation
Problem:
When we calculate the approximate volume of Planet Earth we get V =
The calculator will give 1 · 083206917 × 1012 .
4
3
× π × 63713
Solution:
If we round this to 2 significant figures we get 1 · 1 × 10 12 .
If we round this to 3 significant figures we get 1 · 08 × 10 12 .
If we round this to 4 significant figures we get 1 · 083 × 10 12 .
Only the decimal part of the number is rounded.
..........................................
Scientific notation practice
Q7: The mass of planet Earth is 5 · 972 × 10 24 kg mass.
Venus is 0·815 of Earth.
Calculate the mass of Venus correct to 4 significant figures.
..........................................
Q8: The radius of Mars is 3390 km.
Volume of sphere = 4 /3 πr 3
Calculate the volume of Mars correct to 3 significant figures.
..........................................
Q9:
Evaluate 10992 × 87412 , giving your answer correct to 3 significant figures.
..........................................
Scientific notation exercise
Q10:
Go online
a) Round 254000000000 to 2 significant figures. Express your answer in scientific
notation.
b) Round 8465149 · 236 to 3 significant figures. Express your answer in scientific
notation.
c) Round 0 · 000000782609 to 4 significant figures. Express your answer in scientific
notation.
© H ERIOT-WATT U NIVERSITY
TOPIC 1. ROUNDING AND SCIENTIFIC NOTATION
d) Round 10067 · 31974 to 3 significant figures. Express your answer in scientific
notation.
..........................................
Q11: A grain of rice weighs 2 · 8 × 10-2 g. Calculate the number of grains of rice in a
1 · 5 kg bag. Give your answer in scientific notation correct to 2 significant figures.
..........................................
Q12: The radius of Saturn is 58,232 km. Surface Area of a Sphere = 4πr 2
a) Calculate the surface area of Saturn giving your answer in scientific notation correct
to 3 significant figures.
b) Give your answer in full (as a normal number) correct to 2 significant figures.
..........................................
Q13: If the mass of a grain of sand 0·5 mm in diameter is 5 · 5 × 10 -5 g, calculate the
number of grains of sand would there be in a 25 kg bag of sand? Give your answer in
scientific notation correct to 3 significant figures.
..........................................
© H ERIOT-WATT U NIVERSITY
9
10
TOPIC 1. ROUNDING AND SCIENTIFIC NOTATION
1.3
Learning Points
Significant figures
A digit is significant if it adds value or accuracy to the number.
•
1532864 = 2000000 to 1 significant figure
•
0 · 00002598 = 0 · 000026 to 2 significant figures
•
23 · 98421 = 24 · 0 to 3 significant figures
•
6 · 32587 × 107 = 6 · 33 × 107 to 3 significant figures
Scientific notation
To perform calculations involving scientific notation you must know how to enter them
into your calculator and how to interpret the answer.
•
8 · 96 × 109 = 8960000000
•
7 · 312 × 10-7 = 0 · 0000007312
© H ERIOT-WATT U NIVERSITY
TOPIC 1. ROUNDING AND SCIENTIFIC NOTATION
1.4
11
End of topic test
End of topic 1 test
Q14:
a) Round 2 · 52 to 2 significant figures.
b) Round 5045 to 3 significant figures.
c) Round 0 · 668141 to 4 significant figures.
d) Round 4 · 00675 to 2 significant figures.
e) Round 2 · 70195 x1019 to 3 significant figures.
..........................................
Q15:
a) Express the number 295330000000 in standard form.
b) Express 0 · 00000000000193 in standard form.
c) Express 1 · 67262178 × 109 in full.
..........................................
Q16:
1. How many seconds are there in 50 years? Give your answer in scientific notation
correct to 3 significant figures.
2. The mass of a proton is said to be 1 · 67262178 × 10 -27 kg.
a) Calculate in scientific notation, to 2 significant figures the mass of 2 ×
106 protons.
b) Calculate correct to 1 significant figure the number of protons in a mass of
1 × 10-25 kg.
..........................................
© H ERIOT-WATT U NIVERSITY
Go online
12
ANSWERS: TOPIC 1
Answers to questions and activities
1 Rounding and scientific notation
Rounding to decimal places (page 3)
Q1:
£6·54
Q2:
£3·57
Answers from page 4.
Q3:
74.2
Answers from page 4.
Q4:
3.00
Rounding calculations exercise (page 6)
Q5:
a) 85 · 9
b) 85 · 88
c) 85 · 882
d) 86
e) 90
Q6:
a) 5 · 5
b) 7920
c) 0 · 5441
d) 24 · 0
e) 5430000
Scientific notation practice (page 8)
Q7: 5 · 972 × 1024 × 0 · 815 = 4 · 86718 × 1024 = 4 · 867 × 1024 kg.
Note: to enter the answer into SCHOLAR online you would type 4·867*10^24
Q8: V = 43 × π × 33903 = 1 · 63 × 1011 km3
Note: to enter the answer into SCHOLAR online you would type 1·63*10^11
Q9:
The calculator gives 9 · 228213324 × 1013 which rounds to 9 · 23 × 1013 .
© H ERIOT-WATT U NIVERSITY
ANSWERS: TOPIC 1
Scientific notation exercise (page 8)
Q10:
a) 2 · 5 × 1011
b) 8 · 47 × 106
c) 7 · 826 × 10-7
d) 1 · 01 × 104
Q11: 5 · 4 × 104
You will need to divide the weight of the bag by the weight of a grain of rice.
Note: to enter the answer into SCHOLAR online you would type 5·4*10^4
Q12:
a) 4·26 × 1010
b) 43000000000
Q13: 4 · 55 × 108
End of topic 1 test (page 11)
Q14:
a) 2 · 5
b) 5050
c) 0 · 6681
d) 4 · 0
e) 2 · 70 × 1019
Q15:
a) 2 · 9533 × 1011
b) 1 · 93 × 10-12
c) 1672621780
Q16:
1. 1 · 5768 × 109
2. a) 3 · 3 × 10-21
b) 60
© H ERIOT-WATT U NIVERSITY
13