Page 5 - 新思维数学学生用书9 样章

1 Number and calculation

How to use this book 1, 4, 9 and 16 are the first four square numbers. They have integer square roots.
1 = 1 and 11 2 = 4 and 4 = 2
=
2
2
1 Number and calculation 3 = 9 and 9 = 3 4 = 16 and 16 = 4
2
2
1 = 1 and 11 How to use this book
1, 4, 9 and 16 are the first four square numbers. They have integer square roots.
2 = 4 and 4 =
=
2
2
2
3 = 9 and 9 = 3 4 = 16 and 16 = 4
2
2
What about 2? Is there a rational number n for which n = 2? Remember that you can write a
2
In this book you will find lots of different features to help your learning.
Number and
rational number as a fraction.
1
1
1
1
1
1 1 2 ( ) 2 = 1 × 1 = 2 so 2 must be a little less than 1 .
2
2
4
2
calculation
5
1
5
A closer answer is 1 because 1 12 ( ) 2 = 2 144 .
12
Questions to find out what you 169 169 2 1
An even closer answer is 1
What about 2? Is there a rational number n for which n = 2? Remember that you can write a 408 because 1 408 ( ) = 2 166464 .
2
know already.
Do you think you can find a fraction which gives an answer of exactly 2 when you square it?
rational number as a fraction. Getting started
( ) 2 = 1 × 1 = 2 so 2 must be a little less than 1 . 1 A calculator gives the answer 2 = 1 414213562. This is a rational number because you can
.
Write as a number:
1
1
1
1
1
1
2
4
2
2
2
b
c
d
a
3
3
64
2
12
5
414213562
write it as a fraction: 1
81 .
( )
1000000000
2 = 256
8
5
A closer answer is 1 because 1 5 2 = 2 1 . 2 Is 1.414213562 × 1.414213562 exactly 2?
12 12 144 In this unit, you will look at numbers such as 2.
Use this fact to work out the value of
( )
An even closer answer is 1 169 because 1 169 2 = 2 1 . a 2 9 b 2 7
5
408 408 166464 3 Here is a multiplication: 15 × 15 2
a
Write the correct answer from this list: 15 15 30 30
10
Do you think you can find a fraction which gives an answer of exactly 2 when you square it? 7 10 7 1.1 Irrational numbers
b 1.1 Irrational numbers
Write the answer to 15 ÷ 15 in index form.
5
2
A calculator gives the answer 2 = 1 414213562. This is a rational number because you can 3 225
.
4
Look at these numbers: 4 −4.5 3000 17
20
Integers are whole numbers. For example, 13, −26 and 100 004 are integers. Tip
a
Which of these numbers are integers?
write it as a fraction: 1 414213562 . You can write rational numbers as fractions. For example, 9 , −3 and Key words
3
4
b In this section you will …
1000000000 Which of these numbers are rational numbers? 4 15 The set of
What you will learn in the unit.
Is 1.414213562 × 1.414213562 exactly 2? 5 18 are rational numbers. irrational number
Write one million as a power of 10.
5
•
learn about the difference between rational numbers rational numbers
11
In this unit, you will look at numbers such as 2. You can write any fraction as a decimal. includes integers.
and irrational numbers
rational number
3
use your knowledge of square numbers to estimate
9 = • 975. −3 4 = −3 26666666... 18 5 = 18 4545454...
.
.
4 15 11 surd
square roots
The fraction either terminates (for example, 9.75) or it has recurring Tip
use your knowledge of cube numbers to estimate
•
1.1 Irrational numbers digits (for example, −3.266666666666… continues with 6s and Square roots
cube roots.
18.45454545454… continues with the digits 4 and 5 repeating).
There are many square roots and cube roots that you cannot write as of negative
numbers do not
fractions. When you write these fractions as decimals, they do not belong to the
Important words to learn.
In this section you will … terminate and there is no recurring pattern. For example, a calculator set of rational
Key words
gives the answer 7 = 2 645751... The calculator answer is not exact. The
.
• learn about the difference between rational numbers decimal does not terminate and there is no recurring pattern. Therefore, or irrational
irrational number
numbers. You
7 is not a rational number.
and irrational numbers Numbers that are not rational are called irrational numbers. 7, 23, will learn more
rational number
3
• use your knowledge of square numbers to estimate 8 3 10 and 45 are irrational numbers. Irrational numbers that are square about these
surde roots are called surds.
square roots roots or cub numbers if you
There are also numbers that are irrational but are not square roots or
continue to study
• use your knowledge of cube numbers to estimate cube roots. One of these irrational numbers is called pi, which is the mathematics to a
cube roots. Greek letter π. Your calculator will tell you that π = 3.141592… You will higher level.
meet π later in the course.
Step-by-step examples showing Worked example 1.1
Do not use a calculator for this question. 7
how to solve a problem. a Show that 90 is between 9 and 10.
b N is an integer and 90 is between N and N + 1. Find the value of N.
3
Answer
8 a 9 = 81 and 10 = 100
2
2
81 < 90 < 100 This means 90 is between 81 and 100.
So 81 < 90 < 100
<
And so 9 < 90 10
b 4 = 64 and 5 = 125
3
3
1 Number and calculation
64 < 90 < 125 and so
3 64 < 3 90 < 3 125
9 Here are four numbers:
So 4 < 3 90 < 5 and N = 4
w = 9.81 × 10 −5 x = 2.8 × 10 −4 y = 9.091 × 10 −5 z = 4 × 10 −4
These questions help you to a Which number is the largest? 9
b
Which number is the smallest?
4
develop your skills of thinking 10 a Explain why the number 65 × 10 is not in standard form.
b
4
Write 65 × 10 in standard form.
c Write 48.3 × 10 in standard form.
6
and working mathematically. 11 Write these numbers in standard form.
a 15 × 10 −3 b 27.3 × 10 −4 c 50 × 10 −9
12 Do these additions. Write the answers in standard form.
a 2.5 × 10 + 3.6 × 10 6 b 4.6 × 10 + 1.57 × 10 5 c 9.2 × 10 + 8.3 × 10 4
6
4
5
13 Do these additions. Write the answers in standard form.
a 4.5 × 10 + 3.1 × 10 −6 b 5.12 × 10 + 2.9 × 10 −5
−5
−6
4 c 9 × 10 + 7 × 10 −8
−8
14 a Multiply these numbers by 10. Give each answer in standard form.
i 7 × 10 5 ii 3.4 × 10 6
iii 4.1 × 10 −5 iv 1.37 × 10 −4
b Generalise your results from part a.
c Describe how to multiply or divide a number in standard form by 1000.
What are the advantages of writing numbers in standard form?
Summary checklist
I can write large and small numbers in standard form.
14