1 Number and calculation
Exercise 1.1
1 Write whether each of these numbers is an integer or an irrational
number. Explain how you know.
a 9 b 19 c 39 d 49 e 99
2 a Write the rational numbers in this list.
1 7 5 −38 160 − 2.25 − 35
12
b Write the irrational numbers in this list.
0.3333… −16 200 1.21 23 3 343
8
3 Write whether each of these numbers is an integer or a surd.
Explain how you know.
a 100 b 3
100 c 1000
d 3
1000 e 10 000 f 3 10 000
4 Is each of these numbers rational or irrational? Give a reason for each
answer.
a 2+ 2 b 2+2
c 4+ 3 4 d 3
4+4
5 Find
a two irrational numbers that add up to 0
b two irrational numbers that add up to 2.
Think like a mathematician
6 a Use a calculator to find
i 8× 2 ii 3 × 12 iii 20 × 5 iv 2 × 18
b What do you notice about your answers?
c Find another multiplication similar to the multiplications in part a.
d Find similar multiplications using cube roots instead of square roots.
7 Without using a calculator, show that
a 7 < 55 < 8 b 4 < 3 100 < 5
8 Without using a calculator, find an irrational number between
a 4 and 5 b 12 and 13.
9 Without using a calculator, estimate
a 190 to the nearest integer
b 3
190 to the nearest integer.
12
1.1 Irrational numbers
10 a Use a calculator to find
i ( 2 + 1) × ( 2 −1) ii ( 3 + 1) × ( 3 − 1) iii ( 4 + 1) × ( 4 − 1)
b Continue the pattern of the multiplications in part a.
c Generalise the results to find ( N + 1) × ( N − 1) where N is a positive integer.
d Check your generalisation with further examples.
11 Here is a decimal: 5.020 020 002 000 020 000 020 000 002…
Arun says:
There is a regular pattern:
one zero, then two zeros,
then three zeros, and so on.
This is a rational number.
a Is Arun correct? Give a reason for your answer.
b Compare your answer with a partner’s. Do you agree? If not, who is correct?
In this exercise, you have looked at the properties of rational and
irrational numbers.
a Are the following statements true or false?
i The sum of two integers is always an integer.
ii The sum of two rational numbers is always a rational
number.
iii The sum of two irrational numbers is always an irrational
number.
b Here is a calculator answer: 3.646 153 846
The answer is rounded to 9 decimal places.
Can you decide whether the number is rational or irrational?
Summary checklist
I can use square numbers and cube numbers to estimate square roots and
cube roots.
I can say whether a square root or the cube root of a positive integer
is rational or irrational.
13
1 Number and calculation
1.2 Standard form
In this section you will … Key words
• learn to write large and small numbers in standard form. scientific notation
standard form
Look at these numbers
4.67 ×10 = 46.7
4.67 ×10 2 = 467 Tip
4.67 ×103 = 4670 4.67 × 102 is the
4.67 ×106 = 4 670 000 same as
4.67 × 100 or
You can use powers of 10 in this way to write large numbers. For 4.67 × 10 × 10
example, the average distance to the Sun is 149 600 000 km. You can
write this as 1.496 × 108 km. This is called standard form. You write a
number in standard form as a × 10n where 1 ⩽ a < 10 and n is an integer.
You can write small numbers in a similar way, using negative integer
powers of 10. For example:
4.67 ×10 −1 = 0.467 Tip
4.67 ×10 −2 = 0.0467
Think of 4.67 × 10−1
−3
4.67 ×10 = 0.004 67 as 4.67 ÷ 10
4.67 ×10 −7 = 0.000 004 67
Small numbers occur often in science. For example, the time for light to
Tip
travel 5 metres is 0.000 000 017 seconds. In standard form, you can write
this as 1.7 ×10 −8 seconds. Standard form is
also sometimes
Worked example 1.2 called scientific
notation.
Write these numbers in standard form.
a 256 million b 25.6 billion c 0.000 025 6
Answer
Tip
a 1 million = 1 000 000 or 10 6
Notice that in
So 256 million = 256 000 000 = 2.56 × 108 every case the
b 1 billion = 1 000 000 000 or 109 decimal point
So 25.6 billion = 25 600 000 000 = 2.56 × 1010 is placed after
the 2, the first
c 0.000 025 6 = 2.56 × 10−5 non-zero digit.
14
