Surds And Indices Page 7
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28EXAMPLE 1
Show that the following are rational numbers.
.
3 _
−3.1
a
b
c
d
e
2
0.637
3
0.
4
4
11 ___
Any terminating or recurring decimal is
3 _
=
a
2
a rational number.
4
4
a __
3 _
, where a and b are integesr; hence 2
This is in the form
is a rational number.
4
b
637
____
0.637 =
b
1000
a __
This is in the form
, where a and b are integers; hence 0.637 is a rational number.
b
3 _
3 =
c
1
a __
This is in the form
, where a and b are integers; hence 3 is a rational number.
b
.
4 _
4 =
d
0.
9
.
a __
This is in the form
, where a and b are integers; hence 0.
4 is a rational number.
b
1 __
−3.1 = −3
e
10
31
__
= −
10
a __
, where a and b are integers; hence −3.1 is a rational number.
This is in the form
b
Exercise 11C
a __
1
Show that the following are rational numbers by expressing them in the form
.
b
.
2 _
1 _
−5
a
b
c
d
e
f
4
0.91
5
0.
7
2.84
3
2
__
.
.
___
√
4 __
−2.6
√
g
h
i
j
k
l
0.
5
3
16
30%
7.3%
9
2
Convert the following rational numbers to decimals.
When a rational number is
converted to a decimal, the decimal
3 _
5 _
a
b
1
5
8
either terminates or recurs.
5 __
2 _
c
d
4
12
3
1 _
2 _
e
f
g
h
69%
6.5%
17
%
7
3
EXAMPLE 2
Convert the following real numbers to decimals and discuss whether they are rational or irrational.
__
__
√
√
a
b
2
5
Using a calculator:
__
__
2 = 1.414 213 562 …
5 = 2.236 067 978 …
√
√
a
b
Since neither decimal terminates or recurs (although we can show answers to only 9 decimal places,
the limit of the calculator display) these numbers cannot be expressed as the ratio of two integers and
hence are not rational. They are irrational numbers.
7
Chapter 11
Surds and indices
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