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28EXAMPLE 3
Determine whether the following real numbers are rational or irrational.
___
__
√
16 ___
√
a
b
6
49
__
6 = 2.449 897 43 …
√
a
Since the decimal neither terminates nor recurs, it cannot be expressed as the ratio of two integers,
__
√
so
6 is an irrational number.
___
√
16 ___
4 __
4 __
4 __
16 ___
=
×
=
b
(since
)
49
7
7
7
49
___
√
a __
16 ___
This is in the form
, where a and b are integers, so
is a rational number.
b
49
3
Determine whether the following real numbers are rational or irrational
___
___
__
__
___
√
√
4 ___
5 ___
√
√
√
a
b
c
d
e
8
9
11
25
16
EXAMPLE 4
__
6 = 2.449 489 743 … Write true or false for the following statements and discuss.
√
Using a calculator
__
__
__
6 = 2.44
6 = 2.449
6 = 2.4494
√
√
√
a
b
c
= 5.9536
a
2
2.44
The statement is false.
= 5.997 601
b
2
2.449
The statement is false.
= 5.999 560 36
c
2
2.4494
The statement is false.
__
√
Because
6 is irrational, its exact value cannot be written as a decimal.
__
√
The values given in parts a, b and c are rational approximations for
6 .
__
2 = 1.414 213 562 … Write true or false for the following statements.
√
4
Using a calculator,
__
__
__
2 = 1.41
2 = 1.414
2 = 1.4142
√
√
√
a
b
c
5
Write rational approximations, correct to 3 decimal places, for the following.
___
___
___
___
____
√
√
√
√
√
a
b
c
d
e
11
15
37
99
151
6 a
Using a ruler and set square, copy the diagram.
P
1
1
O
−3
−2
−1
0
1
2
3
b
Use Pythagoras’s rule to calculate the length of the interval OP
.
1
__
√
c
Using a pair of compasses, with point at O, accurately mark the position of
2 on the number line.
8
Insight Maths 9
Australian Curriculum
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