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7. Rational and irrational numbers
top
a where a and b are
A rational number is one which can be expressed as
b
integers. An irrational number is one which can’t. Fractions, integers, and
2
recurring decimals are rational. Examples of rationals: , 1, 0.25, 3 8 .
3
Examples of irrationals: π , 2 , 0.1234.... (not recurring).
a (to confirm they really are rational)
b
125
1
A terminating decimal: 0.125 =
=
1000
8
A recurring decimal: 0.123 . Call the number x, so x = 0.123123123......
Multiply by a suitable power of 10 so the recurring decimal appears exactly
again: 1000 x = 123.123123.....
= 123 + 0.123123....
123
41
so 1000x = 123 + x , then 999x = 123 and x =
=
.
999
333
(i) Converting rationals to the form
(ii) rationalising a denominator:
2
has a 3 in the denominator, so multiply top and bottom by
3
3
does not change the value of the expression, only the shape):
2
3
6
3
×
= 3
= 6.
3
3
3
(iii)
a b = ab
a
b
=
3 (which
a
b
and the same with cube roots, etc.
To simplify expressions using these:
200 = 100 × 2 = 100 × 2 = 10 2
18
18
=
= 9 = 3
2
2
(iv) Finding irrational numbers in a given area:
e.g. find an irrational number between 5 and 6. Note that most square roots
4
are irrational (except for 16,
, etc) are irrational, so as 5 = 25 and
9
6 = 36 , pick a root in between, e.g. 28 . (Or say π + 2 for example).
Questions
a
:
(i) 0.375
b
50
6
(ii)
(iii) 72
(b) Simplify (i)
2
2
(c) Find an irrational number between 1 and 1.1
Answers
(a) Convert into the form
(a) (i) 0.375 =
375
3
=
1000 8
i i
(ii) 0.3 6
(iv)
6
2
6 2 = 3 2
×
=
2
2
2
(ii) = 50 = 25 = 5
2
(b) (i) =
72 = 36 × 2 = 36 × 2 = 6 2
(iv) = 3 125 × 2 = 3 125 × 3 2 = 5 3 2
(c) e.g.
2 − 0.3,
101
etc
10
250
(ii) x = 0.36363636...... so 100x = 36.363636.....
= 36 + 0.363636 = 36 + x . Therefore 99x = 36 , so x =
(iii)
3
36
4
=
99
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