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RATIONAL AND IRRATIONAL NUMBERS
Recurring decimals
Recurring decimals contain digits that are repeated over and over again.
0.2222222222…2.43535353535…0.142142142142…6.801980198019…
are all examples of recurring decimals
0.2222222222…2.43535353535…0.142142142142…6.801980198019…
are all examples of recurring decimals
Dots are used to show how the decimals recur.
2.43535353535... 2.435
0.2222222222... 0.2
0.142142142... 0.142
6.801980198019... 6.8019
Changing recurring decimals to fractions
1 Change to a fraction.0.7
let x 0.777777...
10x 7.777777...
multiply both sides of the equation by 10
write underneath x 0.777777...
x 0.777777... subtract the two equations
9x 7 divide both sides by 9
x
7
9
Answer: 70.7
9
If 1 digit recurs multiply by 10.If 2 digits recur multiply by 100.If 3 digits recur multiply by 1000.
If 1 digit recurs multiply by 10.If 2 digits recur multiply by 100.If 3 digits recur multiply by 1000.
Changing recurring decimals to fractions
2 Change to a fraction.0.47
let x 0.474747...
100x 47.474747...
multiply both sides of the equation by 100
write underneath x 0.474747...
x 0.474747... subtract the two equations
99x 47 divide both sides by 99
x
47
99
Answer: 470.47
99
If 1 digit recurs multiply by 10.If 2 digits recur multiply by 100.If 3 digits recur multiply by 1000.
If 1 digit recurs multiply by 10.If 2 digits recur multiply by 100.If 3 digits recur multiply by 1000.
Changing recurring decimals to fractions
3 Change to a fraction. 0.125
let x 0.125125125...
1000x 125.125125...
multiply both sides of the equation by 1000
write underneath x 0.125125...
0.125125...x subtract the two equations
999x 125 divide both sides by 999
x
125
999
Answer: 1250.125
999
If 1 digit recurs multiply by 10.If 2 digits recur multiply by 100.If 3 digits recur multiply by 1000.
If 1 digit recurs multiply by 10.If 2 digits recur multiply by 100.If 3 digits recur multiply by 1000.
Changing recurring decimals to fractions
4 Change to a fraction.0.947
let x 0.9474747...
100x 94.7474747...
multiply both sides of the equation by 100
write underneath x 0.9474747...
x 0.9474747... subtract the two equations
99x 93.8 divide both sides by 99
x
93.8
99
Answer: 4690.947
495
938
990
469
495 If 1 digit recurs multiply by 10.If 2 digits recur multiply by 100.If 3 digits recur multiply by 1000.
If 1 digit recurs multiply by 10.If 2 digits recur multiply by 100.If 3 digits recur multiply by 1000.
The set of real numbers can be divided into two sets:
RATIONAL NUMBERS IRRATIONAL NUMBERSand
Numbers that can be written in the form a . b
Numbers that can be written in the form a . b
Numbers that cannot be written in the form a . b
Numbers that cannot be written in the form a . b
Rational numbers include:
all integers all integers
eg
3
5 all fractions all fractions
all mixed numbers all mixed numbers
all terminating decimals all terminating decimals
all recurring decimals all recurring decimals
some square roots some square roots
some cube roots some cube roots
eg 8
8
1
eg 83 2
2
1
eg 25 5
5
1
eg 2
4
5
14
5
eg 0.23
23
100
2eg 0.6
3
Irrational numbers include:
some square roots some square roots
some cube roots some cube roots
some trig ratios some trig ratios
eg 3
eg 53
eg sin20
1 Which of these numbers are irrational numbers?
2.1 33 3.8 sin30 5
1
4cos40
Answer: cos40 , 33 and
2 Write each of these numbers in the correct place on the Venn diagram.
4
9 36
4
9 4.9
Rational numbers
Integers
4
9 36
4
9
4.9
2
3
2
3
3 Is x rational or irrational for this triangle?
10 cmx cm
24 cm
Using Pythagoras x2 102 242
x2 100 576
x2 676
x 26
Answer: x is rational
4 Is x rational or irrational for this triangle?
4 cm
x cm
12 cm
Using Pythagoras x2 42 122
x2 16 144
x2 128
x 11.317085...
Answer: x is irrational
