Rational and
Irrational Numbers
Learning Outcomes


I can distinguish between rational and irrational numbers
I can see the significance of recurring and non-recurring
decimals

I can understand what is meant by a surd

I can use the rules of surds ( eg. √a x√b=√ab)

I can rationalise the denominator of surds

I can expand and simplify surds in brackets
Rational Numbers
A rational number is a number that can be written as a ratio. That means it
can be written as a fraction, in which both the numerator (the number on
top) and the denominator (the number on the bottom) are whole numbers.
 8 is a rational number because it can be written as the fraction 8/1.
 3/4 is a rational number because it can be written as a fraction.
 0.125 is a rational number because
0.125 = 125/100 = 1/8
i.e. can be written as a fraction
 1.75 is a rational number because
1.75 = 7/4
i.e. can be written as a fraction
Irrational Numbers
All numbers that are not rational are considered irrational. An irrational
number can be written as a decimal, but not as a fraction.
An irrational number has endless non-repeating digits to the right of the
decimal point. Here are some irrational numbers:
π = 3.141592…
√2 = 1.414213…
Rational and
Irrational Numbers
1. Which one of these equations has solutions which are irrational?
A 3x2 = 4
B 3x2 = 4/3
C 3x2 = 3/4
2. Decide whether each of the numbers in the table is rational or irrational; and
put a tick in the appropriate box.
number
Rational
Irrational
√7
3√8
3
1.
27
5
.1
10
5
10.
10
22
7
0.8
1
π
Rational and
Irrational Numbers
D = b2 – 4ac
D is evaluated in each of the following cases:
i. b = 3, a = 1, c = 1.25
ii. b = 10π, a = 8π, c = 2π
iii. b = 12, a = 4, c = 9
iv. b = -6, a = -1, c = -4
3.
Which two of these values of D are irrational?
Rational and
Irrational Numbers
4. Which two of the numbers a, b, c in the answers below are rational?
5cm
O
Area of circle = a cm2
8 cm
W
Z
X
Y
6 cm
Length of EF = b cm
Length of WY, the
diagonal of rectangle
WXYZ = c cm
Rational and
Irrational Numbers
5. Which two of the following are rational?
i.
√5
v. √5-2
ii. 4π
vi. √41/4
iii.
vii. √(22+23)
3√100
iv. √0.4
viii. √(32+33)
Recurring Decimals
A recurring decimal is a decimal fraction which repeats itself forever.
.
1/3 gives the recurring decimal fraction 0.333333... = 0.3
.
.
1/7 gives the recurring decimal fraction 0.142857142857142857... = 0.142857
The recurring pattern is usually indicated by placing a dot over the first digit of the
recurring pattern, and another dot over the final digit of the recurring pattern. Also
known as periodic decimal. The period of a recurring decimal is the number of
digits which forms the repeating pattern.
Recurring Decimals
Example
Express 0.12 as a fraction.
..
①
0.12 = 0.12121212…
Let
x = 0.1212121212…
② We want to move the decimal point to the right, so that the first "block" of repeated
digits appears before the decimal point. Remember that multiplying by 10 moves
the decimal point 1 position to the right.
x = 0.1212121212…
100 x = 12.1212121212…
③
Now we can subtract our original number, x, from both sides to get rid of everything
after the decimal point on the right:
100 x = 12.1212121212…
x = 0.1212121212…
99 x = 12
x = 12
99
Recurring Decimals
1. When written in decimal form, which two of the following are recurring
decimals.
i. √6
iv. 1/4π
ii. √4/49
v. 3√12
iii. √1.5
vi. 4/17
Recurring Decimals
2. Express as a fraction:
.
.
i. 0.688
.
ii. 1.3
.
.
iii. 0.52123
Surds
Surds are numbers left in 'square root form' (or 'cube root form' etc). They are
therefore irrational numbers. The reason we leave them as surds is because in
decimal form they would go on forever and so this is a very awkward way of writing
them.
Addition and Subtraction of Surds
When adding and subtracting surds we need the numbers being square
rooted (or cube rooted etc) to be the same.
Examples:
5√7 - 2√ 7 = 3√ 7
√12 + √75 = √3√4 + √3√25
= 2√3 + 5√3
= 7√3
Surds
Multiplication and Division
√a √b = √ab
√a x √a = a
√a =
√b
a
b
Examples:
√3 x √12 = √36
= 6
√12 =
√6
12 = √2
6
(1 + √3)(2 + √3)
= 2 + √ 3 + 2 √3 + √3 √3
= 2 + 3 √3 +3
= 5 + 3 √4
Surds
Simplify the following:
1. √ 20 =
2. √4 + √4 =
3. √5 x √12 =
4. √16 =
√2
5. (1 + √3)(2 + √3) =
6. (√18 - √2)2 =
Rationalising the
Denominator
It is untidy to have a fraction which has a surd denominator. This can be 'tidied up'
by multiplying the top and bottom of the fraction by a particular expression. This is
known as rationalising the denominator.
Example:
1
√2
1+2
1 - √2
1 x √2
√2
√2
=
=
=
(1 + 2)(1 + √2)
(1 - √2)(1 + √2)
√2
√2 x √2
=
=
√2
2
1 + √2 + 2 + 2√2
1 + √2 - √2 - 2
=
3 + 3√2
-1
Rationalising the
Denominator
Rationalise the following:
(i)
8
√2
=
(ii)
12
√3
=
(iii)
(iv)
2
.
=
3 + √2
=
1+ √3
1 - √3
Additional Notes
Rational and
Irrational Numbers
Learning Outcomes:
At the end of the topic I will be able to


I can distinguish between rational and irrational numbers
I can see the significance of recurring and non-recurring
decimals

I can understand what is meant by a surd

I can use the rules of surds ( eg. √a x√b=√ab)

I can rationalise the denominator of surds

I can expand and simplify surds in brackets
Can
Do
Revise
Further
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