ζ

Dr Frost

GCSE – Irrational Numbers and Surds

Objectives: Appreciate the difference between a rational and
irrational number, and how surds can be manipulating both within
brackets and fractions.
Learning Objectives
By the end of this topic, you’ll be able to answer the
following types of questions:
Types of numbers

Real numbers are any

Real Numbers possible decimal or
whole number.

Rational Numbers Irrational Numbers

are all numbers which are real numbers which

can be expressed as are not rational.
some fraction involving
integers (whole
numbers), e.g. ¼ , 3½, -7.
Rational vs Irrational
Activity: Copy out the
Venn diagram, and put
the following numbers
Irrational numbers into the correct set.

Rational numbers

Integers 3 0.7
.
π 1.3
√2 -1

Edwin’s exact
3
4
√9 e
height (in m)
What is a surd?
Vote on whether you think the following are surds or not surds.

Not
a surd 
Surd

Not asurd 
Surd

Not
a surd 
Surd

Not asurd 
Surd

Not
a surd 
Surd
√7
3

Therefore, can you think of a suitable definition for a surd?

A surd is a root of a number that cannot ?

be simplified to a rational number.
Law of Surds

And that’s it!

Law of Surds
Using these laws, simplify the following:

? ?

? ?

?
Expansion involving surds
Work these out with neighbour. We’ll feed back in a few minutes.

? ?

?
Simplifying surds
It’s convention that the number inside the surd is as small as possible, or the
expression as simple as possible.
This sometimes helps us to further manipulate larger expressions.

? ?

? ?
Simplifying surds
This sometimes helps us to further manipulate larger expressions.

? ?

?
Expansion then simplification

Put in the form , where and are integers.

?
Put in the form , where and are integers.

?
Exercises

Edexcel GCSE Mathematics

Page 436 Exercise 26E
Q1, 2
Rationalising Denominators

Here’s a surd. What could we multiply it by such that it’s no

longer an irrational number?

? ?
Rationalising Denominators
In this fraction, the denominator is irrational. ‘Rationalising the
denominator’ means making the denominator a rational number.

What could we multiply this fraction by to both rationalise the

denominator, but leave the value of the fraction unchanged?

? ?

There’s two reasons why we might want to do this:

1. For aesthetic reasons, it makes more sense to say “half of root 2” rather
than “one root two-th of 1”. It’s nice to divide by something whole!
2. It makes it easier for us to add expressions involving surds.
Rationalising Denominators

? ?

2+ √ 2
? = √ 2+1
?
√2
Exercises

Edexcel GCSE Mathematics

Page 436 Exercise 26E
Q3-8

(End at this slide except for Set 1)

Wall of Surd Ninja Destiny

Write in the form , which and are

integers.

Simplify

Rationalise the ?
denominator of

?
Calculate .

?
Rationalising Denominators

What is the value of the following. What is

significant about the result?
?

This would suggest we can use the difference of two squares to

rationalise certain expressions.

What would we multiply the following by to make it rational?

?
Examples
Rationalise the denominator. Think what we need to multiply
the fraction by, without changing the value of the fraction.

5 5 √ 6 −5
= ?
√ 6 −2 2

2 2 √ 5 −2 √ 3
= ?
√ 7+ √ 3 4
Recap

8
=4 √
?2 √ 128=8 ?√ 2
√2

√ 3 = 3 √ 3+ √ 6
?
√ 27 + √ 48=7 √? 3 3 −√2 7
Xbox One vs PS4

The left side of the class is Xbox One.

The right side is PS4.

Work out the question for your console. Raise your hand
when you have the answer (but don’t say it!). The winning
console is the side with all of their hands up first.
Xbox One vs PS4

√ 300=10?√ 3 √ 700=10?√ 7
 Xbox One vs PS4

(6 +√ 3 )( 1−2 √3 )=−11 √ 3 (5 +√ 5)(3 −3 √ 5)=−12 √ 5

? ?
Xbox One vs PS4

√ 3 − √ 2 =5 − 2? √ 6 √ 6+ √ 5 =11+2? √ 30
√ 3+ √ 2 √6 −√5
Difficult Worksheet Questions

Section D, Qa)
Factorise

Section D, Qc)
Factorise