This document provides an overview of irrational numbers and surds for a GCSE maths class. It begins by defining rational and irrational numbers, then discusses surds and how to manipulate them within expressions and fractions. Examples are provided of simplifying surds, expanding expressions involving surds, rationalizing denominators, and solving worksheet questions involving surds. The document aims to help students understand the difference between rational and irrational numbers and how to work with surds.

1. GCSE – Irrational Numbers and Surds
Dr Frost
Objectives: Appreciate the difference between a rational and
irrational number, and how surds can be manipulating both within
brackets and fractions.

2. Learning Objectives
By the end of this topic, you’ll be able to answer the
following types of questions:

3. Types of numbers
Real Numbers
Real numbers are any
possible decimal or
whole number.
Rational Numbers Irrational Numbers
are all numbers which
can be expressed as
some fraction involving
integers (whole
numbers), e.g. ¼ , 3½, -7.
are real numbers which
are not rational.

4. Rational vs Irrational
Activity: Copy out the
Venn diagram, and put
the following numbers
into the correct set.
3 0.7
π
.
1.3
√2 -1
3
4
√9 e
Edwin’s exact
height (in m)
Integers
Rational numbers
Irrational numbers

5. What is a surd?
Vote on whether you think the following are surds or not surds.
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.


Not a surd Surd
 
Not a surd Surd


Not a surd Surd
 
Not a surd Surd
?
3
7


Not a surd Surd

6. Law of Surds
?
?
And that’s it!

7. Law of Surds
Using these laws, simplify the following:
? ?
? ?
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8. Expansion involving surds
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?
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Work these out with neighbour. We’ll feed back in a few minutes.
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9. 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.
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10. Simplifying surds
This sometimes helps us to further manipulate larger expressions.
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11. Expansion then simplification
Put 4 − 2 3 + 8 in the form 𝑎 + 𝑏 2, where 𝑎 and 𝑏 are integers.
= 𝟖 + 𝟓 𝟐
Put 3 + 3 1 + 27 in the form 𝑎 + 𝑏√3, where 𝑎 and 𝑏 are integers.
= 𝟏𝟐 + 𝟏𝟎 𝟑
?
?

12. Exercises
Edexcel GCSE Mathematics
Page 436 Exercise 26E
Q1, 2

13. Rationalising Denominators
Here’s a surd. What could we multiply it by such that it’s no
longer an irrational number?
? ?

14. 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.

15. ? ?
?
Rationalising Denominators
2 + 2
2
= 2 + 1
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16. (End at this slide except for Set 1)
Edexcel GCSE Mathematics
Page 436 Exercise 26E
Q3-8
Exercises

17. Wall of Surd Ninja Destiny
Write 2 + 2 3 + 8 in the
form 𝑎 + 𝑏√2, which 𝑎 and 𝑏 are
integers.
= 𝟏𝟎 + 𝟕 𝟐
Simplify 48
= 𝟒 𝟑
Rationalise the
denominator of
8
2
= 𝟒 𝟐
Calculate 2 + 2 2 − 2 .
= 𝟐
?
?
?
?

18. Rationalising Denominators
What is the value of the following. What is
significant about the result?
3 + 2 3 − 2 = 7
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?
5 − 3 × 5 + 3 = −4
?
?

19. Examples
5
6 − 2
=
5 6 − 5
2
2
7 + 3
=
2 5 − 2 3
4
Rationalise the denominator. Think what we need to multiply
the fraction by, without changing the value of the fraction.
?
?

20. Recap
8
2
= 4 2 128 = 8 2
27 + 48 = 7 3
3
3 − 2
=
3 3 + 6
7
? ?
? ?

21. 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.

22. Xbox One vs PS4
300 = 10 3 700 = 10 7
? ?

23. Xbox One vs PS4
6 + 3 1 − 2 3
= −11 3 ?
5 + 5 3 − 3 5
= −12 5 ?

24. Xbox One vs PS4
3 − 2
3 + 2
= 5 − 2 6
6 + 5
6 − 5
= 11 + 2 30
?
?

25. Difficult Worksheet Questions
Section D, Qa)
Factorise 𝑎 + 𝑏 − 2 𝑎𝑏
Section D, Qc)
Factorise 𝑎 + 𝑏 − 2 𝑎𝑏