ζGCSE – Irrational Numbers and SurdsDr Frost

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 NumbersReal 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.

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

34 √9 eEdwin’s exact

height (in m)

Integers

Rational numbers

Irrational numbers

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

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3√7 Not a surd Surd

Law of Surds

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And that’s it!

Law of SurdsUsing these laws, simplify the following:

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Expansion involving surds

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Work these out with neighbour. We’ll feed back in a few minutes.

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Simplifying surdsIt’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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Simplifying surdsThis sometimes helps us to further manipulate larger expressions.

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Expansion then simplification

Put in the form , where and are integers.

Put in the form , where and are integers.

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Exercises

Edexcel GCSE MathematicsPage 436 Exercise 26EQ1, 2

Rationalising Denominators

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

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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?

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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.

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Rationalising Denominators

2+√2√2

=√2+1?

(End at this slide except for Set 1)

Edexcel GCSE MathematicsPage 436 Exercise 26EQ3-8

Exercises

Wall of Surd Ninja Destiny

Write in the form , which and are integers.

Simplify

Rationalise the denominator of

Calculate .

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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?

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Examples

5√6−2

=5√6−52

2√7+√3

=2√5−2√34

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

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Recap

8√2

=4√2 √128=8√2

√27+√48=7 √3√33−√2

=3√3+√67

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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√3+√2

=5−2√6 √6+√5√6−√5

=11+2√30??

Difficult Worksheet Questions

Section D, Qa)Factorise

Section D, Qc)Factorise