GCSE Irrational Numbers and Surds
Appreciate the difference between rational and irrational numbers, manipulate surds within brackets and fractions, identify types of numbers, define surds, apply laws of surds, simplify and expand surds, and understand the convention of simplifying surds for further manipulation.
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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 possible decimal or whole number. Real Numbers Rational Numbers Irrational Numbers are real numbers which are not rational. are all numbers which can be expressed as some fraction involving integers (whole numbers), e.g. , 3 , -7.
Rational vs Irrational Activity: Copy out the Venn diagram, and put the following numbers into the correct set. Irrational numbers Rational numbers 3 0.7 Integers . 1.3 2 -1 3 4 9 e Edwin s exact 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 37 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 4 2 3 + 8 in the form ? + ? 2, where ? and ? are integers. = ? + ? ? ? Put 3 + 3 1 + 27 in the form ? + ? 3, 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 2 + 2 form ? + ? 2, which ? and ? are integers. 3 + 8 in the ? = ?? + ? ? Simplify 48 = ? ? ? Rationalise the denominator of 8 2 = ? ? ? Calculate 2 + 2 2 2 . = ? ?
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 ?
Examples Rationalise the denominator. Think what we need to multiply the fraction by, without changing the value of the fraction. 6 2=5 6 5 5 ? 2 7 + 3=2 5 2 3 2 ? 4
Recap 8 2= 4 2 ? 128 = 8 2 ? 3 2=3 3 + 6 3 ? 27 + 48 = 7 3 ? 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 700 = 10 7 ? 300 = 10 3 ?
Xbox One vs PS4 5 + 5 3 3 5 = 12 5 6 + 3 1 2 3 = 11 3 ? ?
Xbox One vs PS4 6 + 5 6 5= 11 + 2 30 3 2 3 + 2= 5 2 6 ? ?
Difficult Worksheet Questions Section D, Qa) Factorise ? + ? 2 ?? Section D, Qc) Factorise ? + ? 2 ??
