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Real Numbers

Surds
● A surd is an irrational number that is represented by a

root sign or a radical sign, for example:  
● Surds: roots of numbers that do not have an exact
answer, so they are irrational numbers
...

● Note: An irrational number written in surd form gives an
exact value of the number; whereas the same number
written in decimal form (for example,
to 4 decimal places) gives an approximate value
...
 
● Besides π a large part of irrational numbers are made up
of surds, a special group of numbers that involve roots
...
So for
example,√4 is not a surd as it can be simplified down to 2
...
One such method is called proof by
contradiction
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Therefore the original assumption must be
false
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The next worked
example sets out to prove that is irrational
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For

example, √250 can be rewritten as 5√10
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Therefore, we can say that 5√10 still has the exact value √250
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When we do this, we will need to use a calculator
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8113883
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In reality,
5√10 is an irrational number! We may not need to be this accurate as the
calculator though, and we are often asked to round to 2 decimal places
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81
...
81
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One way is to consider the nearest integer value as a way to check our
workings
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Well, with irrational numbers, it's much the same
idea
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For example
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If we use a calculator, we
find that the answer is 1
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Therefore √3, doesn’t
have a nice simple pattern, but continue with what looks like a
random string of numbers forever
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7320508757
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7320508757 
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So it is safe to say that 1) Rational + Irrational = Irrational
And
2) Rational × Irrational = Irrational

Exceptions to the rules
● Please Note: There is one important exception to the
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●
●
●
●

rule Rational × Irrational = Irrational
That is when you multiply an irrational number by zero
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There are in fact exceptions to both of these rules
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Irrational + Irrational makes a rational
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2
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Hint: Think
about the definition of √3
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● To simplify a surd (if it is possible), it should be rewritten as a
product of two factors, one of which is a perfect square, that
is, 4, 9, 16 25 36 49 64 81, 100 and so on
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For example, 
 could be written as
 
; however, √8 can be further simplified to 
that is 
If, however, the largest perfect square had
been selected and 
had been written as,
the same answer would be obtained in fewer steps
...
Assume

that x and y are positive real numbers
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Assume

that x and y are positive real numbers
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Note: If a question asks you to
calculate to a given amount of decimal places, you should keep exact
values throughout your calculation and only round at the end!
Complete Lesson 5 - Exact answers vs Decimal approximation and
Lesson 6 – Simplifying Surds

Addition and subtraction of surds
● Surds may be added or subtracted only if they are alike
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Assume that a and b are positive
real numbers
...
For example, 
where a and b are positive real numbers
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Once this has been done and a mixed surd has
been obtained, the coefficients are multiplied with each
other and then the surds are multiplied together
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Assume that x and y are positive
real numbers
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Consider
the following examples:

● Observe that squaring a surd produces the number under




the radical sign
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● When a surd is squared, the result is the number (or
expression) under the radical sign; that is, 
where a is a positive real number
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, where a and b are whole

● When dividing surds it is best to simplify them (if possible)

first
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Example
● Divide the following surds, expressing answers in

the simplest form
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Example - Solution

Complete Lesson 9 – Multiplying and Dividing
Surds

Expanding Binomial Surd Expressions
● Example : Expand and simplify (√3−√10)(√3+√12)

● Complete Lesson 10 – Surds and the binomial expansion

Rationalising denominators using
conjugate surds

● This fact is used to rationalise denominators containing a

sum or a difference of surds
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● Two examples are given below:

Example

Example - Solution

Complete Lesson 11 – Rationalising the denominator



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