Aims:• To be to be able to classify types of numbers

• To be able to write a surd in its simplest form

• To be able to add, subtract and multiply surds

SURDS Lesson 1

Types of numberWe can classify numbers into the following sets:

The set of natural numbers, : Ν Positive whole numbers {0, 1, 2, 3, 4 …}

The set of integers, : Positive and negative whole numbers {0, ±1, ±2, ±3 …}

The set of rational numbers, : Numbers that can be expressed in the form , where n and m are integers. All fractions

and all terminating and recurring decimals are rational numbers; for example, ¾, –0.63, 0.2.

nm

The set of real numbers, : All numbers including irrational numbers; that is, numbers that cannot be expressed in

the form , where n and m are integers. For example, and .nm

2

Q

Numbers written in this form are called surds.

When the square root of a number, for example √2, √3 or √5 ,is irrational, it is often preferable to write it with the root sign.

2, 3 or 5,

Manipulating surdsWhen working with surds it is important to remember the following two rules:

You should also remember that, by definition, √a means the positive square root of a.a

a ab b

=

and

ab a b= ×

Also: × =a a a

Simplifying surds

Start by finding the largest square number that divides into 50.

We can do this using the fact that For example:

ab a b= × .

We are often required to simplify surds by writing them in the form .a b

Simplify by writing it in the form 50 .a b

Simplifying surds

Simplify the following surds by writing them in the form a√b.

1) 45 2) 98 33) 40

.a b

Simplifying surds

Adding and subtracting surdsSurds can be added or subtracted if the number under the square root sign is the same. For example:

Simplify 45 + 80.

Start by writing and in their simplest forms.45 80

Basic multiplying and dividing surds

105

181233

232

Expanding brackets containing surdsSimplify the following:

1) (4 2)(1+ 3 2) 2) ( 7 2)( 7 + 2)

Problem 2) demonstrates the fact that (a – b)(a + b) = a2 – b2.

In general:

( )( + )a b a b a b

Do exercise 2A page 30 (Do a, c, e questions from each number)