REAL

NUMBERS

{1, 2, 3, 4, . . . }

If you were asked to count, the numbers you’d say are called counting numbers. These numbers can be expressed using set notation.

These are also called the natural numbers.

{0, 1, 2, 3, 4, . . . }If we include 0 we have the set of whole numbers.

{ …, -3, -2, -1, 0,1, 2, 3, . . . }

Include the opposites of the whole numbers and you have the set of integers.

rational numbers

Whole numbers are a subset of integers and counting numbers are a subset of whole numbers.

integers

whole numbers

counting numbers

If we express a new set of numbers as the quotient of two integers, we have the set of rational numbers

This means to divide one integer by another or “make a fraction”

rational numbers

There are numbers that cannot be expressed as the quotient of two integers. These are called irrational numbers.

integers

whole numbers

counting numbers

2

irrational

numbers

The rational numbers combined with the irrational numbers make up the set of real numbers.

REAL NUMBERS

Translating English to Maths

sum of two numbers

difference between two numbers

The product of two numbers

the quotient of two numbers

is =

ab

a - b

a + b

ba

ORDER OF OPERATIONS

When there is more than one symbol of operation in an expression, it is agreed to complete the operations in a certain order. A mnemonic to help you remember this order is below.

B I M D A Srackets

ndicesultip

lication

ivisiondditio

n

ubtraction

Do any simplifying possible inside of brackets starting with innermost brackets and working out

Apply IndicesComplete multiplication and division from left to rightComplete addition and subtraction from left to right

423532 2

BIMDASBIMDAS

brackets – combine these first

42322 2

BIMDAS

indices – apply the indice now

42342

BIMDAS

complete multiplication and division, left to right

468

BIMDAS

complete addition and subtraction, left to right

10

423532 2

COMMUTATIVE PROPERTY

The operations of both addition and multiplication are commutative

abba 3223 When adding, you can “commute” or trade the terms places

abba 3223

When multiplying, you can “commute” or trade the factors places

ASSOCIATIVE PROPERTY

cbacba 321321

When adding, you can “associate” and add any terms first and then add the other term.

cbacba 432432

When multiplying, you can “associate” and multiply any factors first and then multiply the other factor.

The operations of both addition and multiplication are associative

DISTRIBUTIVE PROPERTY

The operation of multiplication distributes over addition

acabcba 4323423

The distributive property also holds for a factor that is multiplied on the left.

acabacb

2423243

abba A positive

times a negative is

NEGATIVE

abba A negative

times a positive is

NEGATIVE

aa The negative of a negative

POSITIVE

CAUTION: Remember that the value for a and/or b could also be positive or negative.

b

a

b

a

b

a

A positive

divided by a negative or

A negative divided by a positive is

NEGATIVE

b

a

b

a

A negative

divided by a negative is POSITIVE

Acknowledgement

I wish to thank Shawna Haider from Salt Lake Community College, Utah USA for her hard work in creating this PowerPoint.

www.slcc.edu

Shawna has kindly given permission for this resource to be downloaded from www.mathxtc.com and for it to be modified to suit the Western Australian Mathematics Curriculum.

Stephen CorcoranHead of MathematicsSt Stephen’s School – Carramarwww.ststephens.wa.edu.au