Natural Numbers The counting numbers

1, 2, 3, 4, 5, . . .

IntegersInclude all the whole numbes and zero

. . . -3, -2, -1, 0, 1, 2, 3, . . .

Rational NumbersInclude all the integers plus fractions

Real Numbers

Include all the Rational Numbers plus numbers that cannot be written as fractions

N

Z

Q

R

FactorA factor of a number divides exactly into that number

eg: Factors of 14 are:

1, 2, 7 and 14

Prime NumberA number with exactly TWO factors: (1 and itself)

2, 3, 5, 7, 11, 13, 17, . . .

Prime FactorA factor of a number which is also a prime number is called a prime factor

Prime FactorA factor of a number which is also a prime number is called a prime factor

eg 1: Write 24 as a product of Prime Factors

Prime FactorA factor of a number which is also a prime number is called a prime factor

24

÷ 2

12

÷ 2

6

÷ 2

3

÷ 3

1

24 = 2 x 2 x 2 x 3

= 2 x 33

Keep dividing by prime numbers

until you get to an answer of 1

eg 2: Write 315 as a product of Prime Factors

Prime FactorA factor of a number which is also a prime number is called a prime factor

315

÷ 3

105

÷ 3

35

÷ 5

7

÷ 7

1

315 = 3 x 3 x 5 x 7

= 3 x 5 x 72

Keep dividing by prime numbers

until you get to an answer of 1

eg 1: Write 357 as a product of Prime Factors

Prime FactorA factor of a number which is also a prime number is called a prime factor

357

÷ 3

119

÷ 7

17

357 = 3 x 7 x 17

Keep dividing by prime numbers

until you get to an answer of 1

÷ 17

1

Questions to TryWrite each of these numbers as a product of prime factors.

= 2 x 7= 2 x 2 x 5= 3 x 11= 2 x 19= 5 x 11= 2 x 2 x 2 x 2 x 2 x 2= 2 x 5 x 7= 2 x 2 x 2 x 3 x 5= 2 x 3 x 3 x 7= 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2

1.2.3.4.5.6.7.8.9.10.

14203338556470120126512

How well did you

do?

REAL NUMBERS

Rational Numbers Irrational Numbers

These include all the whole numbers and

numbers which can be written as a fraction.

ALL decimals which recur or terminate can be written as fractions.

These are all the and numbers which can NOT be written as a

fraction.

Examples:pi

Square roots of primes and multiples of primes.

Writing a terminating decimal as a fraction

(1) Write all the digits after the decimal point

(2) Draw a line under these

(3) Put a 0 under each digit

(4) Put a 1 in front of the 0s

(5) Simplify if possible

Examples

0.3737001

= 0.2132130001

= 0.0130130001

=

131000

=

Writing a recurring decimal as a fraction

If the recurring part starts straight after the decimal point, then it’s easy . . .

If the fraction has 1 recurring digit, it’s that digit over 9

If the fraction has 2 recurring digits, it’s those digits over 99

If the fraction has 3 recurring digits, it’s those digits over 999, and so on.

Examples

0.779= 0.38

3899

= 0.462462999

=

154333

=

. . . . .

Writing a recurring decimal as a fraction

If the recurring part doesn’t start straight after the decimal point, then we can express the decimal as a fraction by using methods as illustrated in the following examples:

0.73 x = 0.73

66

90

.

10x = 7.3

90x = 66

100x = 73.3

x =

.

.

.

11

15=

Scale up the above so that recurring partstarts straight after decimal point

Scale the above to line up recurring parts

Subtract the two equations above

Writing a recurring decimal as a fraction

If the recurring part doesn’t start straight after the decimal point, then we can express the decimal as a fraction by using methods as illustrated in the following examples:

0.58 x = 0.58

53

90

.

10x = 5.8

90x = 53

100x = 58.8

x =

.

.

. Scale up the above so that recurring partstarts straight after decimal point

Scale the above to line up recurring parts

Subtract the two equations above

Writing a recurring decimal as a fraction

If the recurring part doesn’t start straight after the decimal point, then we can express the decimal as a fraction by using methods as illustrated in the following examples:

0.658 x = 0.658

652

990

. .

10x = 6.58

990x = 652

1000x = 658.58

x =

. .

. .

. .

326

495=

Scale up the above so that recurring partstarts straight after decimal point

Scale the above to line up recurring parts

Subtract the two equations above

Writing a recurring decimal as a fraction

If the recurring part doesn’t start straight after the decimal point, then we can express the decimal as a fraction by using methods as illustrated in the following examples:

0.174 x = 0.174

173

990

. .

10x = 1.74

990x = 173

1000x = 174.74

x =

. .

. .

. . Scale up the above so that recurring partstarts straight after decimal point

Scale the above to line up recurring parts

Subtract the two equations above

Writing a recurring decimal as a fraction

If the recurring part doesn’t start straight after the decimal point, then we can express the decimal as a fraction by using methods as illustrated in the following examples:

0.369 x = 0.369

366

990

. .

10x = 3.69

990x = 366

1000x = 369.69

x =

. .

. .

. . Scale up the above so that recurring partstarts straight after decimal point

Scale the above to line up recurring parts

Subtract the two equations above

61

165=