## Types of Numbers

This section introduces a selection of types of numbers and mathematical operations.

### Prime Numbers prime number has only two factors − itself and 1.

e.g.

 Prime Number 2 7 29 97 Factors 2, 1 7, 1 29, 1 97, 1

The first prime number and the only EVEN prime number is 2. All of the other prime numbers are ODD

1 is not a prime number.

The prime numbers less than 100 are given in the table below:

 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59 61 67 71 73 79 83 89 91 97

Is there any pattern to these numbers? It would appear not!

### Exponents

An exponent, sometimes called an index or power is used to show how many times a number is to be multiplied by itself.

e.g. 3 × 3 × 3 × 3 can be written 34
103 means 10 × 10 × 10

When a number is multiplied by itself once, we say it is "squared"

e.g. 32 is read "three squared" and means 3 × 3 which equals 9

When a number is multiplied by itself twice, we say it is "cubed"

e.g. 43 is read "four cubed" and means 4 × 4 × 4 which equals 64

### Factors

The factors of a number are the numbers that divide into it without any remainder.

e.g. Because 3 divides into 30 EXACTLY, that means that 3 is a FACTOR of 30.

Some numbers have lots of factors.

Here are some numbers and their factors:

 Number 8 45 72 Factors 1, 2, 4, 8 1, 3, 5, 9,15, 45 1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36, 72

The highest common factor (HCF) of numbers is the highest factor shared by the numbers.

e.g.Using the table above you can see that the highest common factor of 45 and 72 is 9.

Using the table, what would the HCF of 72 and 100 be?

### Prime Factors

Numbers can be written as the product of prime numbers. e.g. 30 = 2 × 3 × 5

To work out the prime factors of numbers factor "trees" can be used.

 Number 20 42 72 Factor Tree Prime factors underlined.   Prime Factors 20 = 5 × 2 × 2 = 5 × 22 42 = 7 × 3 × 2 72 = 3 × 3 × 2 × 2 × 2 = 32 × 23

### Composite Numbers

Composite numbers are whole numbers greater than one with more than two factors. i.e. The numbers that are NOT prime numbers.

e.g. 12 is a composite number as it has six factors {1, 2, 3, 4, 6, 12}

The composite numbers less than 10 are shown below.

 Composite number 4 6 8 9 10 Factors 1, 2, 4 1, 2, 3, 6 1, 2, 4, 8 1, 3, 9 1, 2, 5, 10

### Multiples

The multiples of a natural number are formed by multiplying the number by 1, 2, 3, ...
Multiples go on forever!

The first seven multiples of some numbers are shown below.

 Multiples of 4 4, 8, 12, 16, 20, 24, 28,... Multiples of 11 11, 22, 33, 44, 55, 66, 77,... Multiples of 20 20, 40, 60, 80, 100, 120, 140,...

The lowest common multiple of numbers is the lowest multiple that is shared by the numbers.

e.g. The lowest common multiple (LCM) of 4 and 6 is 12.
The lowest common multiple (LCM) of 3 and 5 is 15.

### Reciprocals

The reciprocal of a number is equal to The reciprocal of 0 cannot be calculated. i.e. The fraction 1 0 cannot be calculated.

To find the reciprocal of a fraction, turn the fraction upside down.
e.g.The reciprocal of 25 is 52

A number multiplied by its reciprocal always equals 1.
e.g. 34 × 43 = 1

Most calculators have a reciprocal button or 1/x.

### Squares

The square of a number is the number multiplied by itself.
e.g. 5 2 = 5 × 5 = 25

The first 4 square numbers are shown below:

 Square numbers 1 × 1 = 1 2 × 2 = 4 3 × 3 = 9 4 × 4 = 16 Square    ### Square Roots

The square root of a number, shown by the sign , is the positive number that, when multiplied by itself, gives the number.

e.g. and The square roots of the square numbers up to 100 are shown below:

 Number 1 4 9 16 25 36 49 64 81 100 Square root 1 2 3 4 5 6 7 8 9 10