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

### Sets

Although set theory and operations are not now included in this course it is useful to know what a set is, and about the various number sets.

**Notation**

A set is a collection of objects. These objects, called the members or elements of the set, are enclosed in braces {...} and separated by commas.

A set can be described in words: A = { the first three natural numbers}

or by listing each member: A = {1, 2, 3}

The numbers we use can be chosen from several sets.

**The set of natural numbers.**N = {1, 2, 3, 4, ...}

**The set of whole numbers.**W = {0, 1, 2, 3, ...}

**The set of integers.**I = {... -2, -1, 0, 1, 2, 3,...}

**The set of rational numbers**Q =

e.g.

**The set of irrational numbers**Irrational numbers are numbers that cannot be written as rational numbers.

**The set of real numbers**

**Number lines**

The number sets can be shown on number lines.

Natural numbers ( Whole numbers (includes zero) Integers (positive and negative whole numbers) Rational numbers (all except irrationals) Real numbers (all numbers) |

### Multiples

The **multiples** of a natural number are formed by multiplying the number by 1, 2, 3, ...

e.g. Multiples of 5 are 5, 10, 15, 20, ...

Multiples of 11 are 11, 22, 33, 44, ...

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.

### Factors

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

e.g. Factors of 8 = {1, 2, 4, 8}

Factors of 30 = {1, 2, 3, 5, 6, 10, 15, 30}

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

e.g. Highest common factor of 8 and 12 is 4.

Highest common factor of 12 and 18 is 6.

**Divisibility of Numbers**

There are some useful "tricks" to help you decide whether a number is a factor of another number. This is called **divisibility**.

Click the numbers below to find out what they are.

## 2 |
## 3 |
## 4 |
## 5 |
## 6 |
## 7 |
## 8 |
## 9 |
## 10 |
## 11 |
## 12 |

### Prime Numbers

A **prime number **has only two unique factors − itself and 1.

The only even prime number is 2.

1 is not a prime number.

The first eight prime numbers are {2, 3, 5, 7, 11, 13, 17, 19}

### Prime Factors

Natural numbers can be written as the product of prime numbers.

e.g. 30 = 2 × 3 × 5

12 = 2 × 2 × 3 = 2^{2} × 3

### Composite Numbers

Composite numbers are numbers with more than two factors. i.e. The non-prime numbers.

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

### Square Roots

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

Positive numbers have two square roots, one positive and one negative.

e.g. if x^{2} = 9 then x = ±√9 = +3 or -3

### Squares

The square of a number is the number multiplied by itself.

e.g. 5 ^{2} = 5 × 5 = 25

### Reciprocals

The reciprocal of a number is equal to

The reciprocal of 0 is not defined. 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 ^{2}⁄_{5} is ^{5}⁄_{2}

A number multiplied by its reciprocal always equals 1.

e.g.^{3}⁄_{4}×^{4}⁄_{3}= 1

Most calculators have a reciprocal button ^{1}⁄_{x} or ^{1}/_{x}

### Standard Form

Numbers written in standard form are shown as the product of a number greater or equal to 1 and less than 10 and a power of 10.

e.g. 327 = 3.27 × 10^{ 2}

0.46 = 4.6 × 10 ^{- 1}

Standard form, sometimes called scientific notation, is often used in science to show very large or very small numbers.