## Types of Numbers

### 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}

### Number sets

The numbers we use can be chosen from several sets.

The set of natural numbers.

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

Number lines The number sets can be shown on number lines.

 N W I Q R
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 ={ab where a and b are integers and b ≠ 0}
e.g. The set of irrational numbers
Irrational numbers are numbers that cannot be written as rational numbers.

e.g. {infinite, non-recurring decimals} The set of real numbers

R = {all numbers on the number line}

### Types of Numbers

Multiples

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

e.g. 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..

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}

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.

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.

### 12

Prime Factors

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

e.g. 30 = 2 × 3 × 5

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, shown by the sign , is the positive number that, when multiplied by itself, gives the number.

e.g. and Cube Roots

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

e.g. Squares

The square of a number is the number multiplied by itself. A square is a number or term to the power of 2.

e.g. 5 2 = 5 × 5 = 25 and (-3) = -3 × -3 = 9

Cubes

The cube of a number is the number multiplied three times. A cube is a number or term to the power of 3.

e.g. 5 3 = 5 × 5 × 5= 125 and (-3) = -3 × -3 × -3 = -27

Reciprocals

The reciprocal of a number is equal to 34 × 43 = 1. The reciprocal of 0 is not defined. i.e. The fraction 10 cannot be calculated.

To find the reciprocal of a fraction, turn the fraction upside down. e.g.The reciprocal of 25 is 5⁄2

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

Prime Numbers

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}

### Calculators and Types of Numbers The calculator buttons below are from a typical scientific calculator.

In some cases the answers given below will need to be rounded.

 Type of Number Button Example Answer Square roots Find square root of 37    6.08276253 Cube roots Find cube root of 29    3.072316826 Squares Find square of 42    1764 Cubes Find cube of 14    2744 Reciprocals Find reciprocal of 16    0.0625 Fractions Caclulate 14⁄5 ÷ 9⁄10           2 Powers Find 45    1024