Types of Numbers

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}

Number sets

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.

e.g. {infinite, non-recurring decimals}

• The set of real numbers

R = {all numbers on the number line}

Number lines

The number sets can be shown on number lines.

 Natural numbers (excludes zero) 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.

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}

Prime Factors

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

e.g. 30 = 2 × 3 × 5
12 = 2 × 2 × 3 = 22 × 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.

e.g. and

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

e.g. if x2 = 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

(-3) = -3 × -3 = 9

Reciprocals

The reciprocal of a number is equal to

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 52

A number multiplied by its reciprocal always equals 1.

e.g. 34 × 43 = 1

Most calculators have a reciprocal button 1x 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.