# Special Numbers and Concepts

This page explains several particular types of numbers and terms used in mathematics:

Like any other subject, mathematics has its own language to some extent. This page will take you one step closer to understanding the language of mathematics.

## Prime Numbers

A prime number can only be divided by itself and 1 (one) to leave a whole number (integer) answer.

A mathematician may say: A prime number is a number that has only two integer divisors: itself and one.

Prime Number Example

Examples of prime numbers include 2, 3, 5, 7, 11, 13, 17, 19, 23, and 29, but there are an infinite amount of larger prime numbers too.

7 is a prime number since it can only be divided by itself or 1 to leave a whole number.

7 ÷ 7 = 1 and 7 ÷ 1 = 7

If you divide 7 by any other number the answer is not a whole number.

7 ÷ 2 = 3.5 or 7 ÷ 5 = 1.4

9 is not a prime number. 9 can be divided by itself, 1 and 3 to leave a whole number.

9 ÷ 9 = 1 and 9 ÷ 1 = 9 and 9 ÷ 3 = 3

Some quick facts about prime numbers:

• 1 is NOT a prime number. A prime number, by definition, has to have exactly two positive divisors. 1 only has one positive divisor (1).
• 2 is the only even prime number, because all other even numbers, of course, divide by 2.
• The 1000th prime number is 7,919.
• Euclid, the Greek mathematician, demonstrated in around 300BC that there are an infinite number of prime numbers.

Prime numbers are important in mathematics and computing. For most of us, however their use is probably limited to interest, and to knowing when you’ve reached the limit of simplifying a fraction. See our page: Fractions for more information on working with fractions.

## Squares and Square Roots

The square of a number is the number that you get if you multiply that number by itself. It is written as a superscripted 2 after the number to which it applies, so we write x2, where x is any number.

For example, if x were 5:
52 = 5 x 5 = 25.

Square numbers are used in area calculations as well as elsewhere in mathematics.

Suppose you want to paint a wall which is 5 metres high by 5 metres wide. Multiply 5m × 5m to give you 25m2 . If this is said aloud it would be ‘twenty five metres squared’. You would need to buy enough paint for 25m2. You might see this referred to as ’25 square metres’ as well, which is correct. However, a 25m square is not the same thing at all – this would be 25m x 25m = 625m2.

See our page: Calculating Area for more

The square root of a number is the number that is squared to obtain that number. The square root symbol is √

Square roots are easier to understand with examples:

√25 = 5, i.e. 5 is the square root of 25 since 5 x 5 =25
√4 = 2, i.e. 2 is the square root of 4 since 2 x 2 =4

Not all numbers have a square root that is an integer. For example, √13 is 3.60555.

## Orders, Exponents, Indices and Powers

In a square number, the superscript 2 is the ‘order’ of x, i.e. the number of times x is multiplied by itself. The order can be any number, positive or negative.

For example:
23 = 2 x 2 x 2 = 8
510 = 5 x 5 x 5 x 5 x 5 x 5 x 5 x 5 x 5 x 5 = 9,765,625

Orders are also called exponents, indices and powers. When said aloud, the first example might be referred to as ‘two to the power three’ and the second would be ‘five to the power ten’ or ‘five exponent ten’. The terms are interchangeable and are sometimes regional. For example, the usual term in North America is ‘exponent’, but in the UK it is more usually indices or powers.

### Standard Form

Orders are used to express very large and very small numbers using a type of mathematical abbreviation known as Standard Form. Standard Form is also sometimes called ‘scientific notation’.

Standard form is written as a x 10n.

In this form, a is a number bigger than or equal to 1 and less than 10.

The order n can be any positive or negative whole number, and is the number of times a must be multiplied by 10 to equal the very large or very small number that we are writing.

For example:

2,000,000 = 2 x 10 x 10 x 10 x 10 x 10 x 10 = 2 x 106.
5 x 10-5 = 0.00005

The use of Standard Form reduces the number of digits we need to write. It also helps eliminate errors – it isn’t easy to accurately read this many zeros:
1.23 x 1012 = 1,230,000,000,000
4 x 10-15 = 0.000000000000004

Warning!

When the power is positive, it tells you how many zeros to add to the number that is being multiplied by 10.

For 2 x 106, add 6 zeros to 2, and get 2,000,000.

However, when the power is negative, the number of zeros after the decimal point is one less than the order.

1 x 10-3 is 0.001

This is because you have to divide by 10 once to move the number itself to the other side of the decimal point.

Another way of looking at it is by counting the number of places we move the decimal point.

For 2.0 x 106, we move the decimal point six places to the right, to give 2,000,000.0. Adding ‘.0’ to the end of the number doesn’t change its value, but helps when counting decimal places.

Similarly, for 1.0 x 10-3, we move the decimal point three places to the left, to give 0.001.

## Factors and Multiples

Factors are numbers that divide or ‘go’ a whole number of times into another.

For example, 2, 3, 5 and 6 are all factors of 30.

Each of them goes into 30 a whole number of times. Another way of describing this using more mathematical language is to say 30 can be divided by 2, 3, 5 and 6 to give integer answers.

Multiples are the numbers that you get when you multiply one number by another.

4, for example, is a multiple of 2.

30 is a multiple of 15, 6, 5, 3 and 2.

## Infinite Numbers (Irrational Numbers)

The phrase ‘infinite numbers’ does not refer to the fact that there are an infinite number of numbers. Instead, it refers to numbers that do not themselves ever end.

The best-known infinite number is probably pi, π, which starts 3.142 and goes on from there. Not even the most powerful computer programme in the world could ever map all of its numbers, because it is infinite.

These numbers are also called irrational numbers.

Finite numbers are numbers that have a finite number of digits. After a certain point, the only number that can be added is zero. 1, 3, 1.5, and 0.625 are all examples of finite numbers.

Recurring numbers are one particular form of infinite numbers. Here, the same one or few digits repeat infinitely in the decimal form of the number.

Some numbers which can be expressed easily as fractions turn out to be recurring numbers in the decimal form.

Examples include 1/3, which is 0.33333 recurring in decimals, and 1/11 which is 0.090909090909 recurring.

## Real, Unreal and Complex Numbers

Real numbers are numbers that actually exist and can have a physical value placed on them.

Real numbers can be positive or negative, and may be integers (whole numbers) or decimals. They may even be infinite numbers, but they can be written as numbers and expressed in numerals.

Imaginary numbers, as their name suggests, do not actually exist, but are a mathematical construct to solve certain problems.

The simplest example is the square root of a minus number. We can only obtain a minus (negative) number by multiplying a negative number by a positive number. If you multiply two negative numbers or two positive numbers, you always get a positive answer. It therefore follows that the square root of a negative number cannot exist.

However, it can in mathematics! The square root of minus one is given the notation i. Actually using it in real world maths problems initially requires a bit of abstract thinking, but it is a very useful concept in some applications.

Complex numbers follow from real and unreal numbers. They are numbers composed of a real number multiplied by an unreal or imaginary number, usually denoted by some multiple of i.

Further Reading from Skills You Need

The Skills You Need Guide to Numeracy This four-part guide takes you through the basics of numeracy from arithmetic to algebra, with stops in between at fractions, decimals, geometry and statistics.