# 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 1, 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:

• 2 is the only even prime number. 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, but for most of us, their use is probably limited to interest, and to knowing when you’ve reached the limit of dividing down 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 x2 (where x is any number).

For example:
52 = 5 x 5 = 25.

Square numbers are used in area calculations as well as other areas of mathematics. Suppose you want to paint a wall which is 5 meters high by 5 meters wide. Multiply 5m × 5m to give you 25m2. You would need to buy enough paint for 25m2.

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 whole square root. For example, √13 is 3.60555.

## Exponents and Powers

Squares are particular types of exponents, also known as powers. For exponents, the superscript number, instead of always being 2 as it is for squares, can be any number, and that tells you how many times to multiply the number itself by.

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

The other way that exponents are often used is to express very large and very small numbers in terms of the number of times that they are multiplied by 10.

For example:
2 x 106 = 2 x 10 x 10 x 10 x 10 x 10 x 10 x 10 = 2,000,000.
5 x 10-5 = 0.00005

The use of exponents here reduces the number of digits that very large and very small numbers are expressed in.

For example:
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 superscript number.

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.

Also note: When expressing a numbers in terms of the number of times it is multiplied by 10, you always need the number itself to be between 1 and 9 inclusive.

## 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.

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.

Unreal numbers or imaginary numbers do not actually exist, but are a mathematical construct to solve certain problems.

The simplest example is the square root of a minus number. This makes sense because you can only obtain a minus number by multiplying a negative number by a positive number. If you multiply two negative numbers or two positive numbers, you get a positive number. 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.

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.

### Not exactly everyday concepts?

Some of the concepts described on this page may be more or less useful in everyday life. However, it never hurts to have a basic understanding of some of the simpler mathematical concepts.

You never know, one day you might meet a mathematician at a party, and now you’ll have something in common…

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