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In this article, we will define what numbers are, and we will explore the main types of numbers that you can find so that you can recognise them more easily. We will also explain what is studied by number theory, the different number systems and the concept of number sequence.

## What are numbers?

Numbers are considered the heart of Mathematics, and rightly so because without numbers Maths would simply not exist.

A **number** is a mathematical concept that represents quantity, which has many applications such as counting, measuring, labelling, and performing calculations that help us to solve problems, among others.

### Examples of numbers

Numbers come in many varieties. Let's see some examples below.

Examples of numbers of different types:

\[-3,0,2,3.8,\dfrac{3}{4},\pi, \text{and} \sqrt{2}\]

As you can see, there are many types of numbers, let's identify each one in the following section so that you can recognise them more easily.

## Types of numbers

Numbers can be classified into different groups according to the types of numbers that they include. Let's take a look.

### Natural and whole numbers

**Natural numbers** are also known as counting numbers, because these are the numbers that you first learn how to count with. They include all positive numbers greater than zero. That is \(1, 2, 3, 4, 5, 6\), and so on.

They are represented with the letter \(\mathbb N\). The set notation for natural numbers is as follows:

\[\mathbb{N}=\{1,2,3,4,5,...\}\]

Whole numbers are closely related to natural numbers, with one main difference, as you can see from the definition below.

**Whole numbers** are basically the** natural numbers plus zero**. They do not include negative numbers, fractions or decimals.

They are represented with the letter \(\mathbb{W}\), and their set notation is shown below:

\[\mathbb{W}=\{0,1,2,3,4,5,...\}\]

All natural numbers are whole numbers, but not all whole numbers are natural numbers, this is the case for zero. Let's see this in a diagram.

Natural and whole numbers can be represented on the number line as follows:

### Integers

The **integer numbers** include all positive numbers, zero and negative numbers. Again, integers do not include fractions or decimals.

They are represented with the letter \(\mathbb{Z}\), and their set notation is as follows:

\[\mathbb{Z}=\{...,-4,-3,-2,-1,0,1,2,3,4,...\}\]

If we expand the previous diagram to include integers, it will look like this:

Looking at the diagram above, we can say that **not all integers are natural and whole numbers**, but **all natural and whole numbers are integers**. On the number line, integers can be represented like this:

Some examples of integer numbers are:

\[-45,12,-1,0,23\space\text{and}\space 946\]

### Rational and irrational numbers

**Rational numbers** include all numbers that **can be expressed as a fraction** in the form \(\dfrac{p}{q}\), where \(p\) and \(q\) are integers and \(q\neq 0\). This group of numbers **includes fractions and decimals**. Rational numbers are represented with the letter \(\mathbb{Q}\).

All integers, natural and whole numbers are rational numbers, as they can be expressed as a fraction with a denominator of \(1\). For example, \(3\) can be expressed as a fraction like this \(\dfrac{3}{1}\).

Some examples of rational numbers are:

\[-5.7,-\dfrac{3}{2},0,\dfrac{1}{2},\space\text{and}\space 0.75\]

Let's now define what we mean by **irrational numbers**.

**Irrational numbers** are numbers that **can't be expressed as a fraction of two integers**. Irrational numbers have **non-repeating decimals** that never end, and have **no pattern** whatsoever. They are represented with the letter \(\mathbb{Q}'\).

Some examples of irrational numbers are:

\[\sqrt{2},\sqrt{3},\sqrt{5}\space \text{and}\space \pi\]

There are numbers with non-terminating decimals that are actually rational. This is the case of numbers with **non-terminating decimals that repeat in a pattern**, as they can be expressed as a fraction of two integers. For example, \(\dfrac{1}{9}=0.\bar{1}\), the bar above the decimal \(1\) means that it repeats forever. So, is a rational number.

Read Convert between Fractions and Decimals to learn more about the different types of decimal numbers.

### Real numbers

**Real numbers** include all the numbers that you can think of, which you can find in the real world, apart from imaginary numbers. Real numbers are represented with the letter \(\mathbb{R}\), and they include all rational and irrational numbers, therefore the set of real numbers can be represented as \(\mathbb{R}=\mathbb{Q}\cup \mathbb{Q}'\).

Some examples of real numbers are:

\[-\dfrac{5}{2},-1,0,0.\bar{3},2.45,\sqrt{5},\space\text{and}\space \pi\]

Apart from real numbers, mathematicians came up with a special type of number to be able to solve the square root of negative numbers, included in simple quadratic equations such as \(x^2+9=0\). If we try to solve this equation using Algebra, we end up with the following:

\[\begin{align}x^2+9&=0\\x^2+9-9&=-9\\x^2&=-9\\\sqrt{x^2}&=\sqrt{-9}\\x&=\sqrt{-9}\end{align}\]

But using only real numbers, we can't go any further than this. This is when imaginary numbers come in to play.

### Imaginary numbers

**Imaginary numbers** are the root of negative numbers.

We know that we can't take the square root of negative numbers, because there is no number that when squared will result in a negative number. In this case, we need to use imaginary numbers. To do this, we say that \(\sqrt{-1}=i\). Let’s see this more clearly with an example.

**Solve \(\sqrt{-9}\) using imaginary numbers.**

\(\sqrt{-9}\) can be written as \(\sqrt{9(-1)}=\sqrt{9}\times\sqrt{-1}\)

If we substitute \(\sqrt{-1}\) with \(i\),

then we can say that

\[\sqrt{-9}=3\times i\]

\[\sqrt{-9}=3i\]

Now, let's explore a few other concepts related to the topic of numbers.

## Number theory

What do we mean by number theory?

**Number theory** is the branch of Mathematics that studies positive integer numbers, their properties and relationships.

The types of numbers studied by number theory, can be classified into some of the categories shown in the table below.

Name | Definition | Examples |

Even | Integer numbers that can be divided by 2. | \(2,4,6,8,10,12,14,...\) |

Odd | Integer numbers that cannot be divided by \(2\). | \(1,3,5,7,9,11,13,...\) |

Square | Integer numbers that result from multiplying a number by itself. | \(1,4,9,16,25,36,49,...\) |

Cube | Integer numbers that result from multiplying a number by itself 3 times. | \(1,8,27,64,125,216,343,...\) |

Prime | This type of numbers have only 2 factors, because they are only divisible by itself and 1. | \(2,3,5,7,11,13,17,...\) |

Composite | This type of numbers have more than 2 factors. | \(4,6,8,9,10,12,14,...\) |

Perfect | Integer numbers that result from the sum of their proper divisors. | \(6\) is a perfect number, because if you add the divisors of \(6\), which are \(1\), \(2\), and \(3\), you get \(6\) as a result. Other examples include \(28, 496, 8128, ...\) |

Fibonacci | Series of integers, in which each number results from adding the two preceding numbers, starting from \(1\). | \(1,1,2,3,5,8,13,...\) |

## Number sequence

When you see a list of numbers in order, and you can identify a pattern between them, then you have found a **number sequence**.

A **number sequence **is a list of numbers in either ascending or descending order that follows a pattern or rule to obtain the following number (term) in the sequence.

Number sequences can be **finite**, if they have an end, and** infinite, **if they have no end.

Here are some examples to help you recognise a number sequence.

**Identify if the following lists of numbers represent a sequence:**

a) 1, 3, 5, 7, 9, ...

Yes, this is a number sequence, because it is a list of numbers in ascending order, and there is a consistent pattern to find the following term in the sequence (add 2).

b) 4, 0, 3, 1, 7, ...

No, this is not a number sequence, because it is a list of numbers in **no specific order** whatsoever.

c) 2, 4, 6, 8, 0, ...

No, this is not a number sequence, because **the pattern is not consistent**. Even though the 2^{nd}, 3^{rd} and 4^{th} terms are obtained by adding 2 to the previous term, the zero (0) in the 5^{th} term breaks the pattern.

Read Sequences to learn more about the different types of number sequence.

## Number systems

**Number systems** are systems that represent numbers using a specific set of digits and letters.

The most common number system, that we use on a regular basis, is the **decimal system**, which uses the digits 0 to 9. In the table below, you can also see other types of number systems that are mostly used by computers.

Name | Base | Digits/Letters | Example |

Decimal | \(10\) | \((0 - 9)\) | \(15\) |

Binary | \(2\) | \((0, 1)\) | \(15\) in binary:\((1111)_2\) |

Octal | \(8\) | \((0 - 7)\) | \(15\) in octal:\((17)_8\) |

Hexadecimal | \(16\) | \((0 - 9, A - F)\) | \(15\) in hexadecimal:\((F)_{16}\) |

The table below shows you the equivalence between the different number systems for the numbers from 0 to 15:

Decimal | Binary | Octal | Hexadecimal |

0 | 0000 | 0 | 0 |

1 | 0001 | 1 | 1 |

2 | 0010 | 2 | 2 |

3 | 0011 | 3 | 3 |

4 | 0100 | 4 | 4 |

5 | 0101 | 5 | 5 |

6 | 0110 | 6 | 6 |

7 | 0111 | 7 | 7 |

8 | 1000 | 10 | 8 |

9 | 1001 | 11 | 9 |

10 | 1010 | 12 | A |

11 | 1011 | 13 | B |

12 | 1100 | 14 | C |

13 | 1101 | 15 | D |

14 | 1110 | 16 | E |

15 | 1111 | 17 | F |

This is just an introduction to the different types of number systems, you can also convert numbers between the different number systems and perform arithmetic operations with them, but that goes beyond the scope of this article.

## Number - Key takeaways

- Natural numbers, also known as counting numbers, are all numbers starting from \(1\).
- Whole numbers are all natural numbers plus zero.
- Integer numbers include all negative and positive whole numbers.
- Rational numbers can be expressed as a fraction of two integers, while irrational numbers can't.
- Real numbers include all rational and irrational numbers.
- Real numbers are all the numbers that you can think of, apart from imaginary numbers.
- Number theory is the branch of Mathematics that studies positive integer numbers, their properties and relationships.
- A number sequence is a list of numbers in either ascending or descending order that follow a pattern or rule to obtain the following number (term) in the sequence.
- The four main types of number systems are: decimal, binary, octal and hexadecimal.

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##### Frequently Asked Questions about Number

What is a number in Maths?

A number is a mathematical concept that represents quantity, which has many applications such as counting, measuring, labelling, and performing calculations that help us to solve problems, among others.

What are examples of numbers?

Some examples of numbers of different types include:

-3, 0, 2, 3.8, 3/4, π, and √2.

What are the different types of numbers?

- Natural
- Whole
- Integers
- Rational
- Irrational
- Real
- Imaginary

What is a number sequence?

A number sequence is a list of numbers in either ascending or descending order that follow a pattern or rule to obtain the following number (term) in the sequence.

Is Pi a real number?

π is an irrational number, since it has non-repeating decimals that never end, and have no pattern whatsoever. Real numbers include irrational numbers, therefore, π is a real number.

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