If we think about it for a moment, numbers are everywhere in our daily lives. They help us to think logically and to keep track of the things we do. For example, numbers help us with simple tasks like calculating the time that it takes you to get from home to your place of work, the amount of money that you need to pay for your shopping, and the amount of bags that you need to carry your shopping home. In addition to this, they are also especially useful to solve more complex problems in the world of sciences and engineering, like calculating the amount of fuel that will be needed to get a rocket into space, or the number of lorries that a warehouse needs to transport their customer's orders safely and on time.
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.
Representation of natural and whole numbers - StudySmarter Originals
Natural and whole numbers can be represented on the number line as follows:
Natural and whole numbers on the number line - StudySmarter Originals
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:
Representation of integers - StudySmarter Originals
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:
Integer numbers on the number line - StudySmarter Originals
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 \(\dfrac{3}{1}\).
Representation of rational numbers - StudySmarter Originals
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 \(\mathbb{Q}'\).
Representation of irrational numbers - StudySmarter Originals
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
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.
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 2nd, 3rd and 4th terms are obtained by adding 2 to the previous term, the zero (0) in the 5th 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:
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