Number

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.

Number Number

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Contents
Table of contents

    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.

    Number Natural and whole numbers StudySmarterRepresentation of natural and whole numbers - StudySmarter Originals

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

    Number Natural and whole numbers on number line StudySmarterNatural 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:

    Number Integers StudySmarterRepresentation 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:

    Number Integers on the number line StudySmarterInteger 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 \(\dfrac{3}{1}\).

    Number Rational numbers StudySmarterRepresentation 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 \(\mathbb{Q}'\).

    Number Irrational numbers StudySmarterRepresentation 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

    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 \(\mathbb{R}=\mathbb{Q}\cup \mathbb{Q}'\).

    Number Real numbers StudySmarterRepresentation of real numbers - StudySmarter Originals

    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.

    NameDefinitionExamples
    EvenInteger numbers that can be divided by 2.\(2,4,6,8,10,12,14,...\)
    OddInteger numbers that cannot be divided by \(2\). \(1,3,5,7,9,11,13,...\)
    SquareInteger numbers that result from multiplying a number by itself.\(1,4,9,16,25,36,49,...\)
    CubeInteger numbers that result from multiplying a number by itself 3 times. \(1,8,27,64,125,216,343,...\)
    PrimeThis type of numbers have only 2 factors, because they are only divisible by itself and 1. \(2,3,5,7,11,13,17,...\)
    CompositeThis type of numbers have more than 2 factors. \(4,6,8,9,10,12,14,...\)
    PerfectInteger 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 \(28, 496, 8128, ...\)
    FibonacciSeries 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 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.

    NameBaseDigits/LettersExample
    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:

    DecimalBinaryOctalHexadecimal
    0000000
    1000111
    2001022
    3001133
    4010044
    5010155
    6011066
    7011177
    81000108
    91001119
    10101012A
    11101113B
    12110014C
    13110115D
    14111016E
    15111117F

    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.
    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. 

    Test your knowledge with multiple choice flashcards

    A fraction can be defined as a portion or part of a whole number, expression, or anything other than zero. 

    Decimal numbers are in base 2.

    Percentage can be defined as the fraction or ratio of a number in a hundred.

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