Types of Numbers (Maths): Overview, Definition & Examples

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

The importance of numbers

If we think about it for a moment, numbers are everywhere in our daily lives, and 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, but 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.

What are the different 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 N. The set Notation for Natural Numbers is as follows:

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. Whole numbers do not include negative numbers, Fractions or decimals.

They are represented with the letter W, and their set Notation is shown below:

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.

Types of Numbers Natural and whole numbers StudySmarter Representation of natural and whole numbers, Marilú García De Taylor - StudySmarter Originals

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

Types of Numbers Natural and whole numbers on number line StudySmarter Natural and whole numbers on the number line, Marilú García De Taylor - StudySmarter Originals

Please refer to Natural Numbers to learn more about this type of number.

Integers

Integers include all positive numbers, zero and negative numbers. Again, Integers do not include Fractions or decimals.

They are represented with the letter Z, and their set notation is as follows:

Z = {..., -4, -3, -2, -1, 0, 1, 2, 3, 4, ...}

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

Types of Numbers Integers StudySmarter Representation of integers, Marilú García De Taylor - 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:

Types of Numbers Integers on number line StudySmarter Integers on the number line, Marilú García De Taylor - StudySmarter Originals

Some examples of integer numbers are:

-45, -12, -1, 0, 35, and 946

Check out the Integers article for more details and examples.

Rational and irrational numbers

Rational numbers include all numbers that can be expressed as a fraction in the form $\frac{p}{q}$ , 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 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 $\frac{3}{1}$ .

Types of Numbers Rational numbers StudySmarter Representation of rational numbers. Marilú García De Taylor - StudySmarter Originals

Some examples of rational numbers are:

$- 5.5, - \frac{3}{2}, 0, \frac{1}{2} a n d 0.75$

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 Q'.

Types of Numbers Irrational numbers StudySmarter Representation of irrational numbers, Marilú García De Taylor - StudySmarter Originals

Some examples of irrational numbers are:

$\sqrt{2}, \sqrt{3}, \sqrt{5} a n d π$

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, $\frac{1}{9} = 0 . \bar{1}$ the bar above the decimal 1 means that it repeats forever. So, $0 . \bar{1}$ is a rational number.

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 R, and they include all rational and irrational numbers, therefore the set of real numbers can be represented as $R = Q \cup Q'$ .

Types of Numbers Real numbers StudySmarter Representation of real numbers, Marilú García De Taylor - StudySmarter Originals

Some examples of real numbers are:

$- \frac{5}{2}, - 1, 0, 0 . \bar{3}, \sqrt{5}, π$

Check out the Real Numbers article to expand your knowledge about this topic.

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 $\sqrt{- 1} = i$ .

Solve $\sqrt{- 9}$

\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 $\sqrt{- 9} = 3 \times i = 3 i$

Types of Numbers - Key takeaways

Real numbers are all the numbers that you can think of, apart from imaginary numbers.
Real numbers include all rational and irrational numbers.
Rational numbers can be expressed as a fraction of two integers, while irrational numbers can't.
Integers include all negative and positive whole numbers.
Whole numbers are all natural numbers plus zero.
Natural numbers, also known as counting numbers, are all numbers starting from 1.

Frequently Asked Questions about Types of Numbers

The different types of numbers in Maths include: real, rational, irrational, integers, whole and natural numbers, as well as imaginary 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.

Rational numbers include all numbers that can be expressed as a fraction in the form p/q, where p and q are integers and q ≠ 0. This group of numbers includes fractions and decimals.

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 R, and they include all rational and irrational numbers, therefore the set of real numbers can be represented as R = Q ∪ Q'.

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What are integers?

Integers are numbers without fractional components.

Which of these are examples of an integer?

0 and 2/2

How do you identify integers?

They are positive and negative non-fractional natural numbers, including zero.

Are all natural numbers integers?

Yes

Integer numbers that follow each other in a sequence or in order without gaps are called…

Consecutive

Are these two examples of consecutive integers?

{-3, -2, -1, 0, 1, 2..}
{10, 11, 12, 13, 14..}

Yes

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