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# Types of Numbers

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.

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:

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

### 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:

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:

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 , where p and q are integers and . 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 .

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

Some examples of rational numbers are:

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'.

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

Some examples of irrational numbers are:

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, the bar above the decimal 1 means that it repeats forever. So, 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 .

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

Some examples of real numbers are:

### 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 .

Solve

can be written as , if we substitute with i

then we can say that

## 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'.

## Final Types of Numbers Quiz

Question

What are integers?

Integers are numbers without fractional components.

Show question

Question

Which of these are examples of an integer?

0 and 2/2

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Question

How do you identify integers?

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

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Question

Are all natural numbers integers?

Yes

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Question

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

​Consecutive

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Question

Are these two examples of consecutive integers?

• {-3, -2, -1, 0, 1, 2..}

• {10, 11, 12, 13, 14..}

Yes

Show question

Question

A set of integers is denoted by ...

Z

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Question

What are odd consecutive integers?

These are a set of integers such that each integer in the set differs from the previous integer by a difference of 2 and each integer is an odd number

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Question

What is the difference between whole numbers and integers?

Whole numbers are counting numbers including 0, while integers are positive and negative whole numbers.

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Question

Which of these is a rule to consider when adding and subtracting integers?

Adding two positive integers will always give you a positive integer.

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Question

Which of these is an integer multiplication rule?

The product of two positive integers will always be a positive integer.

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Question

What is integer division?

Integer division is the division where the fractional part is discarded.

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Question

Values that can be expressed as an infinite decimal expansion are termed what?

Real numbers

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Question

Which of the following is not a real number?

None of the following

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Question

What are the types of real numbers?

Rational and irrational numbers

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Question

What is the symbol notation for real numbers?

R

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Question

What is not a real number?

Imaginary numbers

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Question

A type of real number, written as the ratio of two integers is?

Rational number

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Question

How are rational numbers expressed?

p/q, where p and q are integers and not equal to 0.

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Question

Which of these is a rational number with infinite decimals?

All the following

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Question

What is an irrational number?

Irrational numbers are a type of real numbers that cannot be written as the ratio of two integers.

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Question

What is the symbol notation for irrational numbers?

Q'

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Question

“The product and sum of two real numbers remain the same even after interchanging the order of the numbers”. What property of real numbers does this appropriately describe?

Commutative property

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Question

Are there real numbers that aren’t either rational or irrational?

No, there aren’t.

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Question

Are negative numbers real numbers?

Negative numbers are real numbers.

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Question

Applying the distributive property of real numbers as a × (b + c) = (a × b) + (a × c), what will be the value of each equation if a = 66, b = -3 and c = 14

66 ×  (-3 + 14) = (66 ×  (-3)) + (66 ×  14)

726 = 726

Show question

Question

Apply the associative to the next numbers as (a+b)+c and a+(b+c)  if a = 0.91, b = 12 and c = 0,  what will be the result of both?

(0.91 +12) +0 = 0.91 + (12 + 0)

12.91 = 12.91

Show question

Question

What is a natural number?

A positive whole number.

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Question

What are the different types of numbers in Maths?

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

Show question

Question

What are 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 ∪ Q'.

Show question

Question

What are imaginary numbers?

Imaginary numbers are the root of negative numbers.

Show question

Question

What are 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.

Show question

Question

What are integers?

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

Show question

Question

What is the difference between natural and whole numbers?

Whole numbers include zero, and natural numbers don't include it.

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Question

Is zero (0) a natural number?

No, natural numbers do not include zero. Zero is a whole number.

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Question

How can you represent the set of all real numbers?

R = Q ∪ Q'

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Question

Is 4 a rational number?

Yes, 4 is a rational number, as it can be expressed as a fraction with a denominator of 1.

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Question

What are natural numbers also known as?

Counting numbers

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