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Real Numbers

Last Updated : 01 Mar, 2024
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Real Numbers are continuous quantities that can represent a distance along a line, as Real numbers include both rational and irrational numbers. Rational numbers occupy the points at some finite distance and irrational numbers fill the gap between them, making them together to complete the real line. So, in other words, real numbers are those numbers that can be plotted on the real line.

Let’s learn about Real Numbers in detail, including their properties, representation on the number line, and decimal expansion.

Real Numbers

Real Numbers Definition

The collection of all Rational numbers contains all the other numbers like natural numbers, integers, rational as well as irrational. Some examples of real numbers are 3 (a whole number), -1 (an integer), 1/2 (a rational number), √2 (an irrational number), π (an irrational number), 2.5 (a decimal number), etc.

The number or the numeral system is the system of representing numbers. There are various types of number systems in maths like binary, decimal, etc. The Number system represents how a number should be written.

Real Numbers Chart

Rational Numbers, Irrational Numbers, and all the subsets of rational numbers all come under real numbers, and the real number chart is used to express all the branches of real numbers. The real number chart is added below:

Real Numbers Chart

Set of Real Numbers

All the numbers known to humans except the imaginary numbers come under the set of the real number. We use R to denote the set of real numbers. We can have various subsets of the real number that denote different types of numbers. Various subsets of the Real number are,

Real Numbers can be divided into the following subsets:

Natural NumbersCounting numbers used in daily life, starting from 1.1, 2, 3, 4, 5, …
Whole NumbersNatural numbers including 0, making a complete set of all positive integers.0, 1, 2, 3, 4, 5, …
IntegersWhole numbers and negative natural numbers, including a neutral number (0).…, -3, -2, -1, 0, 1, 2, 3, …
Rational NumbersNumbers that can be expressed as a fraction p​/q where p and q are integers and q≠0.​ 22/7, 3/2, -11/13, -13/17
Irrational NumbersReal numbers that cannot be expressed as a simple fraction p​/q​, where p and q are integers and q≠0.22​, 33​, π

List of Real Numbers

The list of real numbers is endless because it includes all kinds of numbers like whole, natural, integers, rational, and irrational numbers. Since it includes integers it has negative numbers too. So, there is no specific number from which the list of real numbers starts or ends. It goes to infinity towards both sides of the number line.

Symbol of Real Numbers

We use R to represent a set of Real Numbers and other types of numbers can be represented using the symbol discussed below,

Real Numbers and their Symbols

Natural Numbers


Whole Numbers




Rational Numbers


Irrational Numbers


Real Number Properties

There are different properties of Real numbers with respect to the operation of addition and multiplication, which are as follows:

Properties of Real Numbers

PropertyAddition ExampleMultiplication Example
Commutative Propertya + b = b + aa × b = b × a
Associative Property(a + b) + c = a + ( b + c)(a × b) × c = a × ( b × c)
Distributive Propertya × ( b + c) = a × b + a × ca × (b + c) = a × b + a × c
Identity Propertya + 0 = aa × 1 = a
Inverse Propertya + (−a) = 0a × (1​/a) = 1 (for a≠0)

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Real Numbers on Number Line

A number line contains all the types of numbers like natural numbers, rational numbers, Integers, etc.As shown in the following number line 0 is present in the middle of the number line. Positive integers are written on the right side of zero whereas negative integers are written on the left side of zero, and there are all possible values in between these integers.

Rational Numbers on Number Line

Representation of Rational Numbers on the Number Line

Rational numbers are written between the numbers they lie. For example, 3/2 equals to 1.5, so is noted between 1 and 2. It shows that the number 3/2 lies somewhere between 1 and 2.

Similarly, the Number 13/4 = 3.25 lies between 3 and 4. So we noted it between 3 and 4. Number -50/9 = -5.555. . . , lies between -5 and -6. So we noted it between -5 and -6 on the number line.

Example: Represent the Following numbers on a number line:

  • 23/5
  • 6
  • -33/7


The rational numbers,

  • 23/5
  • 6
  • -33/7

can easily be represented in a number line as,

Representation of Rational Numbers on the Number Line

Real Numbers on the Number Line Example

Irrational Numbers on Number Line

Irrational Numbers can’t be represented on the number line as it is, we need clever tricks and geometry to represent irrational numbers on a number line.

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Decimal Expansion of Rational Numbers

The decimal expansion of a real number is its representation in base equals to 10 (i.e., in the decimal system). In this system, each “decimal place” consists of a digit from 0 to 9. These digits are arranged such that each digit is multiplied by a power of 10, decreasing from left to right.

Let’s Expand 13/4

Decimal Expansion of 13/4

So 13/4 can also be written as 3.25.

Now Let’s take another example. Let’s expand 1/3

Decimal Expansion of 10/3

So 1/3 can also be written as 0.3333…… We can also write it as [Tex]0.\overline3 [/Tex]

Similarly, 1/7 can be written as 0.142857142857142857… or [Tex]0.\overline{142857}  [/Tex]. This is known as the recurring decimals expansion.

Decimal Expansion of Irrational Numbers

Decimal Expansion of Irrational Numbers is non-terminating and non-repeating. We can find the decimal expansion such as √2, √3, √5, etc. using the long division method. The decimal Expansion of √2 is up to three digits after the decimal is calculated in the following illustration.

Decimal Expansion of Irrational Numbers

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Solved Examples on Real Numbers

Here are some example problems on Real Numbers and their properties.

Example 1: Add √3 and √5


(√3 + √5)

Now answer is an irrational number.

Example 2: Multiply √3 and √3.


√3 × √3 = 3

Now answer is a rational number.

So we can say that result of mathematical operations on irrational numbers can be rational or irrational.

Now add a rational number with an irrational number.

Example 3: Add 2 and √5


(2 + √5)

Now answer is an irrational number.

Related :

FAQs on Real Numbers

Define Real Numbers.

Collection of all rational as well as irrational numbers is called real numbers.

What is the Difference between Rational and Irrational Number?

The key difference between rational and irrational number is rational numbers can be represented in the form of p/q, where p and q are integers and q≠0. Whereas irrational numbers can’t be represented in the same form.

Are Whole Numbers and Integers Real Numbers?

Yes, all whole numbers and integers are real numbers as whole numbers and integers are the subsets of the rational numbers.

What is the Symbol for Real Numbers?

The symbol used to represent real numbers is OR R.

What is Decimal Representation of Real Number?

Decimal Representation of a real number can be either terminating, non-terminating but repeating, or non-terminating non-repeating as a real number contains all real numbers as well as irrational numbers.

Are Imaginary Numbers Real Numbers?

No, imaginary Numbers are not real numbers as they can’t be represented on the number line.

Is Pi a Rational Number?

No, Pi (π) is not a rational number. It is an irrational number because it cannot be expressed as a fraction of two integers, and its decimal representation is non-terminating and non-repeating.

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