Real Numbers and their Symbols
Natural Numbers	N
Whole Numbers	W
Integers	Z
Rational Numbers	Q
Irrational Numbers	Q’

Real Number Properties

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

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Properties of Real Numbers
Property	Addition Example	Multiplication Example
Commutative Property	a + b = b + a	a × b = b × a
Associative Property	(a + b) + c = a + ( b + c)	(a × b) × c = a × ( b × c)
Distributive Property	a × ( b + c) = a × b + a × c	a × (b + c) = a × b + a × c
Identity Property	a + 0 = a	a × 1 = a
Inverse Property	a + (−a) = 0	a × (1​/a) = 1 (for a≠0)

Learn More: Properties of Numbers

Real Numbers on a 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 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

Solution:

The rational numbers,

• 23/5
• 6
• -33/7

can easily be represented in a number line as,

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.

Learn More : Representation of √3 on a number line

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

So 13/4 can also be written as 3.25.

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

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.

Learn More : Square root of 2

Solved Examples Problems on Real Numbers

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

Example 1: Add √3 and √5

Solution:

(√3 + √5)

Now answer is an irrational number.

Example 2: Multiply √3 and √3.

Solution:

√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

Solution:

(2 + √5)

Now answer is an irrational number.

People Also Read:

• Numbers
• Natural Numbers
• Whole Numbers
• Integers
• Rational Numbers
• Irrational numbers

Real Numbers Class 10

Real Numbers is a very important topic for class 10. Here are the resources where you can check ncert solutions and notes for real numbers.

Real Numbers Class 10 Notes

Real Numbers Class 10 NCERT Solutions

Practice Problems on Real Numbers

1. Add √2 and √8.

2. Multiply √7 by √14.

3. Add 5 to √9.

4. Multiply √6 and √2.

5. Add 3/2 (a rational number) to √3 (an irrational number)

Real Numbers- FAQs

What is a real number with example?

A real number is any value that can be represented on a number line, including both rational and irrational numbers. Examples include 7, -3, 0.5, π\piπ, and 2\sqrt{2}2​.

Is 0 a real number?

Yes, 0 is a real number. It represents a point on the number line and is considered both a rational number and a real number.

Is 11 a real number?

Yes, 11 is a real number. It is an integer and can be located on the number line, making it part of the set of real numbers.

What are the first 10 real numbers?

The first 10 real numbers, considering integers, are -5, -4, -3, -2, -1, 0, 1, 2, 3, and 4. These numbers are part of the continuous set of 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 ℝ 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.