Class 10 Maths Real Numbers Notes 2025 & Study Material PDF

Real numbers form a fundamental aspect of mathematics; most mathematical operations are based on real numbers. Class 10 teaches the concepts of real numbers because it does not only solve simple math problems but also gives the basis for advanced subjects such as algebra, geometry, and calculus. In short, one should understand the concept of real numbers because real numbers are practically applicable in life. For example, measuring distance or managing financial transactions involves real numbers.

The subject-wise material for Class 10 Chapter 1 Real Numbers is curated with keeping the revised CBSE pattern in mind. The chapter comprises 6 marks in the final exam. At Educart, we have provided detailed study material like experiential activities, formulas, question banks, and other support study material to help students boost their exam preparation.

CBSE Class 10 Maths chapter 1 Notes 2025

Class 10 math Chapter 1 notes cover all the main concepts like Euclid's Division Lemma and Arithmetic Fundamental Theorem. Real numbers include every number other than the complex numbers and the downloadable notes PDFs provided below are detailed and in easy-to-understand language.

<red> ➜   <red>Real Numbers Notes

Class 10 Real Numbers DoE Worksheet

Students can attempt Real Numbers question answer worksheets to prepare themselves as per the exam pattern. Below, we have provided the links to downloadable PDFs of DoE Worksheets for class 10 Mathematics to practice more questions. 

Worksheet 76EM

Worksheet 77EM

CBSE Class 10 Ch 1 Real Numbers Experiential Activities

Below we have provided the links to downloadable PDFs of Experiential Learning Activity for class 10 Mathematics to help students implement their acquired knowledge in the real world.

<red> ➜   <red>Real Numbers  Experiential Activities

Chapter 1 Class 10 Real Numbers Formulas 2025

Below we have provided the links to downloadable PDFs of formulas for class 10 Mathematics Real Numbers to help students solve complex questions and understand the concepts easily. 

<red> ➜   <red>Class 10 Mathematics Formulas(View)

CBSE Class 10 Real Numbers Mind-maps

Below we have provided the links to downloadable PDFs of mind maps for class 10 Mathematics to help students implement their acquired knowledge in the real world.

<red> ➜   <red>Class 10 Mathematics Mind-maps

Class 10 Real Numbers Important Questions 2025

Below we have provided Class 10 Mathematics Important Questions that cover questions from the NCERT textbook like Class 10 Math Chapter 1 exercise 1.2 solutions and many more. 

<red> ➜   <red>Class 10 Mathematics Important Questions(View)

<red> ➜   <red>Class 10 Real Numbers Extra Questions

Class 10 Mathematics Real Numbers Question Bank 2025

Below we have provided Class 10 Mathematics Question Banks that cover every typology question with detailed explanations from various resources in one place.

<red> ➜   <red>Real Numbers CBSE Question Bank

<red> ➜   <red>Real Numbers Kendriya Vidyalaya Question Bank

Maths Class 10 Real Numbers Support Material

Below we have provided Class 10 Mathematics Support Material that covers Case Study-based questions from the various concepts explained in Math NCERT chapters. 

<red> ➜   <red>Real Numbers Practice Test

<red> ➜   <red>Real Numbers Support Material

What are Real Numbers?

Real numbers include those numbers that can be plotted on the number line. It consists of both rational and irrational numbers that come in form of fractions, decimals, and square roots. The real numbers take a very large range of values, positive and negative and zero.

On the number line, negative numbers fall to the left of zero, positive numbers to the right, and zero is in the middle. Real numbers can be divided into several classes, which are given below.

Rational Numbers:

Rational numbers are those which can be represented in the form of a ratio of two integers such that the denominator is non-zero. That is, a number is rational if it can be written in the form p/q, where p and q are integers and q is not zero.

Examples: ½, -¾, 5, 0.25

Irrational Numbers:

Irrational numbers cannot be represented as a common fraction of two integers. Their decimal expansions are non-recurring and non-terminating.

Examples: π, √2, e, etc.

Integers:

Integers are whole numbers that can be positive, negative, or zero. While every integer qualifies as a rational number, not every rational number is an integer. 

Examples include: -3, 0, 5.

Whole Numbers:

Whole numbers are the set of non-negative integers, which also include zero. 

Examples are: 0, 1, 2, 3, 4.

Natural Numbers:

Natural numbers consist of the counting numbers, and they do not include zero. 

Examples are: 1, 2, 3, 4,…..

Number Line Representation

The number line is a representation of the numbers in an order. On this line, real numbers occupy positions with negative numbers to the left and positive numbers to the right of zero. Zero is the midpoint. Rational numbers fall on specific points on the line while irrational numbers are placed loosely between the rational numbers.

Properties of Real Numbers

The most important properties of real numbers include vital arithmetic and algebraic operations. These make it easier for simplification as well as in solving mathematical problems.

Commutative Property:

• Addition: The summation of any two real numbers is the same regardless of whether they are added first or added last.

a + b = b + a

• Multiplication: The product of two real numbers is the same no matter in which order they are multiplied.

a x b = b x a

‍Associative Property:‍

• Addition: The positioning of numbers in addition does not affect the sum.

(a + b) + c = a + (b + c)

• Multiplication: The way numbers are grouped in multiplication does not change the result.

(a x b) x c = a x (b x c)

‍Distributive Property:‍

• The distributive property implies that multiplication is distributive over addition or subtraction.

a x (b + c) = a x b + a x c

‍Identity Elements:‍

• Addition Identity: The identity element for addition is 0, since adding zero to any number does not change its value.

a + 0 = a

• Multiplication Identity: The identity element for multiplication is 1, since multiplying any number with 1 leaves its value unchanged.

a × 1 = aInverse Elements:‍

• Additive Inverse: Every real number has an additive inverse that when added to the given number results in zero.

a + (−a) = 0

• Multiplicative Inverse: Every non-zero real number has a multiplicative inverse, such that when this inverse is multiplied by the number, the product is one.

a x 1/a = 1 (where a is not equal to 0)

‍Closure Property:‍

• Addition: For any two real numbers, the sum is always a real number.

a + b ∈ R

• Multiplication: For any two real numbers, the product is always a real number.

a × b ∈ R

Operations on Real Numbers

Addition and Subtraction:

When we add and subtract real numbers, we follow the rules of arithmetic. We must be careful with signs:

• If the numbers are of the same sign, add their absolute values and retain that common sign.

Example: (+3) + (+2) = +5

• Since the numbers have opposite signs, you shall subtract the absolute value of the number with the low absolute value to the absolute value of the larger number then assign the sign that is on the larger number.

Example: (+5) - (+8) = -3

Multiplication and Division:‍

‍Multiplication: Multiply the absolute values of the numbers and determine the sign using the multiplication rules:

• Positive × Positive = Positive
• Negative × Negative = Positive
• Positive × Negative = Negative

For example: (-3) x (+4) = -12

‍Division: Divide the absolute values of the numbers and determine the sign by the division rules:

• Positive ÷ Positive = Positive
• Negative ÷ Negative = Positive
• Positive ÷ Negative = Negative

For example: (-3) x (+4) = -12

Real Numbers and Their Decimal Expansion

Real numbers can be expressed in decimals. This can either be terminating or non-terminating and recurring depending on their properties.

1. Terminating Decimals: Such decimals have a finite number of digits after the decimal point. Examples are 0.5, 1.75, and -3.25.

2. Non-Terminating, Repeating Decimals: Such decimals are infinite but possess a cyclic sequence of digits.

3. Non-Terminating, Non-Repeating Decimals: It's an infinite expanding decimal without a repeating sequence. Primarily it is an irrational number, such as √2, e, π.

Rationalising Denominators

When dealing with square roots in a fraction, you can rationalise the denominator. That is the process of getting the square root off the denominator. You can multiply the numerator and denominator by a suitable value to get the denominator to be a perfect square.

For instance, if you wanted to rationalize the denominator in 1/√2, you would multiply both the numerator and the denominator by √2:

1/√2 × √2/√2 = √2/2.

Representation of Real Numbers

Real numbers can be represented in many ways:

1. Decimal form: As mentioned earlier, real numbers can be represented in decimal form.

2. Fractional form: Rational numbers are generally represented in fraction form.

3. Radical form: Irrational numbers, especially square roots, are often represented in radical form.

4. Exponential form: The number like (e) or a power of 10 can be represented in exponential form.

Real numbers are the basic concept of mathematics, and their characteristics and operations are essential for the successful execution of Class 10. This category includes both rational and irrational numbers that can be operated upon through various arithmetical operations. Some of the important concepts that have to be acquired by a student in order to establish a firm mathematical foundation are understanding the number line, the properties of real numbers, performing operations on them, and their decimal expansions. Get comfortable with these and you will be well-prepared to advance in the study of mathematics and science.