Class 10 CBSE Maths Chapter 1: Real Numbers – Full Notes

 

📘 Chapter 1: Real Numbers – Class 10 CBSE Maths

Introduction:
In this chapter, we learn about the basic building blocks of mathematics – real numbers. These include rational and irrational numbers, their properties, and how to use the Euclidean Division Algorithm.

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🔢 What are Real Numbers?

Real numbers include:

• Natural Numbers (1, 2, 3...)

• Whole Numbers (0, 1, 2...)

• Integers (...–2, –1, 0, 1, 2...)

• Rational Numbers (e.g. 2/3, –4, 5.6)

• Irrational Numbers (e.g. √2, π)

Together, all these form the set of Real Numbers.

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✅ Properties of Real Numbers

1. Closure Property

   • Rational and irrational numbers are closed under addition, subtraction, and multiplication.

2. Commutative Property

   • a + b = b + a and a × b = b × a

3. Associative Property

   • (a + b) + c = a + (b + c)

4. Distributive Property

   • a × (b + c) = a × b + a × c

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🔁 Euclid’s Division Lemma

If a and b are positive integers (a > b), then:

a = bq + r, where 0 ≤ r < b

Example:
Find HCF of 56 and 72 using Euclid’s division method.

Solution:

72 = 56 × 1 + 16  
56 = 16 × 3 + 8  
16 = 8 × 2 + 0  
So, HCF = 8

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📏 Fundamental Theorem of Arithmetic

Every composite number can be expressed uniquely as a product of prime numbers.

Example:

ini
60 = 2 × 2 × 3 × 5 = 2² × 3 × 5

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✳️ LCM and HCF using Prime Factorisation

Example:
Find LCM and HCF of 12 and 18.

Prime factors:

• 12 = 2² × 3

• 18 = 2 × 3²

👉 HCF = 2 × 3 = 6
👉 LCM = 2² × 3² = 36

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🧠 Rational and Irrational Numbers

• A number is rational if it can be written in the form p/q (q ≠ 0).

• A number is irrational if it cannot be written in that form.

Examples:

• √2, √3, π are irrational.

• 3/5, –7, 0.25 are rational.

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🔁 Decimal Expansion

• Rational numbers have either terminating or non-terminating recurring decimals.

  • Example: 1/2 = 0.5 (terminating), 1/3 = 0.333... (non-terminating recurring)

• Irrational numbers have non-terminating non-recurring decimals.

  • Example: √2 = 1.414213...

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✍️ Summary:

• Real numbers include rational and irrational numbers.

• Euclid’s Lemma is used to find HCF.

• Fundamental Theorem of Arithmetic helps in prime factorisation.

• Decimal expansions help identify rational or irrational numbers.

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