Class 10 Maths Chapter 1 Real Numbers Introduction – Euclid’s Division Algorithm & Fundamental Theorem Explained

Real Numbers – Introduction

CBSE Class 10 Maths

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🔍 What You Already Know (from Class 9):

• Real Numbers = Rational + Irrational numbers
• Rational Numbers: Can be written as p/q (q ≠ 0)
• Irrational Numbers: Cannot be written as p/q
  • Examples: √2, √5, π

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🧠 What You Will Learn in This Chapter:

In Class 10, we go deeper into the properties of real numbers, especially positive integers, through:

1. Euclid’s Division Algorithm
2. Fundamental Theorem of Arithmetic

These concepts form the foundation of number theory, with applications in HCF, irrational numbers, and decimal expansions.

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📍1. Euclid’s Division Algorithm

📌 Definition:

Given two positive integers a and b (a > b), there exist integers q and r such that:

a = bq + r    {where } 0≤r< b

This is basically the long division method you already know!

🔧 Key Use:

• Helps us find the HCF (Highest Common Factor) of two numbers.
• Example: To find HCF of 210 and 45, we apply Euclid’s algorithm step by step until the remainder is 0.

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📍2. Fundamental Theorem of Arithmetic

📌 Statement:

Every composite number can be expressed as a product of prime numbers, and this factorisation is unique, apart from the order of the factors.

Example:

60 = 2 x 2 x 3 x 5 = 2²x 3x 5

🔧 Key Uses:

We use this theorem to:

1. Prove irrationality of numbers like √2, √3, √5, etc.
2. Understand decimal expansions of rational numbers.

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📍3. Application: Irrationality Proofs

✅ You Learn:

• How to prove that √2, √3, √5 are irrational
• Using contradiction method and prime factorisation

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📍4. Application: Decimal Expansion of Rational Numbers

🔍 Key Concept:

If a rational number is of the form , and:

• The prime factorisation of q contains only 2 or 5, then the decimal expansion is terminating.
• If q has any other prime factor, the decimal expansion is non-terminating repeating.

✏️ Example:

• → Denominator 8 = → Only 2 → Terminating
• → Denominator 7 → Not 2 or 5 → Repeating

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📝 Summary Points:

Concept	Meaning	Use
Euclid’s Division Algorithm		Find HCF of two numbers
Fundamental Theorem of Arithmetic	Every number has a unique prime factorisation	Used in irrationality proofs and decimal expansions
Irrationality Proof	√2, √3, √5 can't be expressed as p/q	Uses contradiction and prime factors
Decimal Expansion	Based on prime factorisation of denominator	Terminating or repeating decimals

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🔁 Quick Recap:

✅ Real numbers = Rational + Irrational
✅ Use Euclid’s Algorithm for HCF
✅ Every composite number = Unique product of primes
✅ Decimal of p/q depends on factors of q:

• Only 2 or 5 → Terminating
• Other primes → Repeating

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📚 Practice Questions:

1. Use Euclid’s Division Algorithm to find HCF of 252 and 105.
2. Prime factorise 84.
3. Prove that √3 is irrational.
4. Determine if has a terminating decimal.
5. What kind of decimal expansion does have?

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