🧠 Introduction

Real numbers include rational and irrational numbers. This chapter helps you understand Euclid’s Lemma, HCF & LCM, prime factorization, and decimal expansions of real numbers.

🔹 Euclid’s Division Lemma

If a and b are two positive integers, then there exist unique integers q and r such that:

a=bq+r, where 0≤r<ba = bq + r,\ \text{where } 0 \leq r < b

📌 This lemma is useful for finding the HCF of two numbers.

🔸 Fundamental Theorem of Arithmetic

Every composite number can be expressed uniquely (excluding order) as a product of prime numbers.

✅ Example:

60=22×3×560 = 2^2 \times 3 \times 5

🔸 HCF and LCM Using Prime Factorization

For any two positive integers:

HCF×LCM=Product of the numbers\text{HCF} \times \text{LCM} = \text{Product of the numbers}

🔸 Rational & Irrational Numbers

  • Rational Numbers: Can be written as pq\frac{p}{q}, where q ≠ 0.

  • Irrational Numbers: Non-terminating, non-repeating decimals (e.g., √2, π)

🔸 Decimal Expansions of Rational Numbers

To decide whether the decimal expansion terminates or not:

  • If the denominator (after simplifying) has only 2 or 5 as its prime factors → Terminating

  • Otherwise → Non-terminating repeating

✅ Examples:

  • 38=0.375\frac{3}{8} = 0.375 → Terminating

  • 17=0.142857...\frac{1}{7} = 0.142857... → Repeating

✅ Key Formulas & Summary

  • Euclid’s Lemma: a=bq+ra = bq + r

  • HCF × LCM = Product of numbers

  • Rational numbers have either terminating or repeating decimals

  • Irrational numbers are non-terminating and non-repeating

📚 Next Chapter: Chapter 2 – Polynomials »

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