📘 Class 10 Maths Chapter 1: Real Numbers – Notes & Summary

 

🧠 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.

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🔹 Euclid’s Division Lemma

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

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

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

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🔸 Fundamental Theorem of Arithmetic

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

✅ Example:

$$60 = 2^2 \times 3 \times 5$$

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🔸 HCF and LCM Using Prime Factorization

For any two positive integers:

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

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🔸 Rational & Irrational Numbers

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

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

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🔸 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:

• $\frac{3}{8} = 0.375$ → Terminating

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

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✅ Key Formulas & Summary

• Euclid’s Lemma: $a = bq + r$

• HCF × LCM = Product of numbers

• Rational numbers have either terminating or repeating decimals

• Irrational numbers are non-terminating and non-repeating

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📚 Next Chapter: Chapter 2 – Polynomials »

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