Real Numbers Chapter 1 | Mathematics

Complete NCERT notes, important formulas, Euclid's Division Lemma, Fundamental Theorem of Arithmetic, HCF & LCM, irrational numbers, and key points for CBSE Class 10 Board Exam 2025-26.

📖 Intro 🔢 Euclid's Lemma 🏛️ Fundamental Theorem 📊 HCF & LCM 🔴 Irrational Numbers 🔢 Decimal Expansions 📐 All Formulas 📝 Examples ⭐ Key Points

📖 Introduction to Real Numbers

Real Numbers: The set of all rational and irrational numbers together form the set of real numbers (denoted by ℝ).

📌 Number System Hierarchy

ℕ ⊂ 𝕎 ⊂ ℤ ⊂ ℚ ⊂ ℝ

ℕ = Natural Numbers (1,2,3...)
𝕎 = Whole Numbers (0,1,2,3...)
ℤ = Integers (...,-2,-1,0,1,2...)
ℚ = Rational Numbers (p/q form)
ℝ = Real Numbers

🔢 Euclid's Division Lemma

📐 Lemma Statement

For any two positive integers a and b, there exist unique integers q and r such that:
a = bq + r, where 0 ≤ r < b

Here: a = Dividend, b = Divisor, q = Quotient, r = Remainder

📝 Example 1

17 = 5 × 3 + 2

Here a=17, b=5, q=3, r=2

📝 Example 2

125 = 8 × 15 + 5

Here a=125, b=8, q=15, r=5

💡 Euclid's Division Algorithm: To find HCF of two positive integers, repeatedly apply the lemma until remainder becomes zero. The last divisor is the HCF.

🏛️ Fundamental Theorem of Arithmetic

📜 Theorem Statement

Every composite number can be expressed as a product of primes, and this factorization is unique (except for the order).

✨ Example

32760 = 2³ × 3² × 5 × 7 × 13

This representation is unique.

🔢 Prime Factorization

84 = 2² × 3 × 7

120 = 2³ × 3 × 5

📊 HCF and LCM

📐 Important Formula

HCF(a, b) × LCM(a, b) = a × b

🔍 Finding HCF using Prime Factorization

HCF = Product of smallest power of common prime factors

Example: 24 = 2³ × 3, 36 = 2² × 3² → HCF = 2² × 3 = 12

🔍 Finding LCM using Prime Factorization

LCM = Product of greatest power of each prime factor

Example: 24 = 2³ × 3, 36 = 2² × 3² → LCM = 2³ × 3² = 72

💡 Verification: HCF(24,36) × LCM(24,36) = 12 × 72 = 864 = 24 × 36 ✓

🔴 Irrational Numbers

Irrational Numbers: Numbers that cannot be expressed in the form p/q where p and q are integers and q ≠ 0. Their decimal expansions are non-terminating and non-repeating.

📌 Important Irrational Numbers

√2, √3, √5, √6, √7, √10, π, e

🔬 Proof that √2 is Irrational

Assume √2 = p/q (in lowest terms)
→ 2 = p²/q² → p² = 2q²
→ p² is even → p is even (p = 2k)
→ 4k² = 2q² → q² = 2k² → q is even
→ p and q both even, contradicting lowest terms assumption.
∴ √2 is irrational.

📌 Sum of rational & irrational

Rational + Irrational = Irrational

Example: 2 + √3 is irrational

📌 Product of rational & irrational

Rational (≠0) × Irrational = Irrational

Example: 5√2 is irrational

🔢 Decimal Expansions of Rational Numbers

Denominator Form	Type of Decimal	Example
2ᵐ × 5ⁿ	Terminating	7/8 = 0.875, 3/20 = 0.15
Other than 2ᵐ × 5ⁿ	Non-terminating Repeating	1/3 = 0.333..., 2/7 = 0.285714...

📌 Important Rule

A rational number p/q has terminating decimal expansion if and only if q = 2ᵐ × 5ⁿ (m, n are non-negative integers)

✅ Terminating Examples

1/2 = 0.5 (q=2)

3/5 = 0.6 (q=5)

7/25 = 0.28 (q=5²)

🔁 Non-Terminating Repeating

1/3 = 0.\overline{3}

2/9 = 0.\overline{2}

5/6 = 0.8\overline{3}

📐 All Important Formulas

1️⃣ Euclid's Division Lemma

a = bq + r, 0 ≤ r < b

2️⃣ HCF × LCM Formula

HCF(a,b) × LCM(a,b) = a × b

3️⃣ Prime Factorization

N = p₁^{a₁} × p₂^{a₂} × ... × pₖ^{aₖ}

4️⃣ HCF (Prime Factor Method)

HCF = p₁^{min(a₁,b₁)} × p₂^{min(a₂,b₂)} × ...

5️⃣ LCM (Prime Factor Method)

LCM = p₁^{max(a₁,b₁)} × p₂^{max(a₂,b₂)} × ...

6️⃣ Decimal Termination Rule

q = 2ᵐ × 5ⁿ → Terminating

📝 NCERT Solved Examples

Example 1: Find HCF of 135 and 225

225 > 135 → 225 = 135 × 1 + 90
135 = 90 × 1 + 45
90 = 45 × 2 + 0
∴ HCF = 45

Example 2: Find LCM and HCF of 6 and 20

6 = 2 × 3, 20 = 2² × 5
HCF = 2, LCM = 2² × 3 × 5 = 60
Verification: HCF × LCM = 2 × 60 = 120 = 6 × 20

Example 3: Prove √5 is irrational

Assume √5 = p/q (lowest terms) → p² = 5q² → p² divisible by 5 → p divisible by 5 → p=5k → 25k²=5q² → q²=5k² → q divisible by 5 → p and q both divisible by 5, contradiction.
∴ √5 is irrational.

⭐ Key Points for Board Exam

Euclid's Division Lemma is used to find HCF of two positive integers.
Fundamental Theorem of Arithmetic guarantees unique prime factorization for composite numbers.
HCF × LCM = Product of two numbers (only for two numbers).
√p is irrational if p is a prime number.
Sum/Difference of rational and irrational is always irrational.
Product/Quotient of rational (≠0) and irrational is always irrational.
Terminating decimal → denominator = 2ᵐ × 5ⁿ
Non-terminating repeating decimal → denominator has prime factors other than 2 and 5.
Decimal expansion of a rational number is either terminating or non-terminating repeating.

📌 Quick Revision Table

Concept	Formula/Statement
Euclid's Lemma	a = bq + r, 0 ≤ r < b
Fundamental Theorem	N = p₁^{a₁} × p₂^{a₂} × ...
HCF × LCM	HCF(a,b) × LCM(a,b) = a × b
Irrational Numbers	√2, √3, √5, √7, π, e
Terminating Decimal	q = 2ᵐ × 5ⁿ

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