Class 10 Mathematics - Real Numbers

1. Euclid's Division Lemma

For any two positive integers a and b:

a = bq + r where 0 ≤ r < b

Example:

Find HCF of 455 and 42

• 455 = 42 × 10 + 35
• 42 = 35 × 1 + 7
• 35 = 7 × 5 + 0

HCF = 7

2. Fundamental Theorem of Arithmetic

Every composite number can be expressed as a product of prime numbers, and this factorization is unique.

Example:

• 420 = 2² × 3 × 5 × 7
• 360 = 2³ × 3² × 5

3. Finding HCF and LCM

Using Prime Factorization:

• HCF = Product of smallest powers = 2 × 3 = 6
• LCM = Product of highest powers = 2² × 3² = 36

Relation:

HCF × LCM = Product of two numbers

4. Irrational Numbers

Numbers that cannot be expressed as p/q form.

Examples:

• √2, √3, √5, π

Proving √2 is irrational:

Assume √2 = p/q (in lowest form)

• 2 = p²/q²
• p² = 2q² (p is even)
• Let p = 2m
• 4m² = 2q²
• q² = 2m² (q is also even)

Contradiction! Both can't be even in lowest form.

5. Decimal Expansions

Condition for Terminating:

Denominator = 2ⁿ × 5ᵐ form

✔️ Important Points

1. HCF × LCM = a × b
2. √2, √3, √5, √7 are irrational
3. Sum/Product of rational and irrational = irrational
4. Terminating decimals: denominator has only 2 and 5

Good luck with your exams! 📚