Real Numbers Revision Notes - Class 10 Mathematics

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📌 Key Points

Euclid's Division Lemma: For positive integers a and b, a = bq + r where 0 ≤ r < b
Euclid's algorithm: Repeatedly apply division lemma until remainder = 0. The last divisor is HCF.
Fundamental Theorem of Arithmetic: Every composite number = unique product of primes
Rational Numbers: Can be expressed as p/q (p, q integers, q ≠ 0)
Rational Decimals: Terminating (0.5, 0.25) or non-terminating repeating (0.333...)
Irrational Numbers: Cannot be expressed as p/q. Examples: √2, √3, π, e
Irrational Decimals: Non-terminating, non-repeating (π = 3.141592...)
Proving √p (p prime) is irrational: Assume √p = a/b, square both sides, show p divides both a and b (contradiction)
Real Numbers = Rational + Irrational. Every real number belongs to exactly one category.
Terminating decimal: Denominator (in lowest form) has only factors of 2 and 5
Non-terminating repeating: Denominator (in lowest form) has factors other than 2 and 5
HCF × LCM = a × b (for any two positive integers a and b)
HCF of numbers: Product of common prime factors with lowest powers
LCM of numbers: Product of all prime factors with highest powers
For three numbers: Find HCF and LCM step-by-step (pairwise if needed)
Operations on irrationals: Rational + Irrational = Irrational; Rational × Irrational (non-zero) = Irrational
Between any two rationals, infinitely many irrationals exist (and vice versa)
√n is irrational if n is not a perfect square. √4 = 2 (rational), √5 is irrational.
Prime factorization method: Divide by smallest prime repeatedly until quotient becomes 1
Coprime/Co-prime numbers: Two numbers with HCF = 1. Example: 7 and 10

📘 Important Definitions

Euclid's Division Lemma

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

HCF (Highest Common Factor)

The largest positive integer that divides both a and b without remainder.

LCM (Least Common Multiple)

The smallest positive integer that is divisible by both a and b.

Rational Number

A number that can be expressed in the form p/q where p and q are integers and q ≠ 0.

Irrational Number

A number that cannot be expressed in the form p/q where p and q are integers.

Prime Number

A natural number greater than 1 that has exactly two factors: 1 and itself.

Composite Number

A natural number greater than 1 that has more than two factors.

Coprime Numbers

Two numbers whose HCF is 1. They have no common factor except 1.

Resistivity (ρ)

A measure of a material's resistance to electrical current. It depends only on the material and temperature.

Perfect Square

A number that is the square of an integer. Example: 4 = 2², 9 = 3², 16 = 4².

🔢 Formulas & Laws

Euclid's Division Lemma

a = bq + r, where 0 ≤ r < b

Used to find HCF using division algorithm

HCF and LCM Relationship

HCF(a, b) × LCM(a, b) = a × b

Always verify HCF and LCM using this relationship

Rational Number

p/q where p, q ∈ ℤ, q ≠ 0

Can be expressed as fraction with integer numerator and denominator

Prime Factorization

n = p₁^a₁ × p₂^a₂ × ... × pₖ^aₖ

Every composite number has unique prime factorization

HCF from Prime Factorization

Product of common prime factors with lowest powers

For 12 = 2² × 3 and 18 = 2 × 3², HCF = 2 × 3 = 6

LCM from Prime Factorization

Product of all prime factors with highest powers

For 12 = 2² × 3 and 18 = 2 × 3², LCM = 2² × 3² = 36

Decimal Expansion (Rational)

Either terminating or non-terminating repeating

1/4 = 0.25 (terminating); 1/3 = 0.333... (repeating)

Converting Repeating Decimal

For 0.a₁a₂...aₙ repeating: x = (a₁a₂...aₙ)/(10ⁿ - 1)

For 0.23̄: x = 23/99

Terminating Decimal Condition

p/q terminates iff q = 2^m × 5^n (in lowest form)

1/8 terminates (8 = 2³), but 1/6 doesn't (6 = 2 × 3)

Sum/Difference of Rational and Irrational

Rational ± Irrational = Irrational

Example: 2 + √2 is irrational

Product (Non-zero Rational × Irrational)

Rational × Irrational = Irrational

Example: 3 × √2 = 3√2 is irrational

Irrationality of √p (p prime)

√p cannot be expressed as p/q (proved by method of contradiction)

√2, √3, √5, √7 are all irrational

⚠️ Common Mistakes

✗ Wrong: Confusing rational and irrational numbers

✓ Correct: Rational: can be expressed as p/q; Irrational: cannot be expressed as p/q

✗ Wrong: Thinking √4 is irrational because it's a square root

✓ Correct: √4 = 2 (rational); only √n for non-perfect square n is irrational

✗ Wrong: Forgetting that remainder must satisfy 0 ≤ r < b

✓ Correct: In Euclid's lemma a = bq + r, the remainder must always be less than divisor

✗ Wrong: Mixing up HCF and LCM applications

✓ Correct: HCF: maximum/largest (e.g., maximum length); LCM: minimum/least (e.g., next occurrence)

✗ Wrong: Thinking 22/7 is irrational because it approximates π

✓ Correct: 22/7 is rational (fraction of integers); π itself is irrational

✗ Wrong: Confusing terminating and repeating decimals as both terminating

✓ Correct: 0.5 is terminating; 0.333... is non-terminating repeating (both are rational)

✗ Wrong: Not verifying HCF × LCM = a × b

✓ Correct: Always use this relationship to check your answers

📝 Exam Focus

These questions are frequently asked in CBSE exams:

Finding HCF using Euclid's division algorithm (2-3 marks)

Proving √2 or √3 is irrational (5 marks)

Finding HCF and LCM using prime factorization (2-3 marks)

HCF-LCM application problems (cutting, tiling, bells) (3-5 marks)

Converting repeating decimals to fractions (2-3 marks)

Identifying rational and irrational numbers (1-2 marks)

MCQs on HCF, LCM, and properties of real numbers (1 mark each)

Proving other numbers (√5, 2√3) are irrational (3-5 marks)

🎯 Last-Minute Recall