Real Numbers — Class 10 (Comprehensive Notes)

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1. What are Real Numbers?

Real numbers include all numbers that can be placed on the number line. They consist of rational and irrational numbers.

  • Rational numbers: Numbers expressible as p/q where p and q are integers and q ≠ 0. Their decimal expansions are terminating or repeating.
  • Irrational numbers: Numbers that cannot be written as p/q. Their decimal expansions are non-terminating and non-repeating (example: √2, π).

2. Euclid's Division Algorithm (Statement)

Given two positive integers a and b (a > b), there exist unique integers q and r such that:

a = bq + r, where 0 ≤ r < b.

This algorithm is used to find the highest common factor (HCF) of two numbers by repeated application.

Example (Use of Euclid's Algorithm)

Find HCF(1071, 462).

  1. 1071 = 462 × 2 + 147
  2. 462 = 147 × 3 + 21
  3. 147 = 21 × 7 + 0

So HCF = 21.

3. Fundamental Theorem of Arithmetic (FTA)

Every integer greater than 1 is either a prime or can be written as a product of primes — and this factorization is unique up to order.

This theorem is crucial to prove properties about rational and irrational numbers.

4. Decimal Expansions & Rationality

Properties:

  • Every rational number has a decimal expansion that either terminates or repeats eventually.
  • Every real number with a non-terminating, non-repeating decimal expansion is irrational.

Conversion tips

To convert a repeating decimal to fraction: use algebraic manipulation (let x = 0.ā̅ etc.).

5. Classic proof: √2 is irrational

Sketch of proof by contradiction:

  1. Assume √2 = p/q in lowest terms (p and q integers with gcd(p,q)=1).
  2. Then 2 = p²/q² ⇒ p² = 2q² ⇒ p² is even ⇒ p is even.
  3. Write p = 2k ⇒ p² = 4k² ⇒ 4k² = 2q² ⇒ q² = 2k² ⇒ q² even ⇒ q even.
  4. Thus p and q are both even — contradiction to gcd(p,q)=1. Hence √2 is irrational.

6. Important Properties (Useful for proofs)

  • If p is prime and p divides ab, then p divides a or p divides b.
  • Uniqueness of prime factorization implies rules about divisibility and rationality.
  • Every terminating decimal can be expressed as a rational number whose denominator is a power of 10 (reduced form).

7. Solved Examples (Quick)

Example 1. Show that 0.333... (= 0.̅3) is rational.

Solution. Let x = 0.333... Then 10x = 3.333... Subtract: 9x = 3 ⇒ x = 1/3.

Example 2. Prove that 0.101001000100001... is irrational.

Idea. This decimal does not follow a repeating pattern; one can show that no fraction p/q can match all digits, so it is non-terminating & non-repeating ⇒ irrational.

Example 3. Use FTA to write 360 as product of primes.

360 = 2³ × 3² × 5

8. Practice Questions

  1. Use Euclid's algorithm to find HCF(252, 198).
    Answer: 18 (use repeated division).
  2. Express 0.727272... as a rational number.
    Answer: 8/11.
  3. Prove that √3 is irrational.
  4. Write 546 as product of primes.

9. Summary (Quick revision)

  • Real numbers = Rational ∪ Irrational.
  • Euclid's algorithm finds HCF using repeated division.
  • FTA guarantees unique prime factorisation for integers > 1.
  • Irrational numbers have non-terminating, non-repeating decimals; classic example: √2.