1. Introduction to Real Numbers

A real number is any number that can be found on the number line. It includes:

  1. Natural Numbers (N) = {1, 2, 3, 4, ...}
  2. Whole Numbers (W) = {0, 1, 2, 3, ...}
  3. Integers (Z) = {..., -3, -2, -1, 0, 1, 2, 3, ...}
  4. Rational Numbers (Q) = Numbers that can be written as p/q, where p and q are integers and q ≠ 0. Example: 3/5, -7/2, 0.75
  5. Irrational Numbers (I) = Numbers that cannot be written as p/q. Their decimal expansion is non-terminating and non-repeating. Example: √2, Ï€, e
  6. Real Numbers (R) = Set of all rational and irrational numbers.

2. Euclid’s Division Lemma & Algorithm

Euclid’s Division Lemma

For any two positive integers a and b, there exist unique integers q and r such that:


a = bq + r, \quad 0 \leq r < b

Euclid’s Division Algorithm

Used to find the HCF (Highest Common Factor) of two numbers.
Steps:

  1. Divide a by b (where a > b).
  2. Find remainder r.
  3. Replace a with b and b with r, then repeat until r = 0.
  4. The last non-zero remainder is HCF.
  • Example: Find HCF of 455 and 42
    • HCF = 7

3. Fundamental Theorem of Arithmetic

  • Every composite number can be expressed as a product of prime numbers in a unique way (except for the order of factors).
  • Example:
    • 60 = 2 × 2 × 3 × 5
    • 84 = 2 × 2 × 3 × 7

Application: Used to find the HCF and LCM of numbers.

  • HCF (Highest Common Factor): Product of common prime factors.
  • LCM (Least Common Multiple): Product of highest powers of prime factors.

Example: Find HCF & LCM of 60 and 84

  • 60 = 2² × 3 × 5
  • 84 = 2² × 3 × 7
  • HCF = 2² × 3 = 12
  • LCM = 2² × 3 × 5 × 7 = 420

Relation:


\text{HCF} \times \text{LCM} = \text{Product of Numbers}

12 \times 420 = 60 \times 84 ]

4. Revisiting Rational & Irrational Numbers

Rational Numbers and Decimal Expansion

A rational number in decimal form is either:

  1. Terminating (e.g., 1/4 = 0.25)
  2. Non-Terminating but Repeating (e.g., 1/3 = 0.3333...)

Irrational Numbers

Numbers with non-terminating, non-repeating decimal expansions.
Examples: √2 = 1.414213..., Ï€ = 3.141592653...

Proof of Irrationality

  • Proof that √2 is irrational:
    • Assume √2 is rational, so it can be written as p/q (in lowest terms).
    • Squaring both sides: 2 = p²/q² → p² = 2q²
    • This implies p² is even, so p is even. Let p = 2m.
    • Substituting: (2m)² = 2q² → 4m² = 2q² → q² = 2m²
    • This means q² is even, so q is also even.
    • Since p and q are both even, they have a common factor 2, contradicting our assumption that p/q is in lowest terms.
    • Hence, √2 is irrational.

Similarly, √3, √5, etc., are irrational.

5. Decimal Representation of Rational & Irrational Numbers

6. Laws of Exponents for Real Numbers

For any real numbers a and b, and integers m and n:

  4. (for a ≠ 0)

Example:


2^3 \times 2^4 = 2^{3+4} = 2^7 = 128

7. Important Theorems & Applications

  1. If a prime number p divides a², then p also divides a.
  2. There are infinitely many prime numbers. (Proof by contradiction using Euclid’s theorem)
  3. If a number has a terminating decimal expansion, its denominator (in simplest form) must be of the form .

Example:

  • 7/8 = 0.875 (Denominator 8 = 2³, so terminating)
  • 1/3 = 0.3333... (Denominator 3 is not in the form , so non-terminating)

8. Summary

✔ Real numbers include both rational and irrational numbers.
✔ Euclid’s division algorithm helps find HCF.
✔ Fundamental Theorem of Arithmetic states every number has a unique prime factorization.
✔ Rational numbers have terminating or repeating decimals.
✔ Irrational numbers have non-terminating, non-repeating decimals.
✔ A number has a terminating decimal if its denominator has only 2 and/or 5 as prime factors.