Introduction
Irrational numbers are real numbers that cannot be written as a simple fraction or ratio.
Definition
A number is irrational if it cannot be written in the form p/q, where p and q are integers and q ≠
0.
Decimal Expansion
The decimal expansion of an irrational number is Non-Terminating Non-Repeating (Non-Recurring).
Example: 0.101001000100001... (The pattern does not repeat cyclically).
Famous Irrational Numbers
- √2 (Square root of 2): Approximately 1.414... Use in calculating the diagonal of a square with side 1.
- √3: Approximately 1.732...
- π (Pi): The ratio of circle's circumference to its diameter. Approx 3.14159...
- e (Euler's Number): Approx 2.718...
Note: √4 = 2 (Rational), √9 = 3 (Rational). Only square roots of non-perfect squares are irrational.
Operations with Irrational Numbers
- Sum/Difference of a rational and an irrational is Irrational. (e.g., 2 + √3)
- Product/Quotient of a non-zero rational and an irrational is Irrational. (e.g., 2√3)
- Sum, Difference, Product, or Quotient of two irrationals may be Rational or
Irrational.
(e.g., √2 × √2 = 2 → Rational; √2 × √3 = √6 → Irrational)