What is an Irrational Number?
A number is called irrational if it cannot be written in the form p/q, where p and q are integers and q ≠ 0.
Common Irrational Numbers
• Square roots of non-perfect squares: √2, √3, √5, √7, √11
• Cube roots of non-perfect cubes: ∛2, ∛3, ∛4
• The famous mathematical constant: π (pi) ≈ 3.14159...
• Non-terminating non-recurring decimals: 0.101001000100001...
Real Numbers Collection (R)
The collection of all rational numbers (Q) and irrational numbers (I) together forms the set of Real Numbers (R). Every real number is represented by a unique point on the number line.
Locating √2 on the Number Line (Pythagoras Method)
1. Draw a line and mark origin O as 0 and unit distance OA = 1 unit.
2. Construct a perpendicular AB = 1 unit at point A.
3. By Pythagoras Theorem in right △OAB:
OB = √2 units
4. With O as centre and radius OB = √2, draw an arc intersecting the number line at point P. Point P represents √2.
Key Takeaways
Irrational numbers have non-terminating non-recurring decimal expansions.
Sum, difference, product, or quotient of a rational and irrational number is always irrational.
Pythagoras theorem is used step-by-step to construct square root spiral (√2, √3, √4, √5...).