Answer:
Here are three reasons why π (pi) is considered irrational:
1. Non-repeating decimal: Irrational numbers are defined as numbers that cannot be expressed as a fraction of two integers. Pi's decimal representation never repeats or terminates, meaning it goes on indefinitely without a repeating pattern. This non-repeating nature makes it impossible to represent pi as a simple fraction.
2. Proof by contradiction: One of the earliest proofs of π's irrationality was by Johann Lambert in 1768. He showed that if π were rational, then it would imply the existence of a polynomial equation with integer coefficients having π as a root. However, such a polynomial cannot exist because π is transcendental (meaning it is not a root of any non-zero polynomial equation with integer coefficients). This contradiction demonstrates that π cannot be rational.
3. Circle circumference and diameter relationship: Pi is intrinsically linked to the geometry of circles, specifically the relationship between a circle's circumference and its diameter. No matter the size of the circle, this ratio remains constant and is equal to π. Since this ratio cannot be expressed as a fraction, π must be irrational.
