Yes, pi squared (π²) is irrational.
Understanding Irrationality of π²
A number is considered irrational if it cannot be expressed as a fraction p/q, where p and q are integers and q ≠ 0. The famous number pi (π) is a well-known irrational number. But what about pi squared?
The reference provided states: "Proof: π is transcendental, meaning that it is not the root of any polynomial equation with integer coefficients. Hence, π² is transcendental and irrational too."
Why is π² Irrational?
The irrationality of π² stems from the fact that π is a transcendental number.
- Transcendental Numbers: These numbers are not the root of any polynomial equation with integer coefficients. Examples include π and e.
- Irrational Numbers: All transcendental numbers are irrational, but not all irrational numbers are transcendental (e.g., √2 is irrational but algebraic).
Since π is transcendental, squaring it doesn't change this fundamental property. Therefore, π² also cannot be expressed as a fraction of two integers, making it irrational. In fact, it is also transcendental.
Examples and Implications
- The value of π² is approximately 9.869604401. Because it is irrational, its decimal representation goes on forever without repeating.
- Because π² is transcendental, constructions involving areas of circles can never be perfectly converted into squares using only a compass and straightedge (squaring the circle).
Summary Table
| Property | π | π² |
|---|---|---|
| Irrational | Yes | Yes |
| Transcendental | Yes | Yes |
| Algebraic | No | No |
