The Irrationality of π

Introduction:
The value of π (pi) is a fundamental mathematical constant that represents the ratio of a circle's circumference to its diameter. It is an irrational number, which means it cannot be expressed as a fraction of two integers. In this response, we will explain why π is irrational in detail.

Definition of an Irrational Number:
An irrational number is a real number that cannot be expressed as a simple fraction or ratio of two integers. It is a non-repeating and non-terminating decimal.

Proof by Contradiction:
To prove that π is irrational, we can use a method called proof by contradiction. This involves assuming the opposite of what we want to prove and then showing that it leads to a contradiction.

Assuming π is Rational:
Let's assume that π is rational, meaning it can be expressed as a fraction a/b, where a and b are integers with no common factors other than 1, and b is not equal to 0.

Expressing π as a Fraction:
If π is rational, we can express it as a fraction a/b. Therefore, π = a/b.

Squaring Both Sides:
Now, let's square both sides of the equation to get rid of the square root.

π^2 = (a/b)^2
π^2 = a^2 / b^2

Cross-Multiplying:
Next, we can cross-multiply the equation.

π^2 * b^2 = a^2

π^2 * b^2 is Rational:
Since a^2 and b^2 are integers, the product π^2 * b^2 is also a rational number.

Deducing π is Rational:
If π^2 * b^2 is rational, it means that π^2 itself is rational. Therefore, π must also be rational.

Contradiction:
However, this contradicts the assumption that π is rational. We assumed that π is rational, but we have reached a contradiction by proving that it is also rational. Hence, our initial assumption is incorrect.

Conclusion:
Through the proof by contradiction, we have shown that assuming π is rational leads to a contradiction. Therefore, π cannot be expressed as a fraction or ratio of two integers. Hence, π is an irrational number.

Summary:
- π is a fundamental mathematical constant representing the ratio of a circle's circumference to its diameter.
- An irrational number cannot be expressed as a fraction of two integers.
- By using the proof by contradiction, we assume π is rational, but reach a contradiction, proving that π is irrational.
- Thus, π cannot be expressed as a simple fraction or ratio of two integers, making it an irrational number.

22/7 is not its exact value. We assume that value unless stated to take it 3.14...