Why π Is Not 3.14: Understanding Approximation, Infinite Series, and Ramanujan’s Formula

Pi, written as π, is one of the most famous numbers in mathematics. In school, students often use 3.14 or 22/7 whenever a problem mentions π. These values are useful shortcuts, but they are not the same as π itself. This research paper explores the difference between an approximation and an exact mathematical constant. It begins with a simple calculus limit that shows why replacing π with 22/7 can lead to the wrong answer. The paper then explains the meaning of π as the ratio of a circle’s circumference to its diameter, and it discusses why π is irrational and transcendental. Next, it studies the Basel Problem, where Euler connected π to the infinite sum of reciprocal squares. Finally, the paper examines Srinivasa Ramanujan’s famous infinite series for 1/π, which shows how deeply π is connected to patterns, symmetry, and advanced number theory. The main argument is that π should not be treated as just 3.14. It is an exact, infinite, and powerful idea that connects geometry, calculus, number theory, and mathematical creativity.