Here’s the infamous “proof” that shows that π is equal to 4.
The zig-zagged polygon, whose perimeter always remains 4, does appear to approach the circle as we repeat these steps to infinity. Ergo, π is equal to 4!
The problem, as Vi Hart explains in this amusing math doodle video, is that while the area of the polygon does approach the area of the circle, the actual perimeter of the polygon is much larger than the circumference of the circle. Confused? Think of it this way: if you put a jumbled up 10 feet long rope into a 1 foot long container, you wouldn’t say that the rope is now 1 foot long, would you? You would take out the rope, and extend it fully to measure its actual size.
See some more discussion here. Also, here’s another video that “proves” — using a similar approach — that π is equal to 2, and the square root of 2 (another one of my favorite irrational numbers) is also equal to 2!
By the way, if π were actually equal to 4, all circles would be squares. What a terrible world that would be! Similarly, if π were equal to 3, all circles would be hexagons. (For a hexagon, the ratio of its circumference to its diameter is equal to 3.)
Oh, and happy pi day! Don’t forget to have some delicious pie, and as you eat it, marvel at the glory of this magnificent, transcendental, and most importantly, irrational number.