Tuesday, April 13, 2010

The Math of Pi Made Easy.

Why the hell don't they teach math this way?
Blogged without permission from http://opinionator.blogs.nytimes.com/2010/04/04/take-it-to-the-limit/?src=me&ref=general. Go read the damn thing yourself.

If you keep moving halfway to the wall, will you ever get there?
Something about this one was deeply frustrating, the thought of getting closer and closer and yet never quite making it. Another concern was the thinly veiled presence of infinity. To reach the wall you’d need to take an infinite number of steps, and by the end they’d become infinitesimally small. Whoa.

Questions like this have always caused headaches. Around 500 B.C., Zeno of Elea posed a set of paradoxes about infinity that puzzled generations of philosophers, and that may have been partly to blame for its banishment from mathematics for centuries to come. In Euclidean geometry, for example, the only constructions allowed were those that involved a finite number of steps. The infinite was considered too ineffable, too unfathomable, and too hard to make logically rigorous.

But Archimedes, the greatest mathematician of antiquity, realized the power of the infinite. He harnessed it to solve problems that were otherwise intractable, and in the process came close to inventing calculus — nearly 2,000 years before Newton and Leibniz.

Thanks to him, we have Pi.

Let’s recall what we mean by pi. It’s a ratio of two distances. One of them is the diameter, the distance across the circle through its center. The other is the circumference, the distance around the circle. Pi is defined as their ratio, the circumference divided by the diameter.

circle with diameter and circumference indicated

If you’re a careful thinker, you might be worried about something already. How do we know that pi is the same number for all circles? Could it be different for big circles and little circles? The answer is no, but the proof isn’t trivial.

The key to thinking mathematically about curved shapes is to pretend they’re made up of lots of little straight pieces. That’s not really true, but it works … as long as you take it to the limit and imagine infinitely many pieces, each infinitesimally small. That’s the crucial idea behind all of calculus.

Here’s one way to use it to find the area of a circle. Begin by chopping the area into four equal quarters, and rearrange them like so.

Four quarters of a circle on left, then rearranged on right

The strange scalloped shape on the bottom has the same area as the circle, though that might seem pretty uninformative since we don’t know its area either. But at least we know two important facts about it. First, the two arcs along its bottom have a combined length of πr, exactly half the circumference of the original circle (because the other half of the circumference is accounted for by the two arcs on top). Second, the straight sides of the slices have a length of r, since each of them was originally a radius of the circle.

Next, repeat the process, but this time with eight slices, stacked alternately as before.

Circle showing eight slices

The scalloped shape looks a bit less bizarre now. The arcs on the top and the bottom are still there, but they’re not as pronounced. Another improvement is the left and right sides of the scalloped shape don’t tilt as much as they used to. Despite these changes, the two facts above continue to hold: the arcs on the bottom still have a net length of πr, and each side still has a length of r. And of course the scalloped shape still has the same area as before — the area of the circle we’re seeking — since it’s just a rearrangement of the circle’s eight slices.

As we take more and more slices, something marvelous happens: the scalloped shape approaches a rectangle. The arcs become flatter and the sides become almost vertical.

Circle with many slices

In the limit of infinitely many slices, the shape is a rectangle. Just as before, the two facts still hold, which means this rectangle has a bottom of width πr and a side of height r.

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