[tech4all] Visualizing the area of a circle using the power of infinity
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. In the coming weeks we'll delve into the great ideas at the heart of calculus. But for now I'd like to begin with the first beautiful hints of them, visible in ancient calculations about circles and 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.  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. Here's an intuitive argument. Imagine using a photocopier to reduce an image of a circle by, say, 50 percent. Then all distances in the picture — including the circumference and the diameter — would shrink in proportion by 50 percent. So when you divide the new circumference by the new diameter, that 50 percent change would cancel out, leaving the ratio between them unaltered. That ratio is pi. Of course, this doesn't tell us how big pi is. Simple experiments with strings and dishes are good enough to yield a value near 3, or if you're more meticulous, 3 and 1/7th. But suppose we want to find pi exactly or at least approximate it to any desired accuracy. What then? This was the problem that confounded the ancients. Before turning to Archimedes's brilliant solution, we should mention one other place where pi appears in connection with circles. The area of a circle (the amount of space inside it) is given by the formula  Here A is the area, π is the Greek letter pi, and r is the radius of the circle, defined as half the diameter.  All of us memorized this formula in high school, but where does it come from? It's not usually proven in geometry class. If you went on to take calculus, you probably saw a proof of it there, but is it really necessary to use calculus to obtain something so basic? Yes, it is. What makes the problem difficult is that circles are round. If they were made of straight lines, there'd be no issue. Finding the areas of triangles, squares and pentagons is easy. But curved shapes like circles are hard. 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.  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.  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.  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.  But now the problem is easy. The area of a rectangle equals its width times its height, so multiplying πr times r yields an area of πr2 for the rectangle. And since the rearranged shape always has the same area as the circle, that's the answer for the circle too! What's so charming about this calculation is the way infinity comes to the rescue. At every finite stage, the scalloped shape looks weird and unpromising. But when you take it to the limit — it becomes simple and beautiful, and everything becomes clear. That's how calculus works at its best. Excerpt from http://opinionator. |
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posted by Bharath @ 11:43 PM
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