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Baltimore Baltimore is far more than what you know about it from television. Edmonton Northern exposure, at long last. Can calculus be used to describe how electrical signals propagate along nerves or to predict the flow of traffic on a highway? By pursuing this ambitious agenda, always in cooperation with other parts of science and technology, calculus has helped make the world modern.

Using observation and experiment, scientists worked out the laws of change and then used calculus to solve them and make predictions. For example, in Albert Einstein applied cal- culus to a simple model of atomic transitions to predict a remarkable effect called stimulated emission which is what the s and e stand for in laser, an acronym for light amplification by stimulated emission of radiation.

He theorized that under certain circumstances, light passing through matter could stimulate the production of more light at the same wavelength and moving in the same direction, cre- ating a cascade of light through a kind of chain reaction that would result in an intense, coherent beam.

A few decades later, the predic- tion proved to be accurate. The first working lasers were built in the early s. Since then, they have been used in everything from compact-disc players and laser-guided weaponry to supermarket bar-code scanners and medical lasers.

The laws of change in medicine are not as well understood as those in physics. Yet even when applied to rudimentary models, cal- culus has been able to make lifesaving contributions. The insights provided by the model overturned the pre- vailing view that the virus was lying dormant in the body; in fact, it was in a raging battle with the immune system every minute of every day.

With the new understanding that calculus helped provide, HIV infection has been transformed from a near-certain death sentence to a manageable chronic disease — at least for those with access to combination-drug therapy.

Admittedly, some aspects of our ever-changing world lie beyond the approximations and wishful thinking inherent in the Infinity Principle. In the subatomic realm, for example, physicists can no longer think of an electron as a classical particle following a smooth path in the same way that a planet or a cannonball does. According to quantum mechanics, trajectories become jittery, blurry, and poorly defined at the microscopic scale, so we need to describe the behavior of electrons as probability waves instead of Newtonian trajectories.

As soon as we do that, however, calculus returns triumphantly. In fact, it works spectacularly well. Naturally, the place to start is at infinity.

Shepherds needed to keep track of their flocks. Farmers needed to weigh the grain reaped in the harvest. Tax collectors had to decide how many cows or chickens each peasant owed the king. Out of such practical demands came the invention of numbers. At first they were tallied on fingers and toes. Later they were scratched on animal bones. As their representation evolved from scratches to symbols, numbers facilitated everything from taxation and trade to accounting and census taking.

We see evidence of all this in Meso- potamian clay tablets written more than five thousand years ago: row after row of entries recorded with the wedge-shaped symbols called cuneiform. Along with numbers, shapes mattered too. In ancient Egypt, the measurement of lines and angles was of paramount importance. Its predilection for straight lines, planes, and angles reflected its utili- tarian origins — triangles were useful as ramps, pyramids as monu- ments and tombs, and rectangles as tabletops, altars, and plots of land.

Builders and carpenters used right angles for plumb lines. Yet even when geometry was fixated on straightness, one curve always stood out, the most perfect of all: the circle. We see circles in tree rings, in the ripples on a pond, in the shape of the sun and the moon.

Circles surround us in nature. And as we gaze at circles, they gaze back at us, literally. There they are in the eyes of our loved ones, in the circular outlines of their pupils and irises. Circles span the practical and the emotional, as wheels and wedding rings, and they are mystical too. Their eternal return suggests the cycle of the seasons, reincarnation, eternal life, and never-ending love.

No won- der circles have commanded attention for as long as humanity has studied shapes. Mathematically, circles embody change without change. A point moving around the circumference of a circle changes direc- tion without ever changing its distance from a center.

And, of course, circles are symmetrical. If you rotate a circle about its center, it looks unchanged. That rotational symmetry may be why circles are so ubiquitous. Consider what happens when a raindrop hits a puddle: tiny ripples expand outward from the point of impact. Because they spread equally fast in all directions and because they started at a single point, the ripples have to be circles.

Symmetry demands it. Circles can also give birth to other curved shapes. If we imagine skewering a circle on its diameter and spinning it around that axis in three-dimensional space, the rotating circle makes a sphere, the shape of a globe or a ball. When a circle is moved vertically into the third dimension along a straight line at right angles to its plane, it makes a cylinder, the shape of a can or a hatbox.

INFI NI TY 3 Circles, spheres, cylinders, and cones fascinated the early ge- ometers, but they found them much harder to analyze than trian- gles, rectangles, squares, cubes, and other rectilinear shapes made of straight lines and flat planes.

They wondered about the areas of curved surfaces and the volumes of curved solids but had no clue how to solve such problems.

Roundness defeated them. Infinity as a Bridge Builder Calculus began as an outgrowth of geometry. Back around bce in ancient Greece, it was a hot little mathematical startup devoted to the mystery of curves.

The ambitious plan of its devotees was to use infinity to build a bridge between the curved and the straight. The hope was that once that link was established, the methods and tech- niques of straight-line geometry could be shuttled across the bridge and brought to bear on the mystery of curves.

At least, that was the pitch. At the time, that plan must have seemed pretty far-fetched. In- finity had a dubious reputation. Worse yet, it was nebulous and bewildering. What was it exactly? A number? A place? A concept? Given all the discoveries and technologies that ultimately flowed from calculus, the idea of using infinity to solve difficult geometry problems has to rank as one of the best ideas anyone ever had. Of course, none of that could have been foreseen in bce.

Still, infinity did put some impressive notches in its belt right away. One of its first and finest was the solution of a long-standing enigma: how to find the area of a circle. A Pizza Proof Before I go into the details, let me sketch the argument. The strat- egy is to reimagine the circle as a pizza. The result is a formula for the area of a circle. For the sake of this argument, the pizza needs to be an idealized mathematical pizza, perfectly flat and round, with an infinitesimally thin crust.

Its circumference, abbreviated by the letter C, is the dis- tance around the pizza, measured by tracing around the crust. In particular, r also measures how long the straight side of a slice is, assuming that all the slices are equal and cut from the center out to the crust.

We seem to be going backward. But as in any drama, the hero needs to get into trouble before triumphing. The dramatic tension is building. The first observation is that half of the crust became the curvy top of the new shape, and the other half became the bottom. So the curvy top has a length equal to half the circumference, C 2 , and so does the bottom, as shown in the diagram. The other thing to notice is that the tilted straight sides of the bulbous shape are just the sides of the original pizza slices, so they still have length r.

That length is eventually going to turn into the short side of the rectangle. If we make eight slices and rearrange them like so, our picture starts to look more nearly rect- angular. And the scallops on the top and bottom are a lot less bulbous than they were. They flattened out when we used more slices. As before, they have curvy length C 2 on the top and bottom and a slanted-side length r. To spruce up the picture even more, suppose we cut one of the slanted end pieces in half lengthwise and shift that half to the other side.

Our maneuvers are producing a sequence of shapes that are magically homing in on a certain rectangle. Well, since the slices are standing upright, the height is just the radius r of the original circle. Thus the width is half the circumference, C 2. And since moving the pizza slices around did not change their area, this must also be the area of the original circle! After an unpromising start, the more slices we took, the more rectangular the shape became.

But it was only in the limit of infinitely many slices that it became truly rect- angular. Everything becomes simpler at infinity. Limits and the Riddle of the Wall A limit is like an unattainable goal. You can get closer and closer to it, but you can never get all the way there. For example, in the pizza proof we were able to make the scalloped shapes more and more nearly rectangular by cutting enough slices and rearranging them.

But we could never make them genuinely rectangular. We could only approach that state of perfection. In fact, many of the greatest pioneers of the subject did precisely that and made great discoveries by doing so.

Logical, no. Imaginative, yes. Suc- cessful, very. A limit is a subtle concept but a central one in calculus. Perhaps the closest analogy is the Riddle of the Wall.

If you walk halfway to the wall, and then you walk half the remaining distance, and then you walk half of that, and on and on, will there ever be a step when you finally get to the wall? After you take ten steps or a million or any other number of steps, there will always be a gap between you and the wall.

But equally clearly, you can get arbitrarily close to the wall. What this means is that by taking enough steps, you can get to within a centimeter of it, or a millimeter, or a nanometer, or any other tiny but nonzero distance, but you can never get all the way there.

Here, the wall plays the role of the limit. It took about two thousand years for the limit concept to be rigorously defined. Until then, the pioneers of calculus got by just fine with intuition. From a modern perspective, they matter because they are the bedrock on which all of calculus is built. If the metaphor of the wall seems too bleak and inhuman who wants to approach a wall?

The Parable of. To reinforce the big ideas that everything becomes simpler at infin- ity and that limits are like unattainable goals, consider the following example from arithmetic. I vividly remember when my eighth-grade math teacher, Ms.

Stanton, taught us how to do this. It was memorable because she suddenly started talking about infinity. My parents certainly had no use for it. It seemed like a secret that only kids knew about. On the playground, it came up all the time in taunts and one-upmanship. Its invincible proper- ties made it great for finishing arguments in the schoolyard.

Who- ever deployed it first would win. But no teacher had ever talked about infinity until Ms. Stanton brought it up that day. By comparison, infinite decimals, which had infinitely many digits after the decimal point, seemed strange at first but appeared natural as soon as we started to discuss fractions.

There was no way out of the loop. The three dots at the end of 0. The naive interpretation is that there are literally infinitely many 3s packed side by side to the right of the decimal point. The advantage of this interpretation is that it seems easy and com- monsensical, as long as we are willing not to think too hard about what infinity means.

The more sophisticated interpretation is that 0. Except here, 0. The advantage of this interpretation is that we never have to invoke woolly-headed notions like infinity. We can stick to the finite. But there are other situations in mathematics where the completed infinity interpretation can cause logical mayhem. This is what I meant in the introduction when I raised the specter of the golem of infinity. Sometimes it really does make a difference how we think about the results of a process that approaches a limit.

Pretending that the process actually terminates and that it somehow reaches the nirvana of infinity can occasionally get us into trouble. The Parable of the Infinite Polygon As a chastening example, suppose we put a certain number of dots on a circle, space them evenly, and connect them to one another with straight lines.

Notice that the more dots we use, the rounder the polygons become and the closer they get to the circle. Meanwhile, their sides get shorter and more numerous. As we move progressively further through the sequence, the polygons approach the original circle as a limit.

In this way, infinity is bridging two worlds again. Of course, at any finite stage, a polygon is still just a polygon. It gets closer and closer to being a circle, but it never truly gets there. We are dealing here with potential infinity, not completed infinity.

So everything is airtight from the standpoint of logical rigor. But what if we could go all the way to completed infinity? Would the resulting infinite polygon with infinitesimally short sides actu- ally be a circle? All its corners would be sanded off.

Everything would become perfect and beautiful. A circle is simpler and more grace- ful than any of the thorny polygons that approach it.

So too for the pizza proof, where the limiting rectangle was simpler and more elegant than the scalloped shapes, with their unsightly bulges and cusps. In all these cases, the limiting shape or number was simpler and more symmetrical than its finite approximators. This is the allure of infinity. Everything becomes better there.

Should we take the plunge and say that a circle truly is a polygon with infinitely many infinitesimal sides? Doing so would be to commit the sin of completed infinity. It would condemn us to logi- cal hell. To see why, suppose we entertain the thought, just for a mo- ment, that a circle is indeed an infinite polygon with infinitesimal sides.

How long, exactly, are those sides? Zero length? If so, then infinity times zero — the combined length of all those sides — must equal the circumference of the circle. But now imagine a circle of double the circumference. Infinity times zero would also have to equal that larger circumference as well. So infinity times zero would have to be both the circumference and double the circumference.

What nonsense! There simply is no consistent way to define infinity times zero, and so there is no sensible way to regard a circle as an infinite polygon. Nevertheless, there is something so enticing about this intu- ition. Like the biblical original sin, the original sin of calculus — the temptation to treat a circle as an infinite polygon with infinitesi- mally short sides — is very hard to resist, and for the same reason.

For thousands of years, geometers struggled to figure out the circumference of a circle. If only a circle could be replaced by a polygon made of many tiny straight sides, the problem would be so much easier. By listening to the hiss of this serpent — but holding back just enough, by using potential infinity instead of the more tempting completed infinity — mathematicians learned how to solve the cir- cumference problem and other mysteries of curves.

But first, we need to gain an even deeper appreciation of just how dangerous completed infinity can be. The Sin of Dividing by Zero All across the world, students are being taught that division by zero is forbidden. They should feel shocked that such a taboo exists. Numbers are supposed to be orderly and well behaved.

Math class is a place for logic and reasoning. Dividing by zero is one of them. The root of the problem is infinity. Dividing by zero summons infinity in much the same way that a Ouija board supposedly sum- mons spirits from another realm. The answer to 6 divided by 0. Divide 6 by an even smaller num- ber, say 0. If we dare to divide 6 by a number much closer to zero, say 0. The trend is clear. The smaller the divisor, the bigger the answer.

In the limit as the divisor approaches zero, the answer approaches in- finity. Imagine dividing a 6-cen- timeter line into pieces that are each 0.

Those 60 pieces laid end to end make up the original. If we keep going and take this chopping frenzy to the limit, we are led to the bizarre conclusion that a 6-centimeter line is made up of infinitely many pieces of length zero. Maybe that sounds plausible. After all, the line is made up of infinitely many points, and each point has zero length.

We could just as well have claimed that a line of length 3 centimeters, or Evidently, multiplying zero by in- finity can give us any and every conceivable result — 6 or 3 or The Sin of Completed Infinity The transgression that dragged us into this mess was pretending that we could actually reach the limit, that we could treat infinity like an attainable number.

Back in the fourth century bce, the Greek philosopher Aristotle warned that sinning with infinity in this way could lead to all sorts of logical trouble.

He railed against what he called completed infinity and argued that only potential infinity made sense.

That, Aristotle felt, would lead to nonsense — as it does here, in revealing that zero times infinity can give any answer. And so he forbade the use of completed infinity in mathematics and philosophy. His edict was upheld by mathemati- cians for the next twenty-two hundred years. However, when Mac comes out of the women's bathroom, no one can leave until the bathroom situation is solved.

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