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A historian of mathematics would tell it differently. So would a pure mathemati- cian. What fascinates me as an applied mathematician is the push and pull between the real world around us and the ideal world in our heads.

Phenomena out there guide the mathematical questions we ask; conversely, the math we imagine sometimes foreshadows what actually happens out there in reality. When it does, the effect is un- canny. To be an applied mathematician is to be outward-looking and intellectually promiscuous.

To those in my field, math is not a pris- tine, hermetically sealed world of theorems and proofs echoing back on themselves. We embrace all kinds of subjects: philosophy, poli- tics, science, history, medicine, all of it.

This is a much broader view of calculus than usual. It encom- passes the many cousins and spinoffs of calculus, both within math- ematics and in the adjacent disciplines. Far from it. Science and technology were essential partners — and argu- ably the stars of the show. My point is merely that calculus has also played a crucial role, albeit often a supporting one, in giving us the world we know today.

Take the story of wireless communication. So, obviously, experimental physics was indispensable here. But so was calculus. In the s, a Scottish mathematical phys- icist named James Clerk Maxwell recast the experimental laws of electricity and magnetism into a symbolic form that could be fed into the maw of calculus. Apparently something was miss- ing in the physics. He tried patching it up by including a new term in his equa- tion — a hypothetical current that would resolve the contradiction — and then let calculus churn again.

This time it spat out a sensible result, a simple, elegant wave equation much like the equation that describes the spread of ripples on a pond. A changing electric field would generate a changing magnetic field, which in turn would regenerate the electric field, and so on, each field bootstrapping the other for- ward, propagating together as a wave of traveling energy. And when Maxwell calculated the speed of this wave, he found — in what must have been one of the greatest Aha!

So he used calculus not only to predict the existence of electromagnetic waves but also to solve an age-old mystery: What was the nature of light? Light, he realized, was an electromagnetic wave. A decade later, Nikola Tesla built the first radio communication sys- tem, and five years after that, Guglielmo Marconi transmitted the first wireless messages across the Atlantic. Soon came television, cell phones, and all the rest. Clearly, calculus could not have done this alone.

But equally clearly, none of it would have happened without calculus. Or, per- haps more accurately, it might have happened, but only much later, if at all.

In the case of electromagnetic waves, it was a key first step for Maxwell to translate the laws that had been discovered experimentally into equations phrased in the language of calculus. But the language analogy is incomplete. It lets us transform one equation into another by performing various symbolic operations on them, operations subject to certain rules.

The symbol shuffling is useful shorthand, a convenient way to build arguments too intricate to hold in our heads. To a mathematician, the process feels almost palpable.

We want them to open up and talk to us. Fortu- nately, however, they did have a secret to reveal. With just the right prodding, they gave up the wave equation. At that point the linguistic function of calculus took over again.

In a matter of decades, this revelation would change the world. Calculus is an imaginary realm of symbols and logic; nature is an actual realm of forces and phenomena. Yet somehow, if the translation from reality into symbols is done art- fully enough, the logic of calculus can use one real-world truth to generate another.

Truth in, truth out. Start with something that is empirically true and symbolically formulated as Maxwell did with the laws of electricity and magnetism , apply the right logical ma- nipulations, and out comes another empirical truth, possibly a new one, a fact about the universe that nobody knew before like the existence of electromagnetic waves. In this way, calculus lets us peer into the future and predict the unknown. But why should the universe respect the workings of any kind of logic, let alone the kind of logic that we puny humans can mus- ter?

According to legend, Pythagoras felt it around bce when he and his disciples discovered that music was governed by the ratios of whole numbers. For instance, imagine plucking a guitar string. As the string vibrates, it emits a certain note. Now put your finger on a fret exactly halfway up the string and pluck it again. The ancient Greek musicians knew about the melodic con- cepts of octaves, fourths, and fifths and considered them beautiful.

This unexpected link between music the harmony of this world and numbers the harmony of an imagined world led the Pythago- reans to the mystical belief that all is number. They are said to have believed that even the planets in their orbits made music, the music of the spheres.

The as- tronomer Johannes Kepler had it bad. So did the physicist Paul Di- rac. In the end it pushed them to make their own discoveries that changed the world. Fortunately, a single big, beautiful idea runs through the subject from beginning to end. Once we become aware of this idea, the structure of calculus falls into place as variations on a unifying theme.

Alas, most calculus courses bury the theme under an avalanche of formulas, procedures, and computational tricks. It will guide us on our journey just as it guided the development of calculus itself, conceptually as well as histori- cally. In a nutshell, calculus wants to make hard problems simpler. It is utterly obsessed with simplicity. That might come as a surprise to you, given that calculus has a reputation for being complicated.

Its bulkiness is unavoidable. In fact, it has tackled and solved some of the most difficult and important problems our species has ever faced. Calculus succeeds by breaking complicated problems down into simpler parts. That strategy, of course, is not unique to calculus. The truly radical and distinctive move of calculus is that it takes this divide-and-conquer strategy to its utmost extreme — all the way out to infinity.

Instead of cutting a big prob- lem into a handful of bite-size pieces, it keeps cutting and cutting relentlessly until the problem has been chopped and pulverized into its tiniest conceivable parts, leaving infinitely many of them. The remaining challenge at that point is to put all the tiny answers back together again. Thus, calculus proceeds in two phases: cutting and rebuilding.

In mathematical terms, the cutting process always involves infinitely fine subtraction, which is used to quantify the differences between the parts. Accordingly, this half of the subject is called differential calculus. The reassembly process always involves infinite addition, which integrates the parts back into the original whole. This half of the subject is called integral calculus.

This strategy can be used on anything that we can imagine slicing endlessly. Such infinitely divisible things are called continua and are said to be continuous, from the Latin roots con together with and tenere hold , meaning uninterrupted or holding together. A shape, an object, a liquid, a motion, a time interval — all of them are grist for the calculus mill. Notice the act of creative fantasy here. Soup and steel are not re- ally continuous. More generally, the kinds of entities modeled as continua by cal- culus include almost anything one can think of.

In every case the strategy remains the same: split a complicated but continuous problem into infinitely many simpler pieces, then solve them separately and put them back together. The Infinity Principle To shed light on any continuous shape, object, motion, process, or phenomenon — no matter how wild and com- plicated it may appear — reimagine it as an infinite series of simpler parts, analyze those, and then add the results back together to make sense of the original whole.

The Golem of Infinity The rub in all of this is the need to cope with infinity. As in any tale of hubris, the monster inevitably turns on its maker. The creators of calculus were aware of the danger but still found infinity irresistible.

Sure, occasionally it ran amok, leaving paradox, confusion, and philosophical havoc in its wake. Yet after each of these episodes, mathematicians always managed to subdue the mon- ster, rationalize its behavior, and put it back to work. In the end, everything always turned out fine. The desire to harness infinity and exploit its power is a narrative thread that runs through the whole twenty-five-hundred-year story of calculus.

All this talk of desire and confusion might seem out of place, given that mathematics is usually portrayed as exact and impecca- bly rational. It is rational, but not always initially. Creation is intui- tive; reason comes later.

In the story of calculus, more than in other parts of mathematics, logic has always lagged behind intuition. This makes the subject feel especially human and approachable, and its geniuses more like the rest of us.

Curves, Motion, and Change The Infinity Principle organizes the story of calculus around a meth- odological theme. But calculus is as much about mysteries as it is about methodology. Three mysteries above all have spurred its de- velopment: the mystery of curves, the mystery of motion, and the mystery of change. The fruitfulness of these mysteries has been a testament to the value of pure curiosity.

Puzzles about curves, motion, and change might seem unimportant at first glance, maybe even hopelessly eso- teric. But because they touch on such rich conceptual issues and be- cause mathematics is so deeply woven into the fabric of the universe, the solution to these mysteries has had far-reaching impacts on the course of civilization and on our everyday lives. It all started with the mystery of curves. To keep things as simple as possible, the early geometers typically concentrated on abstract, idealized versions of curved shapes and ignored thickness, roughness, and texture.

The surface of a math- ematical sphere, for instance, was imagined to be an infinitesimally thin, smooth, perfectly round membrane with none of the thick- ness, bumpiness, or hairiness of a coconut shell. Triangles and squares were easy. So were cubes. They were composed of straight lines and flat pieces of planes joined together at a small number of corners. Geometers all over the world — in ancient Babylon and Egypt, China and India, Greece and Japan — knew how to solve problems like these.

But round things were brutal. No one could figure out how much surface area a sphere had or how much volume it could hold. Even finding the circumference and area of a circle was an insurmountable problem in the old days.

There was no way to get started. There were no straight pieces to latch onto. Anything that was curved was inscrutable. So this is how calculus began. Circles and spheres and other curved shapes were the Himalayas of their era.

Like explorers climbing Mount Everest, geometers wanted to solve curves because they were there. The breakthrough came from insisting that curves were actually made of straight pieces. The only hitch was that those pieces would then have to be infinitesimally small and infinitely numerous. Through this fantastic conception, integral calculus was born. This was the earliest use of the Infinity Principle. The story of how it developed will occupy us for several chapters, but its essence is already there, in embryonic form, in a simple, intuitive insight: If we zoom in closely enough on a circle or anything else that is curved and smooth , the por- tion of it under the microscope begins to look straight and flat.

So in principle, at least, it should be possible to calculate whatever we want about a curved shape by adding up all the straight little pieces. Collec- tively, however, and sometimes through bitter rivalries, they eventu- ally began to make headway on the riddle of curves. The quest to solve the mystery of curves reached a fever pitch when it became clear that curves were much more than geometric diversions.

They were a key to unlocking the secrets of nature. They arose naturally in the parabolic arc of a ball in flight, in the elliptical orbit of Mars as it moved around the sun, and in the convex shape of a lens that could bend and focus light where it was needed, as was required for the burgeoning development of microscopes and telescopes in late Renaissance Europe.

And so began the second great obsession: a fascination with the mysteries of motion on Earth and in the solar system. Through observation and ingenious experiments, scientists discovered tanta- lizing numerical patterns in the simplest moving things.

They mea- sured the swinging of a pendulum, clocked the accelerating descent of a ball rolling down a ramp, and charted the stately procession of planets across the sky. The only catch was that nobody could explain the marvelous new pat- terns, at least not with the existing forms of math.

Arithmetic and geometry were not up to the task, even in the hands of the greatest mathematicians. A ball rolling down a ramp kept changing its speed, and a planet revolving around the sun kept changing its direction of travel. Worse yet, the planets moved faster when they got close to the sun and slowed down as they receded from it. There was no known way to deal with motion that kept changing in ever-changing ways.

Earlier mathematicians had worked out the mathematics of the most trivial kind of motion, namely, motion at a constant speed where distance equals rate times time.

But when speed changed and kept on changing continuously, all bets were off. Motion was proving to be as much of a conceptual Mount Everest as curves were. The Infinity Principle came to the rescue, just as it had for curves. This time the act of wishful fantasy was to pretend that mo- tion at a changing speed was made up of infinitely many, infinitesi- mally brief motions at a constant speed.

To visualize what this would mean, imagine being in a car with a jerky driver at the wheel. As you anxiously watch the speedometer, it moves up and down with every jerk.

Nobody can tap the gas pedal that fast. These ideas coalesced in the younger half of calculus, differential calculus. It was precisely what was needed to work with the infinites- imally small changes of time and distance that arose in the study of ever-changing motion as well as with the infinitesimal straight pieces of curves that arose in analytic geometry, the newfangled study of curves defined by algebraic equations that was all the rage in the first half of the s.

Thus, the mysteries of curves and motion collided. They were now both at the center stage of calculus in the mids, banging into each other, creating mathematical mayhem and confusion. Out of the tumult, differential calculus began to flower, but not without controversy.

Some mathematicians were criticized for playing fast and loose with infinity. Others derided algebra as a scab of symbols. With all the bickering, progress was fitful and slow. And then a child was born on Christmas Day. This young mes- siah of calculus was an unlikely hero. Born premature and father- less and abandoned by his mother at age three, he was a lonesome boy with dark thoughts who grew into a secretive, suspicious young man.

Yet Isaac Newton would make a mark on the world like no one before or since. First, he solved the holy grail of calculus: he discovered how to put the pieces of a curve back together again — and how to do it eas- ily, quickly, and systematically. By combining the symbols of algebra with the power of infinity, he found a way to represent any curve as a sum of infinitely many simpler curves described by powers of a variable x, like x 2, x 3, x 4, and so on.

With these ingredients alone, he could cook up any curve he wanted by putting in a pinch of x and a dash of x 2 and a heaping tablespoon of x 3. It was like a master recipe and a universal spice rack, butcher shop, and vegetable garden, all rolled into one.

With it he could solve any problem about shapes or motions that had ever been considered. Then he cracked the code of the universe. Newton discovered that motion of any kind always unfolds one infinitesimal step at a time, steered from moment to moment by mathematical laws written in the language of calculus. With just a handful of differ- ential equations his laws of motion and gravity , he could explain everything from the arc of a cannonball to the orbits of the plan- ets.

Its im- pact on the philosophers and poets of Europe was immense. With the mysteries of curves and motion now settled, calculus moved on to its third lifelong obsession: the mystery of change.

The stock market rises and falls. Are there laws for population growth, the spread of epi- demics, and the flow of blood in an artery? 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. Filter - processing the results in accordance with the specified criteria any word, number by which you can "sift" the data and leave the desired: for example, by genre, by tracker Skrillex - Bangarang -Sebastian[Ub3r].

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