When you start stringing symbols together to get equations, you can wind up with something unexpected. For instance, take the above equation and change the - sign into a + sign. This leaves us with a very innocent-looking equation, 4x + 12 = 0, but the solution to that equation is now - 3, a negative number.
Just as Indian mathematicians accepted zero while Europeans rejected it for centuries, the East embraced negative numbers while the West tried to ignore them. As late as the seventeenth century, Descartes refused to accept negative numbers as roots of equations. He called them “false roots,” which explains why he never extended his coordinate system to the negative numbers. Descartes was a late holdover, a victim of his success in marrying algebra to geometry. Negative numbers had long been useful to algebraists—even Western algebraists. Negative numbers came up all the time in solving equations, such as quadratic equations.
A linear equation like 4x - 12 = 0 is extremely simple to solve, and such problems didn’t entertain algebraists for very long. So they soon turned to more difficult problems: quadratic equations—equations that begin with an x2 term, like x2 - 1 = 0. Quadratic equations are more complicated than regular equations; for one thing, they can have two different roots. For instance, x2 - 1 = 0 has two solutions: 1 and -1. (Substitute -1 or 1 for x in the equation and you’ll see what happens.) Either one of those solutions works; as it turns out, the expression x2 - 1 splits into (x - 1)(x + 1), making it easy to see that if x is 1 or -1, the expression goes to zero.
Though quadratic equations are more complicated than linear equations, there is a simple way to figure out what the roots of a quadratic equation are. It’s the famous quadratic formula, which is the crowning achievement of high-school algebra class. The formula for finding the roots of a quadratic equation ax2 + bx + c = 0 is: . The + sign gives us one root, while the - sign gives us the other. The quadratic formula has been known for centuries; the ninth-century mathematician al-Khowarizmi knew how to solve almost every quadratic equation, though he didn’t seem to consider negative numbers as roots. Not long after that, algebraists learned to accept negative numbers as valid solutions to equations. Imaginary numbers, though, were a little different.
Imaginary numbers never appeared in linear equations, but they began to crop up in quadratic ones. Consider the equation x2 + 1 = 0. No number seems to solve the equation; plugging in –1, 3, –750, 235.23, or any other positive or negative number you could think of doesn’t yield the correct answer. The expression simply will not split. Worse yet, when you try to apply the quadratic equation, you get two silly-sounding answers:
These expressions don’t seem to make any sense. The Indian mathematician Bhaskara wrote in the twelfth century that “there is no square root of a negative number, for a negative number is not a square.” What Bhaskara and others realized was that when you square a positive number, you get a positive number back; 2 times 2 equals 4, for instance. When you square a negative number, you still get a positive number: –2 times –2 also equals 4. When you square zero, you get zero. Positive numbers, negative numbers, and zero all give you nonnegative squares, and those three possibilities cover the whole number line. This means that there is no number on the number line that gives you a negative number when you square it. The square root of a negative number seemed like a ridiculous concept.
Descartes thought that these numbers were even worse than negative numbers; he came up with a scornful name for the square roots of negatives: imaginary numbers. The name stuck, and eventually, the symbol for the square root of –1 became i.
Algebraists loved i. Almost everyone else hated it. It was wonderful for solving polynomials—expressions like x3 + 3x + 1 that have x raised to various powers. In fact, once you allow i into the realm of numbers, every polynomial becomes solvable: x2 + 1 suddenly splits into (x - i)(x + i)—the roots of the equation are +i and -i. Cubic expressions like x3 - x2 + x - 1 split three ways, such as (x - 1)(x + i)(x - i). Quartic expressions—ones with a leading x4 term—always split into four terms, and quintics—ones with a leading x5 term—split five ways. All polynomials of degree n—those that have a leading term of xn—split into n distinct terms. This is the fundamental theorem of algebra.
As early as the sixteenth century, mathematicians were using numbers with i included—the so-called complex numbers—to solve cubic and quartic polynomials. And while many mathematicians saw the complex numbers as a convenient fiction, others saw God.
Leibniz thought that i was a bizarre mix between existence and nonexistence, something like a cross between 1 (God) and 0 (Void) in his binary scheme. Leibniz likened i to the Holy Spirit: both have an ethereal and barely substantial existence. But even Leibniz didn’t realize that i would finally reveal the relationship between zero and infinity. It would take two important developments in mathematics before the true link was uncovered.
Point and Counterpoint
One will then see the simplicity with which these concepts lead to properties already known and to an infinity of others which ordinary geometry does not seem to touch easily.
—JEAN-VICTOR PONCELET
The first development—projective geometry—was born in the turmoil of war. In the 1700s, France, England, Austria, Prussia, Spain, the Netherlands, and other countries were vying for power. Alliances formed and broke over and over again; new territorial disputes erupted over colonies, and countries struggled to dominate trade to and from the New World. France, England, and other countries skirmished throughout the first half of the eighteenth century, and roughly a quarter century after Newton died, a full-scale war erupted. France, Austria, Spain, and Russia fought England and Prussia for nine years.
In 1763 the French capitulated and the Seven Years’ War was over. (Two years of fighting occurred before war was officially declared.) The victory made England the preeminent power in the world, but it came at a great cost. Both France and England were exhausted and in debt—and they would both suffer the consequences: revolutions. A little more than a decade after the end of the Seven Years’ War, the American Revolution began; the revolt would strip England of its richest colony. In 1789, just as George Washington was sworn into office in the newly founded United States, the French Revolution began. Four years later the revolutionaries removed the French king’s head.
A mathematician, Gaspard Monge, signed the revolutionary government’s record of the king’s execution. Monge was a consummate geometer, specializing in three-dimensional geometry. He was responsible for the way architects and engineers draw buildings and machines: they project the design onto a vertical plane and a horizontal plane, preserving all the information needed to reconstruct the object. Monge’s work was so important to the military that much of it was made into a state secret by the revolutionary government and by the Napoleonic government that succeeded it soon afterward.
Jean-Victor Poncelet was a student of Monge’s who learned about three-dimensional geometry as he trained to become an engineer for Napoleon’s army. Unluckily for Poncelet, he entered the army just as Napoleon set off for Moscow in 1812.
While retreating from Moscow, Napoleon’s army was whittled down to almost nothing by a harsh winter and an equally harsh Russian army. At the battle of Krasnoy, Poncelet was left for dead on the battlefield. Still alive, he was captured by the Russians. Moldering in a Russian prison, Poncelet founded a new discipline: projective geometry.
Poncelet’s mathematics was the culmination of the work begun by the artists and architects of the fifteenth century, like Filippo Brunelleschi and Leonardo da Vinci, who discovered how to draw realistically—in perspective. When “parallel” lines converge at the vanishing point in a painting, observers are tricked into believing that the lines never meet. Squares on the floor become trapezoids in a painting; everything gets gently distorted, but it looks perfectly natural to the viewer. This is the property of an infinitely distant point—a zero at infinity.
Johannes Kepler, the man who discovered that planets travel in ellipses
, took this idea—the infinitely distant point—one step further. Ellipses have two centers, or foci; the more elongated the ellipse, the farther apart these foci are. And all ellipses have the same property: if you had a mirror in the shape of an ellipse and you placed a lightbulb at one focus, all the beams of light would converge at the other focus, no matter how stretched-out the ellipses are (Figure 29).
In his mind Kepler stretched an ellipse out more and more, dragging one focus farther and farther away. Then Kepler imagined that the second focus was infinitely far away: the second focus was a point at infinity. All of a sudden the ellipse becomes a parabola, and all of the lines that converged to a point become parallel lines. A parabola is simply an ellipse with one focus at infinity (Figure 30).
Figure 29: Light rays inside an ellipse
You can see this very nicely with a flashlight. Go into a dark room, stand next to a wall, and point the flashlight directly at it. You will get a nice, round circle of light projected on the wall. Now slowly tilt the flashlight upward (Figure 31). You’ll see the circle stretch out into an ellipse that gets longer and longer as you increase the tilt. All of a sudden, the ellipse opens up and becomes a parabola. Thus, Kepler’s point at infinity proved that parabolas and ellipses are actually the same thing. This was the beginning of the discipline of projective geometry, where mathematicians look at the shadows and projections of geometric figures to uncover hidden truths even more powerful than the equivalence of parabolas and ellipses. However, it all depended upon accepting a point at infinity.
Figure 30: Stretching an ellipse yields a parabola.
Figure 31: Flashlight ellipses and parabola
Gérard Desargues, a seventeenth-century French architect, was one of the early pioneers of projective geometry. He used the point at infinity to prove a number of important new theorems, but Desargues’s colleagues couldn’t understand his terminology and concluded that Desargues was nuts. Though a few mathematicians, like Blaise Pascal, picked up on Desargues’s work, it was forgotten.
None of this mattered to Jean-Victor Poncelet. As Monge’s student, Poncelet had learned the technique of projecting diagrams onto two planes, and as a prisoner of war he had a lot of spare time on his hands. He used his stay in prison to reinvent the concept of a point at infinity, and combining it with Monge’s work, he became the first true projective geometer. Upon his return from Russia (carrying a Russian abacus, by then an archaic oddity) he raised the discipline to a high art.* However, Poncelet had no idea that projective geometry would reveal the mysterious nature of zero, because the second important advance, the complex plane, was still needed. We must turn to Germany for this piece of the puzzle.
Carl Friedrich Gauss, born in 1777, was a German prodigy, and he began his mathematical career with an investigation of imaginary numbers. His doctoral thesis was a proof of the fundamental theorem of algebra—proving that a polynomial of degree n (a quadratic has degree 2, a cubic has degree 3, a quartic has degree 4, and so on) has n roots. This is only true if you accept imaginary numbers as well as real numbers.
Throughout his life Gauss worked on an incredible variety of topics—his work on curvature would become a key component of Einstein’s general theory of relativity—but it was Gauss’s way of graphing complex numbers that revealed a whole new structure in mathematics.
In the 1830s Gauss realized that each complex number—numbers that have real and imaginary parts, like 1 - 2i—can be displayed on a Cartesian grid. The horizontal axis represents the real part of the complex number, while the vertical axis represents the imaginary part (Figure 32). This simple construction, called the complex plane, revealed a lot about the way numbers work. Take, for example, the number i. The angle between i and the x-axis is 90 degrees (Figure 33). What happens when you square i? Well, by definition, i2 =-1—a point whose angle is 180 degrees from the x-axis; the angle has doubled. The number i3 is equal to -i—270 degrees from the x-axis; the angle has tripled. The number i4 = 1; we have gone around 360 degrees—exactly four times the original angle (Figure 34). This is not a coincidence. Take any complex number and measure its angle. Raising a number to the nth power multiplies its angle by n. And as you keep raising the number to higher and higher powers, the number will spiral inward or outward, depending on whether the number is on the inside or on the outside of the unit circle, a circle centered at the origin with radius 1 (Figure 35). Multiplication and exponentiation in the complex plane became geometric ideas; you could actually see them happening. This was the second big advance.
Figure 32: The complex plane
Figure 33: i is at 90 degrees
Figure 34: Different powers of i
The person who combined these two ideas was a student of Gauss’s: Georg Friedrich Bernhard Riemann. Riemann merged projective geometry with the complex numbers, and all of a sudden lines became circles, circles became lines, and zero and infinity became the poles on a globe full of numbers.
Figure 35: Spirals inside and outside of the unit circle
Riemann imagined a translucent ball sitting atop the complex plane, with the south pole of the ball touching zero. If there were a tiny light at the north pole of the ball, any figures that are marked on the ball would cast shadows on the plane below. The shadow of the equator would be a circle around the origin. The shadow of the southern hemisphere is inside the circle and the shadow of the northern hemisphere is outside (Figure 36). The origin—zero—corresponds to the south pole. Every point on the ball has a shadow on the complex plane; in a sense, every point on the ball is equivalent to its shadow on the plane and vice versa. Every circle on the plane is the shadow of a circle on the ball, and a circle on the ball corresponds to a circle on the plane…with one exception.
If you’ve got a circle that goes through the north pole of the ball, the shadow is no longer a circle. It is a line. The north pole is like the point at infinity that Kepler and Poncelet imagined. Lines on the plane are simply circles on the sphere that go through the north pole—the point at infinity (Figure 37).
Figure 36: Stereographic projection of the globe
Once Riemann saw that the complex plane (with a point at infinity) was the same thing as a sphere, mathematicians could see multiplication, division, and other, more difficult operations by analyzing the way the sphere deformed and rotated. For instance, multiplying by the number i was equivalent to spinning the sphere 90 degrees clockwise. If you take a number x and replace it with (x - 1)/(x + 1), that is equivalent to rotating the whole globe by 90 degrees so that the north and south poles lie on the equator (Figures 38, 39, 40). Most interesting of all, if you take a number x and replace it with its reciprocal 1/x, that is equivalent to flipping the sphere upside down and reflecting it in a mirror. The north pole becomes the south pole and the south pole becomes the north pole: zero becomes infinity and infinity becomes zero. It’s all built into the geometry of the sphere; 1/0 =? and 1/?= 0. Infinity and zero are simply opposite poles on the Riemann sphere, and they can switch places in a blink. And they have equal and opposite powers.
Figure 37: Lines and circles are the same.
Take all of the numbers in the complex plane and multiply them by two. That is like putting your hands on the south pole and stretching a rubber cover on the sphere away from the south pole and toward the north pole. Multiplying by one-half has the opposite effect. It is like stretching the rubber cover away from the north pole and toward the south pole. Multiplying by infinity is like sticking a needle in the south pole; the rubber sheet all flings upward toward the north pole: anything times infinity is infinity. Multiplying by zero is like sticking a needle on the north pole and everything winds up at zero: anything times zero is zero. Infinity and zero are equal and opposite—and equally destructive.
Figure 38: Riemann sphere
Figure 39: Riemann sphere transformed by i
Figure 40: Riemann sphere transformed by (x - 1)/(x + 1)
Zero and infinity are eternally locked in a struggle to engulf all
the numbers. Like a Manichaean nightmare, the two sit on opposite poles of the number sphere, sucking numbers in like tiny black holes. Take any number on the plane. For the sake of argument, we’ll choose i/2. Square it. Cube it. Raise it to the fourth power. The fifth. The sixth. The seventh. Keep multiplying. It slowly spirals toward zero like water down a drain. What happens to 2i? The exact opposite. Square it. Cube it. Raise it to the fourth power. It spirals outward (Figure 41). But on the number sphere, the two curves are duplicates of each other; they are mirror images (Figure 42). All numbers in the complex plane suffer this fate. They are drawn inexorably toward 0 or toward ?. The only numbers that escape are the ones that are equally distant from the two rivals—the numbers on the equator, like 1, -1, and i. These numbers, pulled by the tug of both zero and infinity, spiral around on the equator forever and ever, never able to escape the grasp of either. (You can see this on your calculator. Enter a number—any number. Square it. Square it again. Do it again and again; the number will quickly zoom toward infinity or toward zero, except if you entered 1 or -1 to begin with. There is no escape.)
The Infinite Zero
My theory stands as firm as a rock; every arrow directed against it will return quickly to its archer. How do I know this? I have studied it…I have followed its roots, so to speak, to the first infallible cause of all created things.
Zero Page 12
