Zero: The Biography of a Dangerous Idea

by Charles Seife

L’Hôpital’s rule states that the value of the fraction was equal to the derivative of the top expression divided by the derivative of the bottom expression. For instance, consider the expression x/(sin x) when x=0; x=0, as does sin x, so the expression is equal to 0/0. Using L’Hopital’s rule, we see that the expression goes to 1/(cos x), as 1 is the derivative of x and cos x is the derivative of sin x. Cos x=1 when x=0, so the whole expression equals 1/1=1. Clever manipulations could also bring l’Hopital’s rule to resolve other odd expressions: ∞/∞, 00, 0∞, and ∞0.

Nobody could explain how those infinitesimals disappeared when squared; they just accepted the fact because making them vanish at the right time gave the correct answer. Nobody worried about dividing by zero when conveniently ignoring the rules of mathematics explained everything from the fall of an apple to the orbits of the planets in the sky. Though it gave the right answer, using calculus was as much an act of faith as declaring a belief in God.

Colin Maclaurin and Brook Taylor, perhaps the best British mathematicians in the era of isolation from the Continent, discovered how to use calculus to rewrite functions in a totally different form.

It was d’Alembert who realized that it was important to consider the journey as well as the destination. He was the one who hatched the idea of limit and solved calculus’s problems with zeros.

When you have an infinity in an expression, or when you divide by zero, all the mathematical operations—even those as simple as addition, subtraction, multiplication, and division—go out the window. Nothing makes sense any longer. So when you deal with an infinite number of terms in a series, even the + sign doesn’t seem so straightforward. That is why the infinite sum of +1 and –1 we saw at the beginning of the chapter seems to equal 0 and 1 at the same time.

Sometimes that limit doesn’t exist. For instance, the infinite sum of +1 and –1 does not have a limit. The value of the partial sums flips back and forth between 1 and 0; it’s not really heading to a predictable destination. But with Achilles’ race, the partial sums go from 1 to 1.5 to 1.75 to 1.875 to 1.9375 and so forth; they get closer and closer to two. The sums have a destination—a limit.

Multiply zero by anything and you get zero. Multiply infinity by anything and you get infinity. Dividing a number by zero yields infinity; dividing a number by infinity yields zero. Adding zero to a number leaves the number unchanged. Adding a number to infinity leaves infinity unchanged.

Zero and infinity are two sides of the same coin—equal and opposite, yin and yang, equally powerful adversaries at either end of the realm of numbers. The troublesome nature of zero lies with the strange powers of the infinite, and it is possible to understand the infinite by studying zero.

Zero is not the only number that was rejected by mathematicians for centuries. Just as zero suffered from Greek prejudice, other numbers were ignored as well, numbers that made no geometric sense. One of these numbers, i, held the key to zero’s strange properties.

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.

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.

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.

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

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.

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.

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.

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

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

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.

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.

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.)

Riemann made it perfectly acceptable to go off to infinity; since infinity is just a point on the sphere like any other point, it was no longer something to be feared. In fact, mathematicians started analyzing and classifying the points where a function blows up: singularities.

This is the definition of the infinite: it is something that can stay the same size even when you subtract from it.