Figure 4: The multiplication rubber band
Anything times zero is zero, so all the tick marks are at zero.
The rubber band has broken. The whole number line has collapsed.
Unfortunately, there is no way to get around this unpleasant fact. Zero times anything must be zero; it’s a property of our number system. For everyday numbers to make sense, they have to have something called the distributive property, which is best seen through an example. Imagine that a toy store sells balls in groups of two and blocks in groups of three. The neighboring toy store sells a combination pack with two balls and three blocks in it. One bag of balls and one bag of blocks is the same thing as one combination package from the neighboring store. To be consistent, buying seven bags of balls and seven bags of blocks from one toy store has to be the same thing as buying seven combination packs from the neighboring shop. This is the distributive property. Mathematically speaking, we say that 7 × 2 + 7 × 3 = 7 × (2 + 3). Everything comes out right.
Apply this property to zero and something strange happens. We know that 0 + 0 = 0, so a number multiplied by zero is the same thing as multiplying by (0 + 0). Taking two as an example, 2 × 0 = 2 × (0 + 0), but by the distributive property we know that 2 × (0 + 0) is the same thing as 2 × 0 + 2 × 0. But this means 2 × 0 = 2 × 0 + 2 × 0. Whatever 2 × 0 is, when you add it to itself, it stays the same. This seems a lot like zero. In fact, that is just what it is. Subtract 2 × 0 from each side of the equation and we see that 0 = 2 × 0. Thus, no matter what you do, multiplying a number by zero gives you zero. This troublesome number crushes the number line into a point. But as annoying as this property was, the true power of zero becomes apparent with division, not multiplication.
Just as multiplying by a number stretches the number line, dividing shrinks it. Multiply by two and you stretch the number line by a factor of two; divide by two and you relax the rubber band by a factor of two, undoing the multiplication. Divide by a number and you undo the multiplication: a tick mark that had been stretched to a new place on the number line resumes its original position.
We saw what happened when you multiply a number by zero: the number line is destroyed. Division by zero should be the opposite of multiplying by zero. It should undo the destruction of the number line. Unfortunately, this isn’t quite what happens.
In the previous example we saw that 2 × 0 is 0. Thus to undo the multiplication, we have to assume that (2 × 0)/0 will get us back to 2. Likewise, (3 × 0)/0 should get us back to 3, and (4 × 0)/0 should equal 4. But 2 × 0 and 3 × 0 and 4 × 0 each equal zero, as we saw—so (2 × 0)/0 equals 0/0, as do (3 × 0)/0 and (4 × 0)/0. Alas, this means that 0/0 equals 2, but it also equals 3, and it also equals 4. This just doesn’t make any sense.
Strange things also happen when we look at 1/0 in a different way. Multiplication by zero should undo division by zero, so 1/0 × 0 should equal 1. However, we saw that anything multiplied by zero equals zero! There is no such number that, when multiplied by zero, yields one—at least no number that we’ve met.
Worst of all, if you wantonly divide by zero, you can destroy the entire foundation of logic and mathematics. Dividing by zero once—just one time—allows you to prove, mathematically, anything at all in the universe. You can prove that 1 + 1 = 42, and from there you can prove that J. Edgar Hoover was a space alien, that William Shakespeare came from Uzbekistan, or even that the sky is polka-dotted. (See appendix A for a proof that Winston Churchill was a carrot.)
Multiplying by zero collapses the number line. But dividing by zero destroys the entire framework of mathematics.
There is a lot of power in this simple number. It was to become the most important tool in mathematics. But thanks to the odd mathematical and philosophical properties of zero, it would clash with the fundamental philosophy of the West.
Chapter 2
Nothing Comes of Nothing
[ THE WEST REJECTS ZERO ]
Nothing can be created from nothing.
—LUCRETIUS, DE RERUM NATURA
Zero clashed with one of the central tenets of Western philosophy, a dictum whose roots were in the number-philosophy of Pythagoras and whose importance came from the paradoxes of Zeno. The whole Greek universe rested upon this pillar: there is no void.
The Greek universe, created by Pythagoras, Aristotle, and Ptolemy, survived long after the collapse of Greek civilization. In that universe there is no such thing as nothing. There is no zero. Because of this, the West could not accept zero for nearly two millennia. The consequences were dire. Zero’s absence would stunt the growth of mathematics, stifle innovation in science, and, incidentally, make a mess of the calendar. Before they could accept zero, philosophers in the West would have to destroy their universe.
The Origin of Greek Number-Philosophy
In the beginning, there was the ratio, and the ratio was with God, and the ratio was God.*
—JOHN 1:1
The Egyptians, who had invented geometry, thought little about mathematics. For them it was a tool to reckon the passage of the days and to maintain plots of land. The Greeks had a very different attitude. To them, numbers and philosophy were inseparable, and they took both very seriously. Indeed, the Greeks went overboard when it came to numbers. Literally.
Hippasus of Metapontum stood on the deck, preparing to die. Around him stood the members of a cult, a secret brotherhood that he had betrayed. Hippasus had revealed a secret that was deadly to the Greek way of thinking, a secret that threatened to undermine the entire philosophy that the brotherhood had struggled to build. For revealing that secret, the great Pythagoras himself sentenced Hippasus to death by drowning. To protect their number-philosophy, the cult would kill. Yet as deadly as the secret that Hippasus revealed was, it was small compared to the dangers of zero.
The leader of the cult was Pythagoras, an ancient radical. According to most accounts, he was born in the sixth century BC on Samos, a Greek island off the coast of Turkey famed for a temple to Hera and for really good wine. Even by the standards of the superstitious ancient Greeks, Pythagoras’s beliefs were eccentric. He was firmly convinced that he was the reincarnated soul of Euphorbus, a Trojan hero. This helped convince Pythagoras that all souls—including those of animals—transmigrated to other bodies after death. Because of this, he was a strict vegetarian. Beans, however, were taboo, as they generate flatulence and are like the genitalia.
Pythagoras may have been an ancient New Age thinker, but he was a powerful orator, a renowned scholar, and a charismatic teacher. He was said to have written the constitution for Greeks living in Italy. Students flocked to him, and he soon acquired a retinue of followers who wanted to learn from the master.
The Pythagoreans lived according to the dicta of their leader. Among other things they believed that it is best to make love to women in the winter, but not in the summer; that all disease is caused by indigestion; that one should eat raw food and drink only water; and that one must avoid wearing wool. But at the center of their philosophy was the most important tenet of the Pythagoreans: all is number.
The Greeks had inherited their numbers from the geometric Egyptians. As a result, in Greek mathematics there was no significant distinction between shapes and numbers. To the Greek philosopher-mathematicians they were pretty much the same thing. (Even today, we have square numbers and triangular numbers thanks to their influence [Figure 5].) In those days, proving a mathematical theorem was often as simple as drawing an elegant picture; the tools of ancient Greek mathematics weren’t pencil and paper—they were a straightedge and compasses. And to Pythagoras the connection between shapes and numbers was deep and mystical. Every number-shape had a hidden meaning, and the most beautiful number-shapes were sacred.