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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.
    â€”L UCRETIUS , D E R ERUM N ATURA
    Z ero 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. *
    â€”J OHN 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