Zero Read Online Free Page B

there was always a nagging fear that at the end of time, disorder and void would reign once more. Zero represented that void.
    But the fear of zero went deeper than unease about the void. To the ancients, zero’s mathematical properties were inexplicable, as shrouded in mystery as the birth of the universe. This is because zero is different from the other numbers. Unlike the other digits in the Babylonian system, zero never was allowed to stand alone—for good reason. A lone zero always misbehaves. At the very least it does not behave the way other numbers do.
    Add a number to itself and it changes. One and one is not one—it’s two. Two and two is four. But zero and zero is zero. This violates a basic principle of numbers called the axiom of Archimedes, which says that if you add something to itself enough times, it will exceed any other number in magnitude. (The axiom of Archimedes was phrased in terms of areas; a number was viewed as the difference of two unequal areas.) Zero refuses to get bigger. It also refuses to make any other number bigger. Add two and zero and you get two; it is as if you never bothered to add the numbers in the first place. The same thing happens with subtraction. Take zero away from two and you get two. Zero has no substance. Yet this substanceless number threatens to undermine the simplest operations in mathematics, like multiplication and division.
    In the realm of numbers, multiplication is a stretch—literally. Imagine that the number line is a rubber band with tick marks on it (Figure 4). Multiplying by two can be thought of as stretching out the rubber band by a factor of two: the tick mark that was at one is now at two; the tick mark that was at three is now at six. Likewise, multiplying by one-half is like relaxing the rubber band a bit: the tick mark at two is now at one, and the tick mark at three winds up at one and a half. But what happens when you multiply by zero?

    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