Zero - Charles Seife [9]
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 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