Zero - Charles Seife [13]
Nowadays the great unsolved problems in mathematics are stated in terms of conjectures that mathematicians are unable to prove. In ancient Greece, however, number-shapes inspired a different way of thinking. The famous unsolved problems of the day were geometric: With only a straightedge and compasses, could you make a square equal in area to a given circle? Could you use those tools to trisect an angle?* Geometric constructions and shapes were the same thing. Zero was a number that didn’t seem to make any geometric sense, so to include it, the Greeks would have had to revamp their entire way of doing mathematics. They chose not to.
Even if zero were a number in the Greek sense, the act of taking a ratio with zero in it would seem to defy nature. No longer would a proportion be a relationship between two objects. The ratio of zero to anything—zero divided by a number—is always zero; the other number is completely consumed by the zero. And the ratio of anything to zero—a number divided by zero—can destroy logic. Zero would punch a hole in the neat Pythagorean order of the universe, and for that reason it could not be tolerated.
Indeed, the Pythagoreans had tried to squelch another troublesome mathematical concept—the irrational. This concept was the first challenge to the Pythagorean point of view, and the brotherhood tried to keep it secret. When the secret leaked out, the cult turned to violence.
The concept of the irrational was hidden like a time bomb inside Greek mathematics. Thanks to the number-shape duality, to the Greeks counting was tantamount to measuring a line. Thus, a ratio of two numbers was nothing more than the comparison of two lines of different lengths. However, to make any sort of measurement, you need a standard, a common yardstick, to compare to the size of the lines. As an example, imagine a line exactly a foot long. Make a mark, say, five and a half inches from one end, dividing the foot into two unequal parts. The Greeks would figure out the ratio by dividing the line into tiny pieces, using, for example, a standard or yardstick half an inch long. One line segment contains 11 of those pieces; the other contains 13. The ratio of the two segments, then, is 11 to 13.
For everything in the universe to be governed by ratios, as the Pythagoreans hoped, everything that made sense in the universe had to be related to a nice, neat proportion. It literally had to be rational. More precisely, these ratios had to be written in the form a/b, where a and b were nice, neat counting numbers like 1, 2, or 47. (Mathematicians are careful to note that b is not allowed to be zero, for that would be tantamount to division by zero, which we know to be disastrous.) Needless to say, the universe is not really that orderly. Some numbers cannot be expressed as a simple ratio of a/b. These irrational numbers were an unavoidable consequence of Greek mathematics.
The square is one of the simplest figures of geometry, and it was duly revered by the Pythagoreans. (It had four sides, corresponding to the four elements; it symbolized the perfection of numbers.) But the irrational is nestled within the simplicity of the square. If you draw the diagonal—a line from one corner to the opposite corner—the irrational appears. As a concrete example, imagine a square whose sides are one foot long. Draw the diagonal. Ratio-obsessed people like the Greeks naturally looked at the side of the square and the diagonal and asked themselves: what is the ratio of the two lines?
The first step, again, is to create a common yardstick, perhaps a tiny ruler half an inch long. The next step is to use that yardstick to divide each of the two lines into equal segments. With a half-inch yardstick we can divide the foot-long side of the square into 24 segments, each half an inch long. What happens when we measure the diagonal? Using the same yardstick, we see that the diagonal gets…well, almost 34 segments, but it doesn’t come out quite evenly. The 34th segment is a wee bit too short; the half-inch