Much ado over nothing

14/04/2000

Much ado over nothing DOES a new millenium have something to do with the attention that zero gets? Suddenly, we have two good books on zero. First, there is Charles Seife's "Zero, the Biography of a Dangerous Idea', brought out by Viking. Second, we have Robert Kaplan's "The Nothing that is, a Natural History of Zero', with illustrations by Robert Kaplan's wife, Ellen Kaplan. This is not a comparative review, it is a review of Kaplan's book.

"If you look at zero you see nothing; but look through it and you will see the world. For zero brings into focus the great, organic sprawl of mathematics, and mathematics in turn the complex nature of things. From counting to calculating, from estimating the odds to knowing exactly when the tides in our affairs will crest, the shining tools of mathematics let us follow the tacking course everything takes through everything else and all of their parts swing on the smallest of pivots, zero.'

Robert Kaplan's book is like a work of historical detection, tracing the origins of zero. It is not only about zero, it is also about notions of "nothing'. A note to the reader states, "If you have had high-school algebra and geometry, nothing in what lies ahead should trouble you, even if it looks a bit unfamiliar at first.' This is not quite true. Even if the essence can be grasped, an understanding of everything in the book requires some familiarity with number theory. The note to the reader also says, "You will find the bibliography and notes to the text on the web' and the web-address is given. I am not sure that this is a good idea. In addition, the notes in the web-site are not particularly illuminating.

But let's stick to the book. What is zero? First, it is a number that signifies nothing. We begin with the Sumerians, 5000 years ago. The Sumerians didn't have a word for nothing and neither did the Greeks. Nor does zero figure in Alexandria, in Sicily or in the work of Archimedes. Several Indians believe that the notion of zero originated in India. This is what the author has to say on this claim. "It does strike me, however, that burdening actual Indian achievements with others' goods ends up diminishing them, and that it is a loss to replace a story rich in the accidents and ambiguities of time with an uplifting tale.' That is, the symbol for zero is squarely ascribed to Greek origins.

However, inventing a symbol for zero (or nothing) doesn't grant zero the status of a number. For that to happen, operations of addition, subtraction, multiplication and division between zero and other numbers have to be defined. Robert Kaplan ascribes this credit to Indian mathematicians Bhaskara, Brahmagupta and Mahavira. Collectively, they established rules for adding, subtracting and multiplying zero and thus, elevated a mere symbol into a number like any other. What about division by zero? The Indian mathematicians got 0/0 and a/0 wrong, where "a' is a positive or negative number, other than zero. When did mathematicians get these right?

Robert Kaplan's book is anecdotal and full of digressions. Although the book's language is easy enough and it is difficult to see how a mathematician could have done a better job, these sudden jumps and shifts sometimes make it difficult to retain the link. Without answering the question about 0/0 and a/0, we therefore jump to the Mayas and their calendar system. From there, we jump to various names for zero. Fibonacci and the invention of double entry book-keeping in Italy are thrown in. All very interesting and the link is clear to the author. "This opened the door to granting equal status to the signs for quantities and for the operations on them, all subject now to yet more abstract operations in turn, and those to others, endlessly, each bearing the peculiar, defining mark of this language: that no matter where in its hierarchy an operation or relation stood, it was expressed by a sign of like standing with the rest, in the matrix of their common grammar.' Clear?

We still have to dispose of 0/0 and a/0 and it gets worse. There is now a digression on exponents and Fermat's little theorem. This is not Fermat's last theorem, which has recently been proved. Fermat's little theorem states the following. If p is a prime and "a' is a number less than p, ap-1 1 is exactly divisible by

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