Entire books have been written about the concept of the number zero. This number was a latecomer to arithmetic, perhaps because it is difficult to visualize zero cubits or zero sheep. Even today, if you pick up a children’s counting book, you will probably not find a page devoted to zero.
The number zero has two different interpretations, one of them a good deal more sophisticated than the other. First, in numbers like 2009 or 90,210, zero is used as a symbol to denote an empty place. That is the function of the zeros. Without the numeral zero, we would not be able to tell those numbers apart from 29 or 921. In a place-value number system, the meaning of “2” depends on where it is; in the number 29 it denotes two tens, but in the number 2009 it denotes two thousands.
Of course cultures that did not use a place-value system, such as ancient Egypt or Rome, did not have this problem and did not need a symbol for an empty place. The Roman numeral MMIX (2009) is easy to distinguish from XXIX (29). Thus it is not surprising that the notion of zero did not arise in those societies. The Babylonians, however, did use a place-value number system, and yet for many centuries it did not occur to them to employ a mark to denote an empty place. Apparently the ambiguity between 2009 and 29 did not trouble them—perhaps because it is usually apparent from context which number is intended. Even today the same thing is true. If someone is telling you what year it is, you expect a number like 2009; if they are telling you how old they are, 29 is more reasonable (the Babylonians actually used a number system based on powers of 60, rather than powers of 10. This does not alter the basic problem of ambiguity. For example, a Babylonian scribe would not be able to distinguish 1,501 (i.e., 25 × 60 + 1) from 90,001 (i.e., 25 × 60² + 1). Both numbers would be written as 25, 1).
Only around 400 BC, near the end of Babylonia’s independent existence and some 1,500 years after the cuneiform number system had first come into use, did scribes start to use two vertical wedges (∧∧) to denote an empty place. This was the first appearance in history of a symbol that meant zero, but it is clear that the Babylonians thought of it only as a placeholder and not as a number itself.
The second, more subtle, concept of zero as an actual entity (as implied by the equation 1 – 1 = 0) arose in India. It appears for the first time in 628 AD, in a book called Corrected Treatise of Brahma, by Brahmagupta.
As is the case for many ancient mathematicians, little information is available about Brahmagupta’s life. He was born in 598 in north central India, and was a member of a mathematical school (in the sense of a loosely knit community of scholars) in Ujjain. He lived not long after the end of the Gupta dynasty (ca. 320–550), a period of prosperity that is often considered a golden age of Indian culture, when much classical Sanskrit literature was written and when astronomers developed very accurate predictions of eclipses and planetary motions.
One thing that stands out clearly in Brahmagupta’s work is his derisory attitude toward his rivals. The very title, Corrected Treatise of Brahma, is an implicit criticism of an earlier astronomical work. Brahmagupta makes comments such as this about his predecessors: “One is not a master through the treatises of Aryabhata, Visnucandra, etc., even when [they are] known [by heart]. But one who knows the calculations of Brahma [attains] mastery.” (Brackets inserted by mathematical historian Kim Plofker, who translated the work.)
Arrogant though he may have been, Brahmagupta clearly understood the nature of zero. He wrote, “[The sum] of two positives is positive, of two negatives negative; of a positive and a negative [the sum] is their difference; if they are equal it is zero.” Thus, zero is obtained by adding a positive number to a negative of equal magnitude; for example, 1 + (–1). This is what is meant by the modern notation 1 – 1. Further, Brahmagupta wrote that adding zero does not change the sign of a number, that 0 + 0 = 0, and that any number times zero gives zero. However, he is not sure about division by zero. Rather tautologically, he writes, “A negative or a positive divided by zero has that as its divisor,” and he states incorrectly that “a zero divided by a zero is zero.” Modern mathematicians would say that any division by zero is undefined.
It is noteworthy that zero goes hand in hand in Brahmagupta’s work with negative numbers. Indeed, the resistance to zero may be explained by the even greater difficulty of visualizing negative cubits or negative sheep. For centuries after Brahmagupta, mathematicians continued to avoid negative numbers in their formulas. For example, the solution of quadratic equations and cubic equations was made unduly complicated by mathematicians’ avoidance of negatives. They perceived the need for several different methods of solution, which we now condense into one formula.
For modern mathematics it is difficult to overstate the importance of zero. It is what mathematicians call an identity element, because when added to any number it does not change that number. Identity elements are as important to mathematics as synonyms are to literature. No one would question why we need both of the words “happy” and “delighted.” They allow us to say essentially the same thing in different ways, possibly revealing slightly different nuances. The availability of zero gives mathematicians the same flexibility. An expression x can be written as x + 0, and from there it can be rewritten as x + 1 – 1 or in many other ways, depending on the requirements of the problem.
In the nineteenth and twentieth centuries, mathematicians discovered many useful algebraic structures besides integers and real numbers, and useful operations besides ordinary addition and multiplication. For example, computers use modular arithmetic; cryptologists use multiplication on elliptic curves; and quantum physicists add and multiply vectors in Hilbert space. All of these operations are variations on the fundamental notions of “plus” and “times,” but they are sometimes very far removed from the addition and multiplication we learn in school. The one thing that they all have in common is an identity element. Thus, Brahmagupta’s contribution to mathematics—the idea of the number zero—is alive and well, even though he might have some trouble recognizing it.
This is an extract from The Story of Mathematics in 24 Equations by Dana Mackenzie (£16.99 at Amazon UK, Modern Books)