Intermediate Algebra (Arnold)

Zero

The use of zero as a placeholder and as a number has a rich and storied history. The ancient Babylonians recorded their work on clay tablets, pressing into the soft clay with a stylus. Consequently, tablets from as early as 1700 BC exist today in museums around the world. A photo of the famous Plimpton_322 is shown in Figure , where the markings are considered by some to be Pythagorean triples, or the measures of the sides of right triangles.

The people of this ancient culture had a sexagesimal (base 60) numbering system that survived without the use of zero as a placeholder for over 1000 years. In the early Babylonian system, the numbers 216 and 2106 had identical recordings on the clay tablets of the authors. One could only tell the difference between the two numbers based upon the context in which they were used. Somewhere around the year 400 BC, the Babylonians started using two wedge symbols to denote a zero as a placeholder (some tablets show a single or a double-hook for this placeholder).

The ancient Greeks were well aware of the Babylonian positional system, but most of the emphasis of Greek mathematics was geometrical, so the use of zero as a placeholder was not as important. However, there is some evidence that the Greeks used a symbol resembling a large omicron in some of their astronomical tables.

It was not until about 650 AD that the use of zero as a number began to creep into the mathematics of India. Brahmagupta (598- 670?), in his work Brahmasphutasiddhanta, was one of the first recorded mathematicians who attempted arithmetic operations with the number zero. Still, he didn’t quite know what to do with division by zero when he wrote

Positive or negative numbers when divided by zero is a fraction with zero as denominator.

Note that he states that the result of division by zero is a fraction with zero in the denominator. Not very informative. Nearly 200 years later, Mahavira (800-870) didn’t do much better when he wrote

A number remains unchanged when divided by zero.

It seems that the Indian mathematicians could not admit that division by zero was impossible.

The Mayan culture (250-900 AD) had a base 20 positional system and a symbol they used as a zero placeholder. The work of the Indian mathematicians spread into the Arabic and Islamic world and was improved upon. This work eventually made its way to the far east and also into Europe. Still, as late as the 1500s European mathematicians were still not using zero as a number on a regular basis. It was not until the 1600s that the use of zero as a number became widespread.

Of course, today we know that adding zero to a number leaves that number unchanged and that division by zero is meaningless,4 but as we struggle with these concepts, we should keep in mind how long it took humanity to come to grips with this powerful abstraction (zero as a number).