Archive for December 13th, 2011

When did algebra begin?

The second in a series. See the November 20 entry, “What is algebra?” for the first.

Two key features of algebra as we understand the word today are:

1. Reasoning about numbers by recognizing patterns across numbers;

2. Solving a problem by introducing a term for an unknown and then, starting with what is known, reasoning to determine its value.

We first see the emergence of both features of algebra in the mathematics of ancient Babylonia, around 2,000 BCE.

Several hundred of the many thousands of Babylonian’s cuneiform-inscribed clay tablets that have been found are devoted to mathematics. They show that those ancient mathematicians had systematic procedures for solving geometric problems involving the determination of lengths and areas of figures. Today, we would solve those kinds of problems using linear and quadratic equations and indeterminate systems of linear equations. Their methods amounted to a form of geometric algebra that could be applied to solve problems beyond overtly geometric examples such as calculating the perimeters or areas of various plane figures or the volumes of solid objects: arithmetic problems arising in trade and commerce, for example, and other financial transactions such as inheritance. In addition, the Babylonians considered problems that seemed to have had no practical application, pursuing them purely for recreation. Although they described their procedures in terms of specific lengths and areas, they did so in a way that made it clear they applied in general, and in that sense they were starting to think algebraically, by recognizing patterns across quantities.

Moreover, some of their writings show the second characteristic feature of algebra, namely introducing an unknown and then reasoning to find its value. In their case, however, the unknown was not numeric but geometric – an unknown line on which they performed geometrical operations to get the answer.

In reasoning with unknown quantities, the Babylonians went further than other early civilizations with a mathematical tradition, such as the Egyptians, the Chinese, and the early Greeks, all of the first millennium BCE. Our knowledge of the mathematics of those peoples comes from works such as the Rhind papyrus, The Nine Chapters of the Mathematical Art, and Euclid’s Elements, respectively. The approach described in those documents was, like that of the Babylonians, fundamentally geometric and exhibited reasoning about patterns of quantities, but we do not find the introduction of an unknown followed by an argument to determine its value.

It is with the work of the Greek mathematician Diophantus  (ca. 210–290 CE) that we first find clearly recognizable algebra, where the unknowns represent numbers whose values are to be determined. Around 250 CE, Diophantus, who lived in Alexandria in Egypt, wrote a multi-volume work, Arithmetica, which its title notwithstanding was an algebra book. Its author used letters (literals) to denote the unknowns and to express equations, but that is a purely notational distinction. He also was one of the first mathematicians to use negative numbers in calculations. He showed how to solve equations by using two techniques called restoration and confrontation. In modern terms, these correspond more or less (but not precisely) to (1) adding a quantity to both sides of an equation to eliminate a negative term on one side, and (2) eliminating like terms from both sides. He used these techniques to solve polynomial equations involving powers up to 6.

Almost four hundred years later, the Indian mathematician Brahmagupta (598–668 CE) likewise displayed recognizable algebra, in his book Brahmasphutasiddhanta, where he described the first complete arithmetic solution (including zero and negative solutions) to quadratic equations.

Following Diophantus and Brahmagupta, the next major step in the development of algebra – and it was huge – took place in the period generally referred to as “Arabic mathematics” or “Muslim mathematics”, a significant outpouring of mathematical activity stretching from the 8th century to the end of the 16th. Indeed, the word algebra itself comes from the Arabic word al-jabr, which occurs in the title of a highly influential book by the Persian mathematician al-Khwārizmī, completed around 830: al-Kitab al-mukhtasar fi hisab al-jabr wa’l-muqābala. The phrase al-jabr wa’l-muqābalah translates literally as “restoration and confrontation,” but more loosely means “solving an equation.”

That period will be the focus of my next article on algebra. I’m Dr. Keith Devlin, a mathematician at Stanford University, an author, the Math Guy on NPR’s Weekend Edition, and an avid cyclist. (The header photo is me halfway up Mt. Baldy in Southern California.)

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