Algebra and Restoration

In the final chapter of The Crest of the Peacock, Joseph explores the mathematical contributions of the Islamic world, covering the eighth through the thirteenth centuries. One of the central players in this history was Muhammad ibn Musa al-Khwarizmi, who was born in present day Iraq around 780.

al-Khwarizmi and al-jabr

Beyond al-Khwarizmi’s mathematical contributions, he gave us two terms: algebra to mean the manipulations necessary to solve an equation, and algorithm, which is derived from the name al-Khwarizmi itself.

Algebra comes from the Arabic word al-jabr meaning “the restoration of broken parts.” The connection to mathematics was that of adding a term to both sides of an equation to remove negative values. For example, if x - 3 = 4, then adding three to both sides would show x = 7.

The corollary to restoration is “reduction,” which translates to al‐muqabala. Reduction would be the subtraction of a term from both sides of an equation to remove positive values. For example, if x + 4 = 9, then subtracting four from both sides would show x = 5.

Restoring a history (and future) of math

Joseph finishes his book with a short summary of 511 pages that precede it. He writes: “No society, however small or remote, has ever lacked the basic curiosity and ‘number sense’ that is part of the global mathematical experience.”

I love this. In a previous post I considered Joseph’s thoughts on the transmission of mathematical ideas. One thing I didn’t mention, however, is my conviction that there are a host of customs and qualities that are shared by humans the world over. This isn’t just my idea, of course, or pulled out of thin air. I first came across it when I was earning a degree in anthropology. Unfortunately, those classes never touched on mathematics as a cultural universal, and I’m embarrassed to admit that I never thought about it much either. So kudos to Joseph for making the case in such a convincing fashion.

As for the future, Joseph writes: “And yet if there is a single universal object, one that transcends linguistic, national, and cultural barriers and is acceptable to all and denied by none, it is our present set of numerals.”

This sentence speaks for itself, though it reminds me of a story. When I traveled to Lebanon a few years ago I was surprised to find license plates that looked like this:

lebanon license plate License plate in Lebanon. Image via Wikipedia.

What were those symbols on the bottom? My guide told me that they were Arabic numerals. I thought ours were Arabic numerals. Later, I learned that there are actually two kinds of Arabic numerals: Hindu-Arabic numerals (0,1,2,3,4,5,6,7,8,9) and Eastern Arabic numerals (٠,١,٢,٣,٤,٥,٦,٧,٨,٩).

The cool thing about the license plates in Lebanon was that they showed the same numbers in both systems. So within a couple days of looking at plates, I had them memorized. I haven’t used that knowledge since, but it was an early education in the fact that while numbers may be universal, their appearance certainly isn’t.