The History of Algebra (It’s Complicated)
A History of Mathematics
The most common question I receive when teaching struggling math students is, “who invented algebra and why did they even bother?” The word “algebra” comes from the Arabic word “al-jabr,” which means “the reunion of broken parts”. This term can be traced back to a text produced by the renowned Arabic mathematician and scientist al-Khwarizmi.
The history of algebra is far more complex, with contributions from practically all major civilisations throughout history. We’re going to travel through time identifying the significant developments contributed by these civilisations.
Ancient Mesopotamian and Egyptians
The ancient Mesopotamians and Egyptians were some of the first to use algebraic concepts, such as using letters to represent unknown quantities.
The Mesopotamians used tables on clay tablets to aid calculations. For example, two tablets found at Senkerah dated to around 2000BC give squares of numbers up to 59. They used the tables in conjunction with formulae such as ab = [(a + b)² - (a - b)²]/4 or [(a + b)² - a² - b²]/2 to simplify multiplication.
The Moscow and Rhind papyri contain examples of how the ancient Egyptians employed algebra to solve problems. These are called aha problems, which involve finding unknown quantities (the aha or stack). In addition to the linear equations seen in the Moscow and Rhind papyrus, we also see more complex quadratic equations in the Berlin papyrus.
Ancient Greece
The Greeks, who followed the Babylonians and Egyptians, also made significant contributions to the development of algebra. The most notable of these contributions come from two mathematicians, Euclid and Diophantus, and their instrumental texts, The Elements and Arithmetica.
Euclid employed the use of what we now call geometric algebra. This involves using geometry to demonstrate mathematical concepts, such as the distributive property, which can be shown by splitting the rectangle.
The latter, Diophantus, wrote a series of books called Arithmetica which dealt with solving equations using algebraic quantities, unlike typical Greek mathematics which used geometric methods. Diophantus seeks to find exact solutions instead of simple approximations and presents what is the first known use of algebraic notation and symbolism, in what we refer to as syncopated algebra.
Ancient India
Algebra was significantly used throughout ancient Indian civilizations, such as in the Bakhshali manuscript, which contained worked examples for calculating square roots or solving linear equations.
Focusing on two ancient Indian mathematicians, Brahmagupta and Bhaskara II. Brahmagupta is an immensely influential mathematician, such as developing the rules for computing with zero and making algebraic contributions. In the Brahma-sphuta-siddhanta, he presented the general solution of the linear equation bx + c = dx + e and also gave a general solution for quadratic equations, which we now recognize as the quadratic formula.
Bhaskara II produced a work on algebra in twelve chapters, called The Bijaganita or the Algebra of the Hindus. It contains many methods for solving and processing algebraic problems and presents the Chakravala method in an 8-line Sanskrit poem, which is a cyclic algorithm for solving indeterminate quadratic equations.
Ancient China
In ancient Chinese civilizations, we have seen many significant developments, which are typically presented as an anonymous collection of results. The most influential book, “The Nine Chapters on the Mathematical Art,” compiled by Lui Hui, presented advanced concepts such as Gaussian elimination over 1500 years before Gauss, as well as other techniques such as Horner’s method.
Arabian Algebra and Al-Khwarizmi
“Finally, we arrive at the Arabic civilization where the book “Al-jabr,” the namesake for algebra, was produced. This was written by the Persian mathematician Muḥammad ibn Mūsā al-Khwārizmī around 820 AD, and he is often cited as the father of algebra.It was the first text to teach algebra in an elementary form for its own sake, rather than just as a tool for solving equations in geometry.
The book gives a systematic approach to solving linear and quadratic equations by reduction and completion, or balancing, methods.The first step is expressing equations in one of six forms, which is the reduction step.nNext, there is a completion or balancing step to remove extraneous terms. Once an equation is in one of the six forms, you can follow the methods in the book to solve for the value of the unknown (x).
This book was so influential because it was translated and spread widely, being used as the primary text in European universities for centuries.
Algebra is often solely attributed to its namesake al-jabr, but there is more to its history. We can see that these concepts were similar to those shared by other mathematicians such as Diophantus or Brahmagupta. This just emphasizes the fact that algebra is not a field that was the result of the discovery of one person throughout history. It was the culmination of contributions made over time by many mathematicians across various civilizations.
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