The Father of Algebra


Diophantus’ greatest accomplishment was Arithmetica. Image: public domain, via Wikimedia Commons.

Diophantus of Alexandria is thought of as “the father of algebra,” a subject that many students in high school have to endure. Though he had an impact and is part of mathematical history, very little is known about Diophantus. “The father of algebra” lived in Alexandria, Egypt, possibly around 200 A.D. to around 290 A.D. One of Diophantus’ contributions in mathematics was his work Arithmetica, which included 13 books but many have been lost and only 6 out of the 13 survived.  He also worked on The Porisms. Like much of his other work, this book was also lost. Much of his work impacted mathematics. Specifically his work on Arithmetica led to “employ algebra as a modern style.”  He was the first to use symbols for unknowns in arithmetic operations.

Diophantus’ Arithmetica helped many mathematicians to construct their own ideas. The 6 books from Arithmetica that have been found each contain problems with their solutions. Most of the problems in Arithmetica are algebraic but Diophantus was not the first one to work with algebraic problems. What is interesting about the problems and solutions in Arithmetica is that Diophantus never used one general idea for solving his problems. He had a different way of solving each problem. The book also contains problems that have determinate (a unique solution) and indeterminate solutions (more than one solution). When looking through Arithmetica many of the solutions are fairly complicated and the steps in the solutions are puzzling. It can be seen that in Diophantus was able to improve some notation in algebra, but how the algebra was solved still needed some improvement.

Diophantus included in his book about 130 algebraic problems. More specifically, any solutions that contained negative or irrational square roots were considered useless to Diophantus because he did not believe in negative numbers. An example of this (that we would see in any high school text book) could be shown with the equation, 4 = 6x + 40. This was considered as an absurd equation because it would lead to a meaningless answer. The solution to this would be -6 and Diophantus did not consider negative numbers as things that were real. When we learn and study about quadratic equations, we know that the result could possibly have two solutions. “There is no evidence to suggest that Diophantus realized that a quadratic equation could have two solutions. However, the fact that he was always satisfied with a rational solution and did not require a whole number is more sophisticated than we might realize today” write J J O’Connor and E F Robertson in Diophantus in Alexandria.

Problems that Diophantus worked with included Pythagorean triples. Image: Gustavb, via Wikimedia Commons.

Diophantus worked on equations including Pythagorean triples. These are triples of numbers, like (3,4,5), for which x2+y2=z2. Image: Gustavb, via Wikimedia Commons.Another type of algebraic problem he would have worked with would have involved two integers such that the sum of their squares is a square. This is what we know today as a Pythagorean triple. An example of this would be x2 + y2 = z2, where x = 3 and y = 4 giving z = 5. Another example would be x = 5 and y =12 giving z = 13. He also worked with two integers such that the sum of their cubes is a square. This would look like x3 + y3 = z2. An example of this would be x = 1 and y = 2, giving z = 3. Diophantus tried to determine if such problems, like these or similar to these, had any solutions or none at all.

Arithmetica contributed to the development of some math ideas, for example, Fermat’s Last theorem. Pierre De Fermat claimed that a generalization of the equation in Arithmetica had no solutions. In the margin of the book he wrote that he “had found a marvelous proof to a proposition, which however the margin is not large enough to contain.” Of course there was no discovery of any proof from Fermat. Though Arithmetica was written in the 3rd century A.D., it impacted the mathematics world most notably in the 90’s when Andrew Wiles proved Fermat’s Last theorem. What can be seen from Diophantus’ work is that math can unfold over time and continue with development.

Though many of Diophantus’ ideas contributed to further ideas in math, some might say that he is not the father of algebra. Many of the methods he used for solving linear and quadratic equations go back to Babylonian and Egyptian mathematics. For this reason, mathematical historian Kurt Vogel writes: “Diophantus was not, as he has often been called, the Father of Algebra. Nevertheless, his remarkable, if unsystematic, collection of indeterminate problems is a singular achievement that was not fully appreciated and further developed until much later.” Either way, Diophantus impacted the math world with his book Arithmetica.