The Father of Algebra: Al-Khwarizmi or Diophantus?

The cover of Diophantus' book Arithmetica. Image: Public domain, via Wikimedia Commons.

The cover of Diophantus’ book Arithmetica. Image: Public domain, via Wikimedia Commons.

The history of algebra is very intriguing because of the many cultures that contributed to its origins. Although there were many ancient civilizations that studied algebra, there are two men that are best know for bringing algebra to our modern day: Al-Khwarizmi and Diophantus. The debate as to who is the “father” of our modern day algebra is still a subject of interest to which I hope to bring some light. I would like to share with you the lives of both of these mathematicians, their works and their legacy.


Much of the life of the Greek mathematician Diophantus is unknown, but we do know that he lived in Egypt sometime after 150 BCE and before 350 CE. From what we have found it seems most likely that he lived during the 3rd century CE. We also have knowledge of his works that were popularized in the 17th and 18th centuries. Arithmetica, one of his greatest works, consists of 13 books of 130 algebraic problems. Out of these books, six were thought to be the only ones to have survived. However, in what’s known as Astan-i Quds, an Arabic manuscript, are thought to be the remaining books of Arithmetica. Even though these manuscripts have been found, some are not convinced of its veracity. The problems found in Arithemtica are known as Diophantine equations. These equations included polynomial equations, linear Diophantine equations, and Diophantine approximations among other Diophantine problems. Other works include Porisms, a collection of lemmas, and many works on polygonal and geometric, all of which helped expand mathematics.

Diophantine polynomial equations are polynomials with a number of unknowns for which only a rational solution is found. These equations usually had many solutions because of their many unknowns. Diophantus generally would only solve for one solution, instead of solving for all or them. A linear Diophantine equation is two sums of monomials of degree zero or more. To solve these equations one would have to use what is called Diophantine analysis. A Diophantine analysis would ask a series of questions, which would help find the solution.

Now the question is, what mark did this make on history? Although it is hard to know exactly who was influenced by Diophantus, we do have knowledge of many mathematicians who were influenced by his work. I would say the most famous work to have come from studying Diophantine equations was from Pierre de Fermat. De Fermat was studying Arithmetica when he scribbled “x^n+y^n=x^n where x, y, z, and n are non-zero integers, has no solution with n greater than 2.” This scribble is better known as Fermat’s Last Theorem, which later inspired algebraic number theorem. Other mathematicians that were inspired by his work are Andrew Wiles, who found proof of Fermats’ theorem, John Chortasmenos, a monk and mathematician, and Wilbur Knorr, a math historian. Above all else he was one of the first people to use symbols in mathematics. This is something we are all used to today in our mathematics from a young age.


A statue of al-Khwarizmi in Uzbekistan. Image: Jori Avlis, via Flickr.

A statue of al-Khwarizmi in Uzbekistan. Image: Jori Avlis, via Flickr.

Abu Abdallah Muhammad ibn Musa al-Khwarizmi, better known as al-Khwarizmi, was a Persian mathematician born in the latter part of the 8th century CE. One of his greatest works was Compendious Book on Calculation by Completion and Balancing. He also had books on arithmetic, astronomy, trigonometry, and geography to name a few. He also helped make the Indian numeric system part of western culture.

His most famous book, as mentioned earlier, is where we get the name algebra. The Arabic name of his book, Hisab al-jabr w’al-muqabala, Kitab al-Jabr wa-l-Muqabala, has the word Al-jabr, which means restoration. Al-jabr was the beginnings of the word algebra. When this book was translated into Latin it was called Liber Algebrae et Almucabola, which indicates clearly the source of algebra. This book expounded on the knowledge of quadratic functions among others. The book has hints of influence from past mathematicians, but the ties with Indian mathematics is most evident. One loss from the Indian mathematics was that of negative numbers. Because negative numbers were not used, equations with negative solutions were not studied. His book used squares, roots, and numbers to describe the equations. It also introduced the forcing of one side to be equal the other, which is what we would use today. This was the completing part. Balancing was done by subtracting the same amount from both sides of the equation. He also dealt with measuring areas and volumes. His work also included the concept of Algorithm, which is used in our everyday lives.

We now know what he taught, but, again, who or what was influenced by his works? In the 12th century, when his book was translated into Latin, Europe began to become familiar with his work. After a few centuries his work helped get Europe out of the dark ages.

Although I believe that both Diophantus and al-Khwarizmi contributed greatly to the math world, I think that al-Khwarizmi should be considered the father of algebra. This is due to the fact that his work is much closer to the algebra that is used today. His work was used for so long and was never lost. His worked helped Europe out of the dark ages, Diophantus did great work but al-Khwarizmi pushed the mathematical world in a great direction.



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