Diophantus’ Contribution in Mathematics

diophantusDiophantus, also known as the ‘father of Algebra,’ was a Greek mathematician, who was born in 200 AD. He is best known for his series of books ‘Arithmetica.’ He was the first to recognize fractions as numbers. Although there are very few facts known about his life, it is believed that he lived for 84 years and died in 284 AD. Some information about his life was gathered by the Greek anthology of numbers and games created by Metrodorus. One of the puzzles states that

‘Here lies Diophantus,’ the wonder behold.

Through art algebraic, the stone tells how old;

‘God gave him his boyhood one-sixth of his life,

One twelfth more as a youth while whiskers grew rife;

And then yet one-seventh ere marriage began;

In five years there came a bouncing new son.

Alas, the dear child of master and sage

After attaining half the measure of his father’s life chill fate took him. After consoling his fate by the science of numbers for four years, he ended his life.”

This puzzle expresses Diophantus’ age in equation form as,

x=x/6 + x/12 + x/7 + 5 + x/2 + 4, where x comes out to be 84. This puzzle also reveals that Diophantus’ son died 4 years before he died. It is believed that Diophantus never used general solutions for his problems. A prominent German mathematician, Hermann Hankel, while sharing his views about Diophantus, said

Our author (Diophantus) not the slightest trace of a general, comprehensive method is discernible; each problem calls for some special method which refuses to work even for the most closely related problems. For this reason it is difficult for the modern scholar to solve the 101st problem even after having studied 100 of Diophantus’s solutions.”

1. Diophantus’ Arithmetica

Arithmetica is a series of thirteen books out of which only 6 survived to date. It is a collection of problems having numerical solutions of both determinate and indeterminate equations, though some of his equations from Arithmetica were found later in Arabic sources. The method for solving these equations is known as Diophantine analysis. His equations are usually algebraic equations having integer coefficients for which there exist integer solutions. Arithmetica has very few things in common with Greek mathematics, since Diophantus believed more in exact solutions rather than approximations, and also, there was no inclusion of geometric methods in this book. Arithmetica contains 150 problems with solutions and also have evidence of the use of identities. Despite having limited algebraic tools, he managed to solve a variety of problems that inspired many mathematicians. The extension given by Pierre de Fermat to the work of Diophantus is the most famous one. Pierre de Fermat is also known as the founder of number theory. Fermat wrote various remarks in the margins of his copy of Arithmetica, providing new solutions and generalization of Diophantus’ method. Fermat theorem is one of his works provided in the margins of the book. The first Latin translation of Arithmetica was given by Bombelli in 1570, but it was never published. In 1621, Bachet gave the most famous Latin translation of Arithmetica, which was the first translation available to the public.


2. Diophantus’ mathematical notation

He was the first one who used mathematical notations, abbreviations for the power of numbers, and relationships and operations that is now known as Syncopated algebra. Before this, everyone used complete equations, which was very lengthy and time-consuming. So, his introduction of symbolism in the field of algebra turned out to be very beneficial. The difference between modern algebra and Diophantus algebra is, how we write coefficients in the equations. An equation in modern notation will be written as  3{x}^{2}+2x+1=0, which, in Diophantus algebra, would be written as {x}^{2}3+{x}^{1}2+{x}^{0}1=0. His concept of symbolism, however, lacked the use of notation of the general number ‘n’, which is used in general expressions. Also, he had a symbol to express only one unknown, and in case of more than one unknowns, he used to express it as ‘first unknown’ and ‘second unknown.’ This points out that he was more focused on particular results rather than the general ones.

3. Diophantine Equation

Diophantine equations are the equations having two or more unknowns, that yields only positive integer solution. He considered two types of problems, linear and exponential equations. Diophantine linear equations are those in which a constant is equated to the sum of two or more monomials. Diophantine exponential equations are those in which unknowns can be expressed in exponents. The simplest example of a linear equation is ax + by=c, where a, b, and c are integers. An example of Diophantine exponential equation is the Fermat equation, {a}^{n}+{b}^{m}={c}^{k}. It was observed that most problems in Arithmetica constitute quadratic equations. He considered three types of quadratic equation, a{x}^{2}+bx=c, a{x}^{2}+c=bx, a{x}^{2}=bx+c By his consideration, it was evident that he did not have any notion for zero, and he rejected negative coefficients by taking a, b, and c, all to be positive integers. He had provided only positive rational solutions to these equations and claimed that those equations are useless that give irrational, square root, and negative solutions. Also, there is no proof in his book that shows that he realized the fact that quadratic equations have two solutions.


4. On Polygonal Numbers

Polygonal numbers represent dots, arranged in a geometrical figure. With the increase in the size of the figure, the number of dots used to construct it also increases in a common pattern. The progression of the polygons is elaborated by the initial dot and successive polygons developing outwards. This concept was introduced by Greek mathematician Hypsicles. Diophantus credits Hypsicles for being the author of Polygonal numbers and discovered that {n}^{th}a-gon is calculated by the formula [n×{2+(n-1)(a-2)}]/2. This formula was used by him to calculate the number of elements in the {n}^{th} term of the polygon having ‘a’ sides. It is, however, believed that before Diophantus, polygonal numbers were used by Pythagoras in his Pythagorean triplet.

polygonal numbers

5. Porisms

Porisms is one out of the lost works of Diophantus. Porisms is believed to be a collection of lemmas. Many scholars claimed that Porisms is a section in Arithmetica or maybe the rest of Arithmetica. Three lemmas of this book are known because Diophantus had mentioned them in Arithmetica. One such lemma is that the difference of the cubes of two rational numbers is equal to the sum of the cubes of two other rational numbers, i.e., given any numbers a, b with a>b, then there exist numbers c and d, all positive and rational such that


6. An Insight to Arithmetica

Diophantus had solved a variety of problems in Arithmetica, like in Book III, he solved problems to find the value that makes two linear expressions simultaneously into squares. For example, to find to make (10x + 9) and (5x + 4) both squares (he calculated x = 28). In Book VI, he solved the problem of finding x, to make (4x+2) a cube and(2x+1) a square simultaneously (he calculated x=3/2). He also claimed, that number of the form (4n+3) or (4n-1) cannot be the sum of two squares and the number of the form (24n+7) cannot be the sum of three squares. However, it is not clear if he knows, that every number can be expressed as the sum of four squares. Mathematician Kurt Vogel writes about Diophantus

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.’

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