Diophantus
Diophantus of
Alexandria was a Greek mathematician. He was the
first Greek mathematician who recognized fractions as numbers. He
allowed positive rational
numbers for the coefficients and solutions. He played a major role in the development of
algebra and was a considerable influence on later number theorists. Diophantine
analysis, which is closely related to algebraic geometry, has experienced a
resurgence of interest in the past half century is due to him.
Little is known
about Diophantus except the fact that he was born around 200 AD in Alexandria
and died at the age of 84 in 284 AD. The claims that Diophantus lived from
about 200 to 284 and spent time at Alexandria are based on detective work in
finding clues to the times he flourished in his and others’ writing. Theon of
Alexandra quoted Diophantus in 365 and his work was the subject of a commentary
written by Theon’s daughter Hypatia at the beginning of 5th century,
which unfortunately is lost. Michael Psellus, an 11th century
Byzantine scholar mentioned that Diophantus dealt with Egyptian arithmetic
which later led to other people working and developing the same branch of
mathematics. The most details we have of Diophantus’s life come from the Greek
Anthology which estimates his life span to be around 84 years. This anthology
says----
His boyhood lasted 1/6th of
his life; he married after 1/7th
more; his beard grew after 1/12th
more, and his son was born 5 years later; the son
lived to half his father's age, and the father died 4 years after the son.
Diophantus is best
known for his famous work Arithmetica, written on the solution
of algebraic equations and on the theory of numbers. The Arithmetica is a
collection of 130 problems giving numerical solutions of determinate equations
(those with a unique solution), and indeterminate equations. The
method of solving indeterminate equation is known as Diophantine analysis. An
equation is indeterminate if it has more than one variable. The Pythagorean
Triplets containing the integers x, y and z satisfying the equation x2
+ y2 = z2 is the example of Diophantine analysis.
Historically, the problem of solving Diophantine
equation has been to find expressions that show the relationship of the
integral values of x and y that satisfy an indeterminate equation ax + by = c, where the coefficients a, b
and c are integers. Such equations have infinite number of solutions.
The
Arithmetica had 13 volumes but only six of the original 13 books were thought to
have survived. However, an Arabic manuscript in the library Astan-i-Quds (The
Holy Shrine Library) in Meshed Iran has a title claiming it is a translation of
Arithmetica and was discovered in 1968. In Arithmetica he considered the
solutions of linear and quadratic equation. Diophantus looked at three types of
quadratic equations ax2 + bx = c, ax2 = bx + c and ax2 + c = bx. Since there was no
symbol of zero when the book was written so Diophantus took all coefficients to
be positive. Diophantus considered negative or
irrational square root solutions "useless", "meaningless",
and even "absurd". There
is no evidence that suggests Diophantus even realized that there could be two
solutions to a quadratic equation. Besides that, he also solved problem from
Simultaneous equations.
Looking at the work of Diophantus it was quite
clear that he was fully aware of the fact that every number can be written as
the sum of four squares which was later proved by Lagrange. Here are two
examples of the problems solved in Arithmetica.
1.
Find four numbers, the sum of every arrangement
three at a time being given: say 22, 24 , 27 and 20. (Answer:- 9, 7 , 4 and 11)
2.
Divide a number, such as 13 which is the sum of
two squares 4 and 9, into two other squares. (Answer:- 324/25 and 1/25)
In the Arithmetica, Diophantus stated
certain theorem without proofs. It is believed that his proofs were found in
his lost book The Porism, a
collection of lemmas. One lemma states
that the difference of the cubes of two rational numbers is equal to the sum of
the cubes of two other rational numbers. For any
given numbers a, b there
exist numbers c, d such that a3 - b3= c3 + d3.
Diophantus'
work has had a large influence in history. Editions of Arithmetica exerted a
profound influence on the development of algebra in Europe in the late
sixteenth and through the seventeenth and eighteenth centuries. Diophantus and
his works have also influenced Arab
mathematics and were of great fame among Arab mathematicians.
Diophantus' work created a foundation for work on algebra and in fact much of
advanced mathematics is based on algebra. European mathematicians did not learn
of the gems in Diophantus's Arithmetica until Regiomontanus wrote in 1463:-No one has yet
translated from the Greek into Latin the thirteen Books of Diophantus, in which
the very flower of the whole of arithmetic lies hid...
Bombelli translated much of the work in 1570
but it was never published. Bombelli did borrow many of Diophantus's
problems for his own Algebra. The most famous Latin translation of
the Diophantus's Arithmetica is
due to Bachet in 1621. It is that edition which Fermat studied. Certainly Fermat was inspired by this work which has
become famous in recent years due to its connection with Fermat's Last Theorem. Diophantus earliest
work was probably his short essay on polygonal numbers, containing ten
propositions in which he employed the classical method in which numbers are
represented by line segments. The topic
on polygonal numbers was of great interest to Pythagoreans.
Diophantus worked before the
introduction of modern algebraic notation, but he moved from rhetorical algebra
to syncopated algebra, where abbreviations are used. Diophantus made important advancement in mathematical notation.
He
was the first person to use algebraic notation and symbolism. Before him
everyone wrote out equations completely. Diophantus introduced an algebraic
symbolism that used an abridged notation for frequently occurring operations,
and an abbreviation for the unknown and for the powers of the unknown.
Mathematical historian Kurt Vogel states: “The
symbolism that Diophantus introduced for the first time, and undoubtedly
devised himself, provided a short and readily comprehensible means of
expressing an equation... Since an abbreviation is also employed for the word
‘equals’, Diophantus took a fundamental step from verbal algebra towards
symbolic algebra.”
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Rajesh Thakur
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