Diophantus:

1
Diophantus
Chapter 5

• The Curtain Closes on
• Greek Mathematics
• Robert Owens Angi Purdon

2
Brief Historical Background

• Ptolemy VII banished all scholars and scientists
  that had not proven their fidelity to
  themAlexandrias loss was the remainder of the
  Mediterraneans gain.

3
Brief Historical Background

• The Golden Age of Greek mathematics ended by
  the close of the third century. By the fourth
  century scholars began to turn their interests to
  theological debates (a consequence of the spread
  of Christianity).

4
Brief Historical Background

• It is interesting to note that although Greek and
  Roman cultures existed simultaneously, no Roman
  mathematicians of note are mentioned in
  historical documents. This is perhaps in part
  due to the widespread Roman contempt for theory
  (illustrated by Cicero). The Romans did,
  however, call on the Greeks for mathematical help
  when they needed it.

Cicero The Greeks held the geometer in the
highest honor accordingly, nothing made more
brilliant progress among them than mathematics.
But we have established as the limits of this art
its usefulness in measuring and counting.
5
Diophantus (in some texts Diophantos)

• As with most of ancient Greek mathematicians,
  little is known about him personally, what little
  we know is that he lived in Alexandria around the
  middle of the third century AD, and was most
  likely a Hellenized Babylonian.
• The remainder of the details of his life we learn
  from a famous epitaphic problem
• God granted him to be a boy for the sixth part
  of his life, and adding a twelfth part to this,
  He clothed his cheeks with down He lit him the
  light of wedlock after a seventh part, and five
  years after his marriage He granted him a son.
  Alas! late-born wretched child after attaining
  the measure of half his father's life, chill Fate
  took him. After consoling his grief by this
  science of numbers for four years he ended his
  life
• Setting this problem up as
• 1/6x 1/12x 1/7x 5 1/2x 4 x This
  implies x 84,
• Personal details can be ascertained from this
  answer.

6
Diophantus Contributions to Mathematics

• We know of three mathematical works written by
  him On Polygonal Numbers (only a small portion
  still exists), Porisms (which remains lost), and
  the most influential, Arithmetica.
• Algebra progressed through three stages
• Rhetorical, in which the solution of an
  equation is
• written in prose from (without symbols or
• abbreviations),
• Syncopated, in which symbols are incorporated
  for the
• more commonly occurring operations and
  quantities, and Symbolic, where solutions are
  expressed in a
• mathematical shorthand composed primarily of
• symbols.
• Diophantus is largely responsible for the
  syncopation of algebra, and the extensive
  exploration of indeterminate algebraic problems.

7
Syncopation of Algebra

• Diophantus syncopated algebra by developing
  symbols for the following expressions an
  unknown, powers, equality, subtraction,
  reciprocals and constants.
• V is designated for an unknown value (x)
• D is designated for an unknown squared (x2)
• k is designated for an unknown cubed (x3)
• DD is a unknown square-square or 4th power
  (x4)
• Dk is a unknown square-cube or 5th power (x5)
• kk is a unknown cube-cube or 6th power (x6)
• y is designated for subtraction (-)
• i is designated for equality ()
• c is designated for reciprocals of powers
• M is designated for a constant

8
Syncopation of Algebra

• Coefficients were ordinary numerals following
  power symbol.
• An example of a term used would be something
  like
• kle k35 35x3
• Since there was no symbol for addition, all
  negative terms were placed together after a
  single subtractive symbol
• k1V8 y D 5 M2
• would be equivalent to
• X3 5x2 8x 2

9
Diophantine Equations

• The expression Diophantine equations is used
  for any equation in one or more unknown that is
  to be solved for the explicit value of the
  unknown(s).
• Diophantus was not the first to solve this type
  of equation, but is given the honor of this
  naming because of his great ability with this
  type of problem (demonstrated in the Arithmetica.
  
• It is fair, however, to say that Diophantus was a
  pioneer in many aspects of number theory, and we
  are all the richer for his developments in this
  area.
• One interesting thing of note, is that although
  rational numbers were considered relevant
  solutions to problems, negative numbers were not.
  Diophantus described a problem such as 2x 17
  7 as absurd.

10
The Arithmetica

• Originally, 13 books only 6 have been preserved.
• Diophantus most famous work.
• There are approximately 190 assorted problems,
  like the Rhind papyrus.

11
Problems from Arithmetica

• Book II, Problem 8. Divide a given square
  number, say 16 into the sum of two squares.
• Let one of the required squares be x2.
• This implies that 16- x2 would yield another
  perfect square.
• Diophantus selected a particular instance of a
  perfect square to set this equal to, one that was
  particularly useful in eliminating the constant
  terms (2x-4)2
• Setting these terms equal to each other, we
  have
• 16 - x2 (2x-4)2 which simplifies to 5x2 16x
• Thus x 16/5.
• Hence, the two squares would be
• (16/5)2 or 256/25
• and 16-256/25, or 144/25

12
Problems from Arithmetica

• Book III, Problem 17. Find two numbers such that
  their product added to either one or to their sum
  gives a square.
• Let the required numbers be x and 4x-1.
• (These numbers are chosen so that they satisfy
  one condition directly, as shown below).
• x(4x-1) x 4x2 (2x)2
• Adding the product to their sum and also to 4x-1
  must yield a square.
• Using the identity a2 b2 (a b)(a b), we
  set a2 4x2 4x 1, and b2 4x2 3x 1, so
  a2 b2 will equal x.
• x is then separated into 2 factors 4x (we will
  call (a b)) and ¼ (we will call (a-b))
• solving this system, we reach a ½(4x ¼) and b
  ½(4x- ¼)
• plugging these values into either a2 or b2
  equation above, we attain x to be 65/224, so the
  other number is 36/224.

13
Problem from Lilavati

• Say quickly, mathematician, what is the
  multiplier by which 221 being multiplied, and 65
  added to the product, the sum divided by 195
  becomes exhausted leaves no remainder?
• This can be expressed as 221y 65 195x,
  rewritten 195x 221y 65
• Using Euclids algorithm to find the gcd of 195
  and 221
• 221 195 1 26
• 195 26 7 13
• 26 13 2
• So 13 is the greatest common divisor of 195 and
  221. Since it also divides 65, a solution
  exists.
• Finding a linear combination, we work backwards
• 13 195 726
• 13 195 7(221-195)
• 13 8 195 (-7)221
• Multiplying the equation through by 5, we see
  that
• 65 40 195 (-35)221
• hence, one (there are many) solution to this
  problem is x 40, y 35.

14
Hypatia

• First woman mathematician of historical
• note.
• Lived 370-415.
• Daughter of Theon learned mathematics,
• medicine and philosophy from him.
• Wrote a commentary on Arithmetica, also on
  Apollonius Conic Sections.
• Lectured at the Museum, attracted many important
  listeners.
• Christian leaders felt threatened by her
  heretical teachings
• One day, returning from teaching, she was
  ambushed, cut using the shells of oysters, then
  dismembered, her remains burnt.
• Considered the last of the great Greek
  Mathematicians.

15
Homework Problems

• 5.3.3. Book I, Problem 27. Find two number such
  that their sum and product are given numbers say
  their sum is 20 and their product is 96. Hint
  Call the numbers 10 x and 10 x. Then one
  condition is already satisfied.
• 5.3.7. Book II, Problem 22. Find two number
  such that the square of either number added to
  the sum of both gives a square. Hint If the
  numbers are taken to be x and x1, then one
  condition is satisfied.
• 5.3.13. Which of the following diophantine
  equations cannot be solved?
• a) 6x 51y 22
• b) 33x 14y 115
• c) 14x 35y 93
• 5.3.15. Determine all solutions in the positive
  integers of the following diophantine equations.
• a) 18x 5y 48
• b) 123x 57y 30
• c) 123x 360y 99
• Homework is due Monday, November 10 and worth a
  quiz grade.

16
Sources Utilized

• Burton, David M., The History of Mathematics, An
  Introduction, p 203-226, McGraw-Hill, New York,
  NY, 2003.
• Eves, Howard, An Introduction to the History of
  Mathematics, p 179-182, Saunders College
  Publishing, Orlando, FL, 1990.
• Eves, Howard, Great Moments in Mathematics, p
  117-120, The Mathematical Association of America,
  1980.
• Eves, Howard, In Mathematical Circles, p 59-62,
  Prindle, Weber Schmidt, Inc., Boston, 1969.
• Grattan-Guinness, Ivor, The Norton History of the
  Mathematicl Sciences The Rainbow of Mathematics,
  p. 80-82, W.W. Norton Company, New York, 1997.
• http//www-gap.dcs.st-and.ac.uk/history/Mathemati
  cians/Diophantus.html
• http//www-gap.dcs.st-and.ac.uk/history/Quotation
  s/Diophantus.html
• http//www.lib.virginia.edu/science/parshall/dioph
  ant.html