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• Main problem: Solve algebraic equations in an algebraic
• E.g. ax2+bx+c=0 can be solved using roots.
• Also: ax3+bx2+cx+d=0 can be solved using iterated roots
  (Ferro, Cardano, Tartaglia)
• There is a two step process to solve (Ferrari)
• There is no formula or algorithm to solve using roots etc
  (Galois).         ax5+bx4+cx3+dx2+ex+f=0
  or higher order equations with general coefficients.
•   ca 2000 BC The Babylonians had collections of solutions of quadratic
    equations. They used a system of numbers in base 60. They also had
    methods to solve some cubic and quartic equations in several unknowns.
    The results were phrased in numerical terms.
•   ca 500 BC The Pythagoreans developed methods for solving quadratic
    equations related to questions about area.
•   ca 500 BC The Chinese developed methods to solve several linear
•   ca 500 BC Indian Vedic mathematicians developed methods of calculating
    square roots.
•   250-230 AD Diophantus of Alexandria made major progress by
    systematically introducing symbolic abbreviations. Also the first to consider
    higher exponents.
•   200-1200 In India a correct arithmetic of negative and irrational numbers
    was put forth.
•   800-900 AD Ibn Qurra and Abu Kamil translate the Euclid’s results from the
    geometrical language to algebra.
•   825 al-Khwarizmi (ca. 900-847) wrote the Condenced Book on the
    Calculation of al-Jabr and al-Muquabala. Which marks the birth of
          algebra. Al-jabr means “restoring” and al-muquabala means                QuickTime™ and a
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          “comparing”. The words algebra is derived from al-jabr and
              the words algorism and algorithm come from the name
               al-Khwarizmi. He also gave a solution to all quadratic equations!
•   1048-1131 Omar Khayyam gave a geometric solution to finding solutions to
    the equation x3+cx=d, using conic sections.
• Scipione del Ferro (1465-1526) found methods to
  solve cubic equations of the type x3+cx=d which
  he passed on to his pupil Antonio Maria Fiore. His
  solution was:

  this actually solves all cubic equations for y3-
  by2+cy-d=0 put y=x+b/3 to obtain x3+mx=n with
  m=c-b2/3 and n=d-bc/3+2b3/2, but he did not                     QuickTime™ and a
                                                        TIFF (Un compressed) decompressor
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  know that.
• Niccolò Tartaglia (1499-1557) and Girolamo
  Cardano (1501-1557) solved cubic equations by
  roots. There is a dispute over priority. Tartaglia
  won contests in solving equations and divulged
  his “rule” to Cardano, but not his method.                     QuickTime™ and a
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  Cardano then published a method for solutions.
• The solutions may involve roots of negative
•   Ludovico Ferrari (1522-1562) gave an
    algorithm to solve quadratic equations.
    1. Start with x4+ax3+bx2+cx+d=0
    2. Substitute y=x+a/4 to obtain
    3. Rewrite (y2+p/2)2=-qy-r+(p/2)2
    4. Add u to obtain
       (y2+p/2+u)2=-qy -r+(p/2)2+2uy2+pu+u2
    5. Determine u depending on p and q such
       that the r.h.s. is a perfect square. Form this
       one obtains a cubic equation
• The algebra of complex numbers appeared in the text
  Algebra (1572) by Rafael Bombelli (1526-1573) when he
  was considering complex solutions to quartic equations.
• François Viète (1540-1603) made the first steps in
  introduced a new symbolic notation.
• Joseph Louis Lagrange (1736-1813) set the stage with
  his 1771 memoir Réflection sur la Résolution Algébrique
  des Equations.
• Paolo Ruffini (1765-1822) published a treatise in 1799
  which contained a proof with serious gaps that the
  general equation of degree 5 is not soluble.
• Niels Henrik Abel (1802-1829) gave a different, correct
• Evariste Galois (1811-1832) gave a complete solution to
  the problem of determining which equations are solvable
  in an algebraic way and which are not.
         Other Developments
• 1702 Leibniz published New specimen of the
  Analysis for the Science of the Infinite about
  Sums and Quadratures. This contains the
  method of partial fractions. For this he considers
  factorization of polynomials and radicals of
  complex numbers.
• 1739 Abraham de Moivre (1667-1754) showed
  that roots of complex numbers are again
  complex numbers.
• In 1799 Gauß (1777-1855) gives the essentially
  first proof of the Fundamental Theorem of
  Algebra. He showed that all cyclotomic
  equations (xn-1=0) are solvable by radicals.
   Areas and Quadratic Equations
• Euclid Book II contains “geometric algebra”
• Definition 1.
   – Any rectangular parallelogram is said to be contained by the two
     straight lines containing the right angle.
• Definition 2
   – And in any parallelogrammic area let any one whatever of the
     parallelograms about its diameter with the two complements be
     called a gnomon
• Proposition 5.
   – If a straight line is cut into equal and unequal segments, then the
     rectangle contained by the unequal segments of the whole
     together with the square on the straight line between the points
     of section equals the square on the half.
  Areas and Quadratic Equations
• Algebraic version: Set AC=CB=a and
  CD=b then (a+b)(a-b)+b2=a2.
• This allows to solve algebraic equations of
  the type ax-x2=x(a-x)=b2, a, b>0 and b<a/2
• Construct the triangle, then get x, c s.t.
• x(a-x)+c2=(a/2)2 and b2+c2=(a/2)2, so

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