Algebra

1
Algebra

• Main problem Solve algebraic equations in an
  algebraic way!
• E.g. ax2bxc0 can be solved using roots.
• Also ax3bx2cxd0 can be solved using iterated
  roots (Ferro, Cardano, Tartaglia)
• There is a two step process to solve (Ferrari)
  ax4bx3cx2dxe0
• There is no formula or algorithm to solve using
  roots etc (Galois). ax5bx4cx3dx2exf0
• or higher order equations with general
  coefficients.

2
History

• 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 equations.
• 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
  Euclids 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
  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 x3cxd,
  using conic sections.

3
History

• Scipione del Ferro (1465-1526) found methods to
  solve cubic equations of the type x3cxd which
  he passed on to his pupil Antonio Maria Fiore.
  His solution was
• this actually solves all cubic equations for
  y3-by2cy-d0 put yxb/3 to obtain x3mxn with
  mc-b2/3 and nd-bc/32b3/2, but he did not 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. Cardano then published a method for
  solutions.
• The solutions may involve roots of negative
  numbers.

4
History

• Ludovico Ferrari (1522-1562) gave an algorithm to
  solve quadratic equations.
• Start with x4ax3bx2cxd0
• Substitute yxa/4 to obtain y4py2qyr0
• Rewrite (y2p/2)2-qy-r(p/2)2
• Add u to obtain (y2p/2u)2-qy
  -r(p/2)22uy2puu2
• Determine u depending on p and q such that the
  r.h.s. is a perfect square. Form this one obtains
  a cubic equation 8u38pu2(2p2-8r)u-q20

5
History

• 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 proof.
• 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.

6
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-10) are solvable by radicals.

7
Euclid 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.

8
Euclid Areas and Quadratic Equations

• Algebraic version Set ACCBa and CDb then
  (ab)(a-b)b2a2.
• This allows to solve algebraic equations of the
  type ax-x2x(a-x)b2, a, bgt0 and blta/2
• Construct the triangle, then get x, c s.t.
• x(a-x)c2(a/2)2 and b2c2(a/2)2, so
• x(a-x)b2