• 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
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
• 250-230 AD Diophantus of Alexandria made major progress by
systematically introducing symbolic abbreviations. Also the first to consider
• 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
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
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• 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
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
• 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.
• 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
• 1739 Abraham de Moivre (1667-1754) showed
that roots of complex numbers are again
• 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