Algebra Main problem Solve algebraic equations in an

Algebra • Main problem: Solve algebraic equations in an algebraic way! • E. g. ax 2+bx+c=0 can be solved using roots. • Also: ax 3+bx 2+cx+d=0 can be solved using iterated roots (Ferro, Cardano, Tartaglia) • There is a two step process to solve (Ferrari) ax 4+bx 3+cx 2+dx+e=0 • There is no formula or algorithm to solve using roots etc (Galois). ax 5+bx 4+cx 3+dx 2+ex+f=0 or higher order equations with general coefficients.

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 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 “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 x 3+cx=d, using conic sections.

History • Scipione del Ferro (1465 -1526) found methods to solve cubic equations of the type x 3+cx=d which he passed on to his pupil Antonio Maria Fiore. His solution was: this actually solves all cubic equations for y 3 by 2+cy-d=0 put y=x+b/3 to obtain x 3+mx=n with m=c-b 2/3 and n=d-bc/3+2 b 3/2, 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.

History • Ludovico Ferrari (1522 -1562) gave an algorithm to solve quadratic equations. 1. 2. 3. 4. Start with x 4+ax 3+bx 2+cx+d=0 Substitute y=x+a/4 to obtain y 4+py 2+qy+r=0 Rewrite (y 2+p/2)2=-qy-r+(p/2)2 Add u to obtain (y 2+p/2+u)2= -qy -r+(p/2)2+2 uy 2+pu+u 2 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 8 u 3+8 pu 2+(2 p 2 -8 r)u-q 2=0

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.

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.

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.

Euclid Areas and Quadratic Equations • Algebraic version: Set AC=CB=a and CD=b then (a+b)(a-b)+b 2=a 2. • This allows to solve algebraic equations of the type ax-x 2=x(a-x)=b 2, a, b>0 and b<a/2 • Construct the triangle, then get x, c s. t. • x(a-x)+c 2=(a/2)2 and b 2+c 2=(a/2)2, so x(a-x)=b 2