1. Chapter 6 Polynomial Equations Algebra Linear Equations and Eliminations Quadratic Equations Quadratic Irrationals The Solution of the Cubic Angle Division Higher-Degree Equations Biographical Notes: Tartaglia, Cardano and Viète

2. 6.1 Algebra Algebra ~ “al-jabr” (Arabic word meaning “restoring”) al-Khwārizmī “Al-jabr w’al mûqabala” 830 CE (“Science of restoring and opposition”) –restoring – adding equal terms to both sides –opposing – setting the two sides equal Note: the word “algorithm” comes from his name Algebra –Indian math: “inside” number theory and elementary arithmetic –Greek math: hidden by geometry –Arabic math. recognizes algebra as a separate field with its own methods Until the nineteenth century algebra was considered as a theory of (polynomial) equations Connection between algebra and geometry: analytic geometry (Fermat, Descartes, 17 th century)

3. 6.2 Linear Equations and Eliminations China (Han dynasty, 206 BCE – 220 CE ): mathematicians invented the method to solve systems of linear equations which is now called “Gaussian elimination” They used counting boards to hold the array of coefficients and to perform manipulations similar to elementary matrix operations Moreover, they discovered that eliminations can be applied to polynomial equations of higher order in two or more variables

4. 6.3 Quadratic Equations Babylon 2000 BCE – algorithm to solve system of the form x + y = p, xy = q which is equivalent to the quadratic eqation x 2 + q = px Steps: Find x and y using Note: this is equivalent to the formula

5. India, 7 th century, Brahmagupta: formula in words expressing general method to solve ax 2 + bx = c: Greek, Euclid’s “Elements”: rigorous basis for the solution of quadratic equations al-Khwārizmī, 9 th century: solution, in which “squares” were understood as geometric squares and “products” as geometric rectangles Example: solve x 2 + 10x = 39 –x 2 and 10x = 5x + 5x –“complete the square”: 25 –the total area = 25 + 39 = 64 –therefore x + 5 = 8 and x = 3 x2x2 x x 5x 25 5 5 Note: we obtained only positive solution!

6. 6.4 Quadratic Irrationals Roots of quadratic equations with rational coefficients are numbers of the form a+√b where a and b are rational Euclid: study of numbers of the form No progress in the theory of irrationals until the Renaissance, except for Fibonacci result (1225): roots of x 3 +2x 2 +10x=20 are not any of Euclid’s irrationals Fibonacci did not prove that these roots are not constructible with ruler and compass (i.e. that it is not possible to obtain roots as expressions built from rational numbers and square roots) Using field extensions it is not hard to show that, say, cube root of 2 is not a quadratic irrational and hence is not constructible (and this could be done using 16 th century algebra) Nevertheless, it was proved only in 19 th century (Wantzel, 1837)

7. 6.5 The Solution of the Cubic First clear advance in mathematics since the time of the Greeks Power of algebra Italy, 16 th century: Scipione del Ferro, Fior, Cardano and Tartaglia Contests in equation solving Most general form of solution: Cardano formula

8. Cardano Formula substitution: x = y – a/3 sub. y = u + v quadratic in u 3 roots:

9. y = u + v Cardano Formula:

10. 6.6 Angle Division France 16 th century: Viète  introduced letters for unknowns  “+” and “-” signs  new relation between algebra and geometry – solution of the cubic by circular (i.e. trigonometric) functions  his method shows that solving the cubic is equivalent to trisecting an arbitrary angle

11. substitution: x = ky Note: Viète tried to find expressions for cos nθ and sin nθ as polynomials in cos θ and sin θ Newton: Note: n is arbitrary (not necessarily integer); if it is an odd integer the above expression is a polynomial

12. Note: Newton’s equation has a solution by n th roots if n is of the form n=4m+1 - de Moivre (1707): This formula is a consequence of the modern version of de Moivres formula:

13. 6.7 Higher-Degree Equations The general 4 th degree (quartic) equation was solved by Cardano’s friend Ferrari This was solution by radicals, i.e. formula built from the coefficients by rational operations and roots linear sub. complete square For any y we have: The r.-h. side Ax 2 +Bx+C is complete square iff B 2 - 4AC = 0 It is a cubic equation in y It can be solved for y using Cardano formulas This leads to quadratic equation for x The final solution for x is a formula using square and cube roots of rational functions of coefficients

14. Equations of order 5 and higher For the next 250 years obtaining a solution by radicals for higher-degree equations ( ≥ 5) was a major goal of algebra In particular, there were attempts to solve equation of 5 th degree (quintic) It was reduced to equation of the form x 5 – x – a = 0 Ruffini (1799): first proof of impossibility to solve a general quintic by radicals Another proof: Abel (1826) Culmination: general theory of equations of Galois (1831) Hermite (1858): non-algebraic solution of the quintic (using transcendental functions) Descartes (1637): (i) introduced superscript notations for powers: x 3, x 4, x 5 etc. and (ii) proved that if a polynomial p(x) has a root a then p(x) is divisible by (x-a)

15. 6.8 Biographical Notes: Tartaglia, Cardano and Viète spent his childhood in poverty received five serious wounds when Brescia was invaded by the French in 1512 one of the wounds to the mouth which left him with a stutter (nickname “Tartaglia” = “stutterer) at the age of 14 went to a teacher to learn the alphabet but ran out of money by the letter “K” taught himself to read and write Nicolo Tartaglia (Fontana) 1499 (Brescia) – 1557 (Venice)

16. moved to Venice by 1534 gave public mathematical lessons published scientific works Tartaglia visited Cardano in Milan on March 25, 1539 and told him about the method for solving cubic equations Cardano published the method in 1545 and Tartaglia accused him of dishonesty Tartaglia claimed that Cardano promised not to publish the method Nevertheless, Cardano’s friend Ferrari tried to defend Cardano 12 printed pumphlets “Cartelli” (Ferrari vs. Cardano) This led to a public contest which was won by Ferrari Other contribution of Tartaglia to Science include a theory describing trajectory of a cannonball (which was a wrong theory), translation of Euclid’s “Elements” (1 st translation of Euclid in a modern language) and translations of some of Archimedes’ works.

17. Cardano entered the University of Pavia in 1520 He completed a doctorate in medicine in 1526 became a successful physician in Milan Mathematics was one of his hobbies Besides the solution of the cubic, he also made contributions to cryptography and probability theory In 1570 Cardano was imprisoned by the Inquisition for heresy He recanted and was released After that Cardano moved to Rome Wrote “The Book of My Life” Giralomo Cardano 1501 (Pavia) – 1576 (Rome)

18. His family was connected to ruling circles in France Viète was educated by the Franciscans in Fontenay and at the University of Poitiers Received Bachelor’s degree in law in 1560 He returned to Fontenay to commence practice Viète was engaged in law and court services and related activities and had several very prominent clients (including Queen Mary of England and King Henry III of France) Mathematics was a hobby François Viète 1540 - 1603

19. During the war against Spain Viète deciphered Spanish dispatches for Henry IV King Philip II of Spain accused the French in using black magic Another famous result of Viète was a solution of a 45 th – degree equation posed to him by Adriaen van Roomen in 1593 Viète recognized the expansion of sin (45 θ) and found 23 solutions