Unsolvable Equation 1

Size: px

Start display at page:

Download "Unsolvable Equation 1"

Lisa Craig
5 years ago
Views:

1 Unsolvable Equation 1

2 Algebra after Cardano s Ars Magna At the end of his magnum opus, Cardano wrote Written in five years, may it last many thousands Bombelli s Algebra, written 15 years later, contained the first attempt to discuss complex numbers systematically by introducing imaginary numbers (multiples of i) and writing complex numbers as, for example, 2 p di m 3 (2 più di meno 3). Nowadays: 2+3i He gave many examples: 1000/(2+11i) 2

3 Analysis of Cardano s formula in Bombelli What happens if you multiply these two identities? If you cube one of them? Can you guess a and b? 3

4 From here Bombelli makes a guess: a=2, b=2 and 4

5 ! Another equation! Can you guess a and b? And what is x? 5

6 The Analytic Art of François Viète Mathematicians of the Renaissance believed that Ancient Greeks had two approaches to mathematics: synthetic mathematics (e.g. Euclid begins with axioms and shows how they can be synthesized to yield more theorems analytic mathematics (mathematician works backwards from the desired result to find a way to arrive at it) Viète ( ) was convinced that ancients discovered their theorems via (hidden and lost) analytic art. After working backwards and discovering the necessary steps to arrive at the result, they would turn around and lay out these steps. A good analogy here is to compare analysis with scaffolding from which a perfected architecture of synthetic proof is constructed. One can admire synthetic proofs but to learn how to prove anything new, one needs to learn analytic art! 6

7 Viéte believed that ancients used algebra as their analytic art. Now a wide consensus is that ancients of course used a lot of empirical thinking and working backwards to derive their results. They would later hide these preliminary analysis behind the facade of rigorous synthetic proofs. But they did not use algebra. Nowadays, we still use three steps outlined by Viète (without using these funny names) both in pure and applied math: zetetic analysis: transforming a problem into equations linking unknown and known quantities poristic analysis: exploring the truth of various statements about equations using logic and symbolic manipulations exegetic analysis: find values for unknowns in equations 7

8 Apart from this philosophy (later much expanded by Descartes, Newton, and others). perhaps the most important important contribution of Viète was introduction of variable coefficients, like in modern notation (essentially finalized by Descartes) 8

9 Girard s ( ) identities 9

10 Newton s ( ) identities 10

11 Despite all advances, no progress was made on solving a quintic: given a general equation as above, is there a formula for x in terms of coefficients and radicals (roots of various degrees?) In fact mathematicians were becoming increasingly convinced that this equation is not solvable in radicals Paolo Ruffini has attempted to prove this in 1798 but the proof was incomprehensible and likely incomplete. 11

12 Niels Henrik Abel ( )

13 Born on August 5, 1802, the son of a pastor and parliament member in the small Norwegian town of Finnø. Father died an alcoholic when Abel was 18, leaving behind nine children (Abel was second oldest) and a widow who also turned to alcohol for solace Abel was shy, melancholy, and depressed by poverty and apparent failure Completely self-taught, Abel entered University of Christiania in 1822 Norway separated from Denmark when Abel was 12, but remained under Swedish control. Only 11,000 inhabitants in Christiania at the time

1823: Abel incorrectly believes that he has solved the general equation of the fifth degree After finding an error, proves that such a solution is impossible To save printing costs, paper is

14 1823: Abel incorrectly believes that he has solved the general equation of the fifth degree After finding an error, proves that such a solution is impossible To save printing costs, paper is published in a pamphlet at his own cost, with the result in summary form 1824: This result together with a paper on the integration of algebraic expressions (now known as Abelian integrals) led to him being awarded a stipend for study trip abroad Abel hoped trip would allow him to marry fiancee who remained in Norway as governess (Christine Kemp)

Abel travels to Berlin, where he stays from September 1825 to February 1826 Encouraged and mentored by the August Leopold Crelle, a promoter of science.

15 Abel travels to Berlin, where he stays from September 1825 to February 1826 Encouraged and mentored by the August Leopold Crelle, a promoter of science. Crelle founds Journal für die reine und angewandte Mathematik (also known as Crelle s Journal), the premiere German mathematical journal, and the very first volume includes several works by Abel Impossibility of solving the quintic equation by radicals Binomial series (contribution to the foundation of analysis) One of the results is Abel s Theorem on Continuity in complex analysis, which clarified and corrected some foundational results of Cauchy

In July 1826 went to Paris where remained until the end of the year Isolated in Paris s more elegant and traditional society; impossible to approach the great men of the Académie, e.g. Cauchy.

16 In July 1826 went to Paris where remained until the end of the year Isolated in Paris s more elegant and traditional society; impossible to approach the great men of the Académie, e.g. Cauchy. Writes Though I am in the most boisterous and lively place in the continent, I feel as though I am in a desert. I know almost nobody. Presented his Memoire sur une classe très étendue de fonctions transcendentes which contains Abel s Theorem on integrals of algebraic functions. Viewed by many as Abel s crowning achievement Increasing financial worries toward the end of 1826

17 1827: Back in Norway, Abel cannot find work In a letter begging friend for a loan, writes I am as poor as a churchmouse Yours, destroyed.! Christmas 1828: holiday with his fiancee in the country, only socks to warm his hands Violent illness (tuberculosis); died on April 6, 1829

But I would not like to part from this ideal type of researcher, such as has seldom appeared in the history of mathematics, without evoking a figure from another sphere who, in spite of his totally

18 But I would not like to part from this ideal type of researcher, such as has seldom appeared in the history of mathematics, without evoking a figure from another sphere who, in spite of his totally different field, still seems related. < > I c o m p a r e h i s k i n d o f productivity and his personality with Mozart's. Thus one might erect a monument to this divinely inspired mathematician like the one to Mozart in Vienna: simple and unassuming he stands there listening, while graceful angels float about, playfully bringing him inspiration from another world.

Instead, I must mention the very different type of memorial that was in fact erected to Abel in Christiania and which must greatly disappoint anyone familiar with his nature.

19 Instead, I must mention the very different type of memorial that was in fact erected to Abel in Christiania and which must greatly disappoint anyone familiar with his nature. On a towering, steep block of granite a youthful athlete of the Byronic type steps over two greyish sacrificial victims, his direction toward the heavens. If needed be, one might take the hero to be a symbol of the human spirit, but one ponders the deeper significance of the two monsters in vain. Are they the conquered quintic equations or elliptic functions? Or the sorrows and cares of his everyday life? The pedestal of the monument bears, in immense letters, the inscription ABEL. - Felix Klein 19

20 Abel s argument Uses reductio ad absurdum - assumes a formula for a solution exists and derives a contradiction Shows that a solution can be expressed in the special form!! Shows that all algebraic functions appearing in this formula can be expressed as rational functions of the solutions Shows that ay rational function of five quantities can take only 1, 2, or 5 or more values when quantities are permuted (but not 3 or 4) From here he shows by careful analysis that any hypothetical formula can not give all five solutions to the equation 20

The Cubic formula and beyond?"

The Cubic formula and beyond?" Which polynomials can we solve using radicals? When was it figured out? Milan vs. Venice" (Cardano and Tartaglia)" 1500ʼs" Doubling the Cube (The Delian Problem)" To rid