- Return of your revised one-page, typed proposal for your paper/project. Final paper/projects are due Monday, at the end of class.
- Problems to return:
- 5.3
- #11
- 5.3
- Sorry, I'm still at work on 4.4/4.5 -- I'll post them on my door
ASAP (hopefully tomorrow)
- It's time for course evaluations. Please take the time to visit
eval.nku.edu, and
give me your feedback. Thanks!
- Remember:
- Monday: class presentations, starting at 4:30
- Wednesday, by 5:00 p.m.: take-home final due.
- Lessons from 7.3.
- Commercial from Debra.
- Lessons from 7.4, etc.
- in Cardan's Liber de Ludo Aleae (Book on Games of Chance) he "broke the ground for a theory of probability more than 50 years before Fermat and Pascal." (p. 323)
- Ars_Magna --
"The Great Art"
- on algebraic equations. "Cardan was no mere plagiarist, but one who combined a measure of honest toil with his piracy." (p. 323)
- We want to examine how to derive the "cubic formula".
- In the process of using "his" formula, Cardan encountered
complex roots ("ghosts of real numbers", as Napier called
them). "Putting aside the mental tortures involved, multiply
by
....": sum 10, product 40.
- "So progresses arithmetic subtlety the end of which, as is said, is as refined as it is useless." (p. 323)
- Let's take a look at Bombelli's cubic,
- Rafael Bombelli (1526-1572)
- First to write the rules of negative numbers
- First to write complex numbers as sums of real and imaginary parts. "From this time on, imaginary numbers lost some of their mystical character...."
- Ludovico Ferrari (February 2, 1522 -- October 5, 1565), "Cardan's Creature"
- Cardan continues to show flashes of decency: he did bring up this poor lad in his household, and, realizing that Ferrari was intelligent, instructed him himself.
- Cardan had tried and failed to solve the quartic, so he turned the problem over to Ferrari; when Ferrari solved it, the result was incorporated into the Ars Magna, with credit to Ferrari.
- Ferrari learned his scruples from Cardan, however, and defended Cardan (quite likely dishonestly) in Cardan's debate with Tartaglia.
- Ultimately (1548) Ferrari and Tartaglia had a contest, and Ferrari stomped him.
- The method:
- Reduce a quartic, to a form without the third degree term (same ol', same ol');
- then add a constraint, which reduces our problem to the solution of a cubic, followed by the solution of a quadratic.
- Postscript:
- Of course, as soon as the quartic was solved, everyone started looking for the solution of the quintic, and then
- looked, and
- looked, and
- looked some more, for about 300 years.
- Finally, Niels Henrik Abel (August 5, 1802 April 6, 1829), basing his work on Paolo Ruffini's, demonstrated that the general quintic cannot be solved by radicals.
- Galois went on to show the conditions under which a quintic can be solved in radicals.
- Both of these mathematicians were among the unluckiest
known to mathematics. That's because they were algebraists!;)
- I was flipping through the rest of the book last night: there's a lot more to know! Timeline of Mathematical History