History of Mathematics
Review Midterm 2
You should have no problems with questions like these:
You should be familiar with the following proofs:
- You should know the main actors behind the theory of
exhaustion (Eudoxus, Euclid, Archimedes), their
contributions, and be familiar with the results and
proofs we we discussed in class and in the homework.
How did Archimedes formulate his results, given that
he had no formulas?
- What are the contributions to Euclid's elements that are
credited to the Pythagorean school?
- What are the
Platonic solids? (you can turn the solids by clicking on
them; don't worry about anything we didn't talk about)
What did Euclid know about them?
- Describe at least one of
paradoxa. Explain the difference between
actual and potential infinity.
- What are the classical unsolved problems of Greek
mathematics, and how and when were they solved? How
are irrational, algebraic, transcendental numbers defined?
What are examples for such numbers?
- Explain the difference between 0 as a symbol and 0 as a number.
- Explain the difference between the geometric multiplication of
numbers (line segments) in Euclid and Descartes. Construction
of sums, products, square roots in Descartes.
- Why were imaginary (complex) numbers `invented'?
- Read the article on the dispute
between Tartaglia and Cardano; again you should know the main
actors (del Fierro, T., C., Ferrari, Bombelli) and their contributions.
- Chronological order of del Ferro, Tartaglia, Cardano Ferrari,
Bombelli, Fermat, Descartes, Gauss.
- results on the circle, area of spiral, segment of parabola
- solving cubic and quartic polynomials