History of Mathematics
History of Mathematics
Review.
Generalities
- You should know the main contributions of the
mathematicians we have discussed: Thales, Pythagoras
and his school, Zeno, Eratosthenes and Aristarchus,
Eudoxos, Euclid, Archimedes, Appolonius, Diophantus;
Tartaglia, Cardano, Bombelli, Ruffini, Euler, Gauss,
Abel, Galois.
- You should be familiar with the most famous works
in the history of mathematics: Euclid's elements,
Archimedes and the method of exhaustion, Diophantus's
Arithmetika, Fibonacci's Liber Abaci (introduction of
Hindu-Arabic numerals to Europe), Cardano's Ars Magna,
Descartes's Geometry (geometric construction of sums,
products, quotients, and square roots) Gauss's Disquisitiones
Arithmeticae (we only talked about the construction of regular
polygons). What are these books famous for? When were
they published? (BTW, all of these books are still in
print.)
- Greek astronomy: circumference of the earth, distance
earth-moon, distance moon-sun.
- Solutions of polynomial equations: contributions of
Babylonians, Chinese and Arabic mathematicians,
Tartaglia, Cardano & Bombelli, Ruffini, Abel, Galois.
- Development of numbers: invention of
negative numbers,
0 as a number and symbol, decimals, complex numbers
(when and why?). Which numbers are constructible? What are
algebraic and transcendental numbers? Examples?
- Possibility of geometrical constructions (regular n-gons,
trisection, duplication of cube, squaring the circle).
- Discovery of non-Euclidean geometry; understand how the non-Euclidean
geometries of Bolyai and Lobachevsky prove that the parallel axiom
cannot be deduced from Euclid's other four postulates.
Here's why: Suppose you can construct a (consistent - this means
that it is free of contradictions, or in other words, that
not every statement is true) geometry in which Euclid's
axioms I-IV hold, but in which the parallel axiom is false.
Then axiom V cannot be deduced from I-IV, because otherwise
V would also be true in the new geometry - which it isn't.
Techniques
- Archimedes computation of the area of a segment of a parabola
(the main idea)
- Solving diophantine equations using sweeping lines
- Solving cubics
- Symmetry group of a regular n-gon
- Euler's summation of 1 + 1/4 + 1/9 + ...