Test 3 Flashcards
What are the main takeaways of math changes that happened in the 19th century
more abstraction and more rigor
What were the top 3 math changes that occured in the 19th century
Arithmetization of algebra, liberation of algebra, liberation of geometry
what is the arithmetization of algebra
resting calculus on rigorous foundtions, axiomitizing the real numbers
compare Cauchy and Weierstrauss
Weierstrauss: was first in law, had late start in mathematics, not a professor until 40, “theory of complex functions by means of power series with absolute rigor”
discovered uniform convergence, generous to students, defined a determinant, very careful, good teacher
Cauchy: was first a civil engineer (late start), was very prolific but was a bit too hasty much of the time, but still cared about rigor
defined the derivative as the limit when delta x approaches 0, introduced characteristic equation determinant of A-lamda I = 0
What is the liberization of algebra
free studies of algebra from number systems, study higher dimentional objects, new kinds of algebra:groups, rings, fields
Who were the main players in the liberization of algebra
Arthur Cayley, George Boole, Evarieste Galois, Niels Abel
Compare and contrast galois and abel
Both discovered unsolvability of quintic by radicals and died young
Galois: developed general solution (groups), proved no polynomial larger than degree four was solvable by radicals, ded of romanticism, lived in france
Abel: developed specific solution, lived in norway, died of poverty
What was the liberation of geometry
discovery of new models where the parallel postulate was not required, freeing of geometric thinking from the ‘real world’ euclidian geometry’
Who were the main players in the liberation of eometry
Carl Gauss, Johannes Bolyai, Nikolaus Lobachevsky, Riemann, Klein
Describe who gauss was, what he thought, and what he did
“greatest mathematician of all time”, “prince of mathematicians”
proved fundamental theorem of algebra in his dissertation (any polynomial of degree n has n roots but they may be complex)
perfectionist, reluctant to publish (“few but ripe”)
said mathematics was queen of sciences and number theory was queen of mathematics
Describe who Fourier was
developed fourier series, all functions cn be represented as infinite sums of sin/cosine, which can be used to represent any function. He used them to study heat
