Monday, November 23, 2015

Bernhard Riemann

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   Bernhard Riemann was the second of six kids, two boys and four girls, born to Friedrich and Charlotte. Their father was their teacher for the first few years of their lives will Bernhard turned ten when a local school teacher took over for their father. 
   in 1840 Bernhard went to Lyceum in Hannover where he was a good student but not exceptional in anyway, he did show interest in mathematics and was allowed to focus more directly on it at school. In 1846 Riemann enrolled in university of Gottingen and studied theology because his father had wanted him to but that didn't stop him from attending mathematics lectures. He did this till he could convince his father into letting him switch to mathematics. He was finally able to and he studied under Moritz Stern and Gauss. 
   This would have been an amazing place to study under Gauss but he was only teaching elementary courses and Riemann still only looked like a good student. The teacher to notice Riemann's potential was Stern which made him move to Berlin university in 1847 to study under Steiner, Jacobi,Dirichlet, and Eisenstein. 
   Moving back to his original University Riemann got his PhD under the supervision of Gauss.He then went on to write his Doctoral thesis which was looked at and accepted bu Gauss in 1851. 
   Gauss recommended Riemann to be appointed to become a lecture at his own home University. In these lectures Riemann have two parts:First he posed the problem of how to define an n-dimensional space and ended up giving a definition of what today we call Riemannian space. In his second part of his lectures he posed deep questions about the relationship of geometry to the world we live in. 
   Once Gauss left Riemann fought for his chair at that University and wasn't given is but two years later he was appointed to professor and in the same year 1857 he had another punished paper. It wasn't till 1859 that Riemann was given the chair of mathematics at Gottingen.
   In 1858 Riemann was visited by Betti, Casorati, and Brischi and they discussed his idea of topology. Later he reported his greatest masterpiece "On the number of primes less then a given magnitude."
   To a more personal side of Riemann in 1862 he married Elise Koch and they had one daughter. In that same year he got tuberculosis. He then traveled to warmer climates to try and get better because in his family everyone died young. He passed away in 1866 while in Selasca, Italy.  After his death  his work published into a book focusing on geometric approach in math. 

Sunday, November 1, 2015

Euler: The Man!

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    From the beginning of Euler he was destined to be a genius, his father Paul Euler having studied theology at University of Basel and studied under Jacob Bernoulli. Not only did Paul learn from Jacob but he and Jacob's brother Johann both lived with Jacob while in school. Once finishing school Euler's father married and soon had Leonard Euler in the town of Basel.
    Euler's first teacher was his father and since he had studied under Jacob Bernoulli he was a well trained math teacher and was able to pass what he knew onto his son. Research shows that Euler wasn't sent to the greatest schools while growing up and self taught mathematics by reading text books. Euler was suppose to follow his father into the church but Johann Bernoulli found the great potential that Euler had and started private tutoring him. Even though his passion was in mathematics Euler followed his father's wishes and began studying theology. After realizing that there wasn't a future for him in the church he got his father's blessing to change his studies and focus on math.
    Euler by the age of 20 in 1726 had countless papers in print, short articles on isochronous curves in a resisting medium and was submitting papers into the Paris Academy Grand Prize and took second place behind Bouguer. After this achievement he was offered a post in St. Petersburg teaching in applications of mathematics and appointed to the mathematical-physical division. In the coming years he was appointed to the senior chair of mathematics and was able to start a family. In 173 he became sick and almost died, while being sick his eye sight started to diminish till he was completely blind in one eye and going blind in the other.
    In 1741 he and his family left St. Petersburg and went to Berlin where he worked at the Academy of Science. Euler spent the next 25 years in Berlin where he wrote 380 articles, books Calculus of variations, planetary orbits, artillery and ballistics, navigation, motion of the moon, and differential calculus. By the year of 1759 Euler assumed leadership of the Berlin Academy. He only stayed there for another 7 years when he decided to move back to St. Petersburg about this time Euler went completely blind, this however didn't stop him from doing his mathematics because he had a remarkable memory and was able to write articles with the help of his students. But in the year 1783 at nearly 77 years old Euler died.

    In his 77 years the mathematics that he had a part in grew and took off into what we know today. Here are just some of the topics he had a part in:

  • Geometry and Trigonometry 
    • considered sin and cos
  • Calculus
    • Differential calculus
    • Introduced 
      • beta 
      • gamma
      • integrating factors
  • Number Theory 
    • if a and b are coprime then a2 + b2 has no divisor of the form 4n - 1
  • Lunar theory with Clairaut
    • Three body problem
    • wave theory of light
    • hydraulics 
    • music
    • acoustics
  • Eulers notations
    • f(x)
    •  i  for the square root on -1
    • π  for pi
    • ∑ for summation
    • finite differences Δand  Δ2y
  • The Basel problem
    • find a closed form for the sum of the infinity series ζ(2) = ∑ (1/n2)
  • And much much more.
   Looking back over Euler's life it is astonishing the things that he was able to accomplish and was able to do even close to the end when he was blind. I can say that I believe that math wouldn't be what we know it to be without Euler and what he gave future mathematics.