Interested in learning a bit more about complex variables and some theorems about them? Then check out this new video series covering some of the basics of complex analysis! Complex analysis is an immensely useful subject that is found in many branches of mathematics. The residue theorem is especially well known as being an extremely important method for the integration of functions (real or complex!). Some other topics discussed include complex differentiation, Cauchy's Integral Formula, Liouville's theorem, the Möbius transformation, and much more!
The advent of graph theory, from the mind of Leonhard Euler, came from a long-standing problem for the people of Königsberg. The problem was that no couple had a long and happy marriage, if they were married in Königsberg. As tradition dictated, a newlywed couple had one chance to travel across Königsberg’s four land masses using each of the seven bridges once and only once. If the two lovers could complete this seemingly simple task, their marriage would be long and happy. Years went by and nobody could complete to task, until Euler constructed a mathematical object that broke the curse of Königsberg… a graph! Watch the proof proposed by Euler below to learn how mathematical abstraction created a whole new field of math, which is now regarded as an important predecessor to topology. Euler’s invention itself is remarkable, but the implications to mathematical philosophy reveals something very deep in the heart of mathematics. Namely, the art of abstraction to gain a better understandin...
This problem of determining the area of a circle, or better defined as the area inside of a circle, was a huge dilemma in the field of mathematics. It was not until the mid 200's BC when Archimedes began to anticipate modern calculus and analysis though concepts of infinitesimals and exhaustion, which he used to solve this major challenge of finding the area of a circle. Archimedes' method of finding the area is described as "squaring the circle", which is trying to find the square that has the same enclosed area as a circle of a given radius. Using this and also using a method where he approximated the area of a circle with other, known shapes such as squares and hexagons, Archimedes was able to determine the area inside of a circle. Take a look at the proof to see how Archimedes came up with the formula we know today:
Interested in learning a bit more about complex variables and some theorems about them? Then check out this new video series covering some of the basics of complex analysis! Complex analysis is an immensely useful subject that is found in many branches of mathematics. The residue theorem is especially well known as being an extremely important method for the integration of functions (real or complex!). Some other topics discussed include complex differentiation, Cauchy's Integral Formula, Liouville's theorem, the Möbius transformation, and much more!
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