Sometimes the greatest revolutions in Mathematics are not theorems or proofs, but changes of notation. The greatest revolution in notation may have been the introduction of the place-value system to represent numbers, replacing other systems like the cumbersome Roman numerals or ambiguous Babylonian numbers. Euler published 800 papers in his lifetime, contributing enormously to the world's mathematical knowledge, but just as significantly, he introduced notation that is now commonplace in mathematics, like the notation f(x) for a function of x. He also introduced the symbols e and i
Euler published the first result in what later became Graph Theory, with this theorem that any graph which contains at most two vertices of odd degree will always contain an Eulerian path, which is a path that traverses each undirected edge exactly once. This result may have been inspired by the famous Konigsburg Bridge Problem, in which the goal is to see if you can find a path that crosses every bridge in Konigsburg without ever going over the same bridge twice. Euler's theorem shows that this is not possible, because in the graph representing Konigsburg, four vertices have degree three.
Euler discovered the closed form for the Fibonacci numbers, based on the Golden Ratio, a mathematical constant that, like e, i , and pi, seems to turn up in the strangest places. Like Gauss would become nearly a century later, Euler was an exceptional mental calculator and used this to his advantage, disproving Fermat's conjecture that all numbers of the form 2^2^n+1 are prime by factoring 2^32+1.
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