Summary of The Proof
This is Joseph Fourier's proof by contradiction. Initially e is assumed to be a rational number of the form a/b. We then analyze a blown-up difference x of the series representing e and its strictly smaller b th partial sum, which approximates the limiting value e. By choosing the magnifying factor to be the factorial of b, the fraction a/b and the b th partial sum are turned into integers, hence x must be a positive integer. However, the fast convergence of the series representation implies that the magnified approximation error x is still strictly smaller than 1. From this contradiction we deduce that e is irrational.
Read more about this topic: Proof That e Is Irrational
Famous quotes containing the words summary and/or proof:
“I have simplified my politics into an utter detestation of all existing governments; and, as it is the shortest and most agreeable and summary feeling imaginable, the first moment of an universal republic would convert me into an advocate for single and uncontradicted despotism. The fact is, riches are power, and poverty is slavery all over the earth, and one sort of establishment is no better, nor worse, for a people than another.”
—George Gordon Noel Byron (1788–1824)
“It comes to pass oft that a terrible oath, with a swaggering accent sharply twanged off, gives manhood more approbation than ever proof itself would have earned him.”
—William Shakespeare (1564–1616)