Math Forum: Proof that e is Irrational

A Proof that e is Irrational

Table of Contents:

The Bridges of Konigsberg
· Euler's Solution
· Solution, problem 1
· Solution, problem 2
· Solution, problem 4
· Solution, problem 5

The Value of Pi
· A Chronological Table of Values
· Squaring the Circle

Prime Numbers
· Finding Prime Numbers

Famous Paradoxes
· Zeno's Paradox
· Cantor's Infinities
· Cantor's Infinities, Page 2

The Problem of Points
· Pascal's Generalization
· Summary and Problems
· Solution, Problem 1
· Solution, Problem 2

Proof of the Pythagorean Theorem

Proof that e is Irrational

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e is one of those special numbers in mathematics, like pi, that keeps showing up in all kinds of important places. For example, in Calculus, the function f(x) = c(e^x) for any constant c is the one function (aside from the zero function) that is its own derivative. It is the base of the natural logarithm, ln, and it is equal to the limit of (1 + 1/n)ⁿ as n goes to infinity. In the proof below, we use the fact that e is the sum of the series of inverted factorials.

Like Pi, e is an irrational number. It is interesting that these two constants that have been so vital to the development of mathematics cannot be expressed easily in our number system. For, if we define an irrational number as a number that cannot be represented in the form p/q, where p and q are relatively prime integers, we can prove fairly easily that e is irrational.

The following is a well known proof, due to Joseph Fourier, that e is irrational.

to a Proof of the Pythagorean Theorem
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August, 1998