The transcendence of e (part 3)

This is the third part in a 3-part blog post in which we prove that is transcendental. Three-step proof that is transcendental Step 1 Step 2 Step 3 Recall that in step 1 and step 2 we proved that for any prime sufficiently large and that is a nonzero integer. In this step we will…

The transcendence of e (part 2)

This is the second part in a 3-part blog post in which we prove that is transcendental. Three-step proof that is transcendental Step 1 Step 2 Step 3 Recall that in step 1 we proved the following lemma. Lemma 1. Suppose is a root of the polynomial . Let be a polynomial and . Then…

The transcendence of e

A real number is called algebraic if it is the root of a polynomial with integer coefficients. Examples of algebraic numbers are (it is the root of ), (), the golden ratio (), and the single real root of the quintic polynomial (which cannot be expressed with radicals). A real number that is not algebraic…