Proving the Irrationality of e Using Calculus

Introduction to the Exponential Function

The number e is one of the most important constants in mathematics and appears in various contexts, including exponential growth and decay, differential equations, and calculus. It is defined as the base of the natural logarithm and can be expressed as an infinite series: $$ e^x sum_{n0}^{infty} frac{x^n}{n!} $$ When x 1, this simplifies to the famous series representation of e $$ e sum_{n0}^{infty} frac{1}{n!} $$

Proof by Contradiction

To prove that e is irrational, we use a proof by contradiction. Suppose, for the sake of contradiction, that e is rational. This means for some positive integers p and q, we can express e as the fraction p/q in lowest terms. Consequently, we have

$$ q! cdot left(frac{p}{q}right) q! cdot e sum_{n0}^{q} frac{q!}{n!} quad {text{etc.}} $$

Since q!p/q is an integer, and the terms up to q! are all integers, we need to analyze the remaining terms beyond q!e.

The terms beyond q!e can be written as the infinite series:

$$ sum_{nq 1}^{infty} frac{q!}{n!} frac{1}{q 1} frac{1}{(q 2)(q 1)} frac{1}{(q 3)(q 2)(q 1)} cdots $$

This series is bounded and can be compared to a geometric series:

$$ frac{1}{q 1} left(1 frac{1}{q 2} frac{1}{(q 2)(q 3)} cdots right) $$

The geometric series inside the parentheses converges to a number less than 1, implying that the sum of the series is less than 1. Therefore, the entire expression on the right is not an integer, leading to a contradiction.

Alternative Proof Using Prime Factors

Consider the assumption that e is rational and can be written as a fraction with an integral denominator. Let the largest prime factor of this denominator be p. Since e is calculated from an infinite series with denominators that are primes and larger than p, the fraction for e must have a denominator divisible by all primes greater than p. This is impossible since the denominator is finite, leading to another contradiction.

High School Level Understanding

Alan Baker's book on Transcendental Numbers includes a proof that is accessible to a high school student familiar with Taylor series. By understanding the series expansion and the properties of rational and irrational numbers, one can grasp the proof that e is indeed irrational.

Conclusion

By using proof by contradiction and analyzing the properties of series and prime factors, we have shown that the number e cannot be expressed as a ratio of two integers. Hence, we conclude that e is irrational. This proof not only deepens our understanding of the number e but also illustrates the power of mathematical reasoning.