Recapitulation

We have reached our final post in this series on the number e. In the first post of the series, we introduced e and looked at the reasons for which it is the base of the natural logarithm and the exponential function. In the second post of the series, we used various techniques to determine that e lies between 2 and 3, a fact that will be important in today’s post. In the previous post, we considered three examples of infinite series for e to show that more advanced mathematics does not guarantee quicker convergence to the value of e. In today’s post, we will demonstrate that e is an irrational number.

Reductio ad Absurdum using Bounds

Since e lies between 2 and 3, e cannot be an integer. We will now use reductio ad absurdum, a method we introduced in another post. So we will begin by assuming that e is a rational number. Hence, suppose

Since e is not an integer, it follows that q cannot be 1. Hence, q>1. Now, we also know that 

Combining the two we get

If we multiply this equation by q! we will get

Distributing the q! into the parentheses on the right side we get

Now the left side, being the product of natural numbers, is a natural number. On the right side, the terms in green are all integers since the numerator q! is necessarily a multiple of the denominators. This means that, for the equation to hold, the terms on the right must add up to an integer. Let us designate this sum as R. Then

Now, since q > 1, it follows that

Taking the reciprocals of each term we get

This means that

Hence, we can conclude that

However, we have seen, when we obtained the lower and upper bounds for e, that the infinite series

Hence, without the leading 1, the sum must be 1. This allows us to conclude that R < 1. However, since all terms in R involve the products and quotients of positive integers, it must follow that R is positive. However, there is no integer between 0 and 1, which means that R cannot be an integer. This means that the terms in red in the equation

do not give an integral sum, which is something that was required for the equation to hold. Since we reached this requirement when we assumed that e is rational, we have reached something absurd, namely that this is possible only if we can find an integer between 0 and 1. This concludes the proof by reductio ad absurdum.

Reductio ad Absurdum using Partial Fractions

I would like to contribute another proof using reductio ad absurdum that I have not seen anywhere else. This uses the concepts surrounding partial fractions. While partial fractions is normally used in the context of algebra, I’d like to explore its implications in the context of arithmetic.

Now, it is clear that

What sets apart the expression in green from the expressions in red is that the one in green only has prime numbers or powers of prime numbers in the denominator. This is not true in the case of the expressions in red, since 6 and 12 do not satisfy the criterion.

So suppose we restrict the splitting of a given fraction into its arithmetic partial fractions where the denominators consist only of prime numbers or the powers of prime numbers. Suppose then that we have a number N such that

where

are distinct prime numbers and

Now any divisor of N can have a particular prime p_k appear from 0 to a_k times. For example, the divisors of 12 are 1, 2, 3, 4, 6, and 12. The prime factorization of 12 is 

Hence, the prime factorization of any divisor of 12 can have 2⁰, as in 1 and 3, or 2¹, as in 2 and 6, or 2², as in 4 and 12. Or it could have 3⁰ as in 1, 2, and 4, or 3¹, as in 3, 6, and 12. In other words, any prime p_k, which appears as a power a_k in the prime factorization of N, can appear as a power from 0 to a_k in divisors of N. Hence, there are a_k ways in which p_k can appear in divisors of N, excluding 1. Hence, the number of divisors of N that are prime numbers or powers of prime numbers is

This means that, if we are able to write 

then splitting it into its arithmetic partial fractions would involve at most P terms. However, P is a finite natural number.

Yet, from the expression

it is clear that e is expressed as the sum of an infinite series. But in the denominators we have all the natural numbers without exception. And we know that there are infinitely many primes in the set of natural numbers. This means that the infinite series for e includes terms that involve infinitely many prime numbers. For example, the term that contains p_k! in the denominator, when split into its arithmetic partial fractions, will contain a finite number of terms that involve all prime numbers and powers of prime numbers that are less than p_k.

However, since there is no limit to p_k, there is no way to combine all the terms into a finite series, such as is required by the arithmetic partial fraction expansion of

Now, we know that 

The inherent pattern involved in the denominators allows us a concise way of combining the infinite terms to give the resulting sum as 2. In general, if we have an infinite set of fractions where the denominators form a discernible pattern, we might be able to combine the terms to give us a determined sum. If the primes themselves appear after some discernible pattern, then too we would possibly be able to combine the infinite terms to yield a determined sum. Since the primes do not appear in any discernible pattern, such a combination is impossible, even in theory.

In other words, we have reached a contradiction, where infinitely many fractions with distinct denominators involving infinitely many unpatterned primes and their powers are combined to produce the sum of a finite number of fractions, therefore yielding a rational number. So once again, by reductio ad absurdum the result is proved and we conclude that e is irrational.

Wrapping Up

We have devoted four posts to the study of the number e. The impetus for this study came from the opening post of this blog in which I introduced Euler’s Identity, stating that it was an example of beauty in mathematics. During the course of these four posts, we have looked at the definition of e. We also saw how e is related to the ideas of compound interest and, therefore, to the ideas of growth and decay that occur in natural systems. We were able, then, to see why e is the ‘natural’ base for logarithms and exponential functions. 

Then we obtained a lower and an upper bound for the value of e. Along the way we introduced some key ideas, notable among which were the limiting process, infinite geometric sequences, and the binomial expansion. This allowed us to obtain an infinite series for e. 

Following this we explored three infinite series for e to determine the speed at which they converge to the value of e. We saw that more advanced mathematics does not guarantee speedier results. This allowed us to conclude that we need to be wary when claims are made that something is based on more rigorous mathematics. 

Finally, we proved that e is irrational. We did this in two ways, both involving reductio ad absurdum, a mathematical strategy for proofs that we had introduced earlier. I introduced a proof that I have not seen elsewhere, though this may just be an indication of my ignorance rather than my ingenuity.

This does not exhaust the study of e. By no means! But this does conclude our exploration at this stage. It has been fruitful and insightful for me. I hope you can say the same.