A Curious Class of Irrational Numbers

Simpler than 𝜋 or e and even simpler than √2

Some logs. Photo by Meritt Thomas on Unsplash

Introduction

A rational number is little more than a real number that’s a fraction, a ratio: 3, 1/2, -1/5 and even 17/12 are all perfectly good examples. When being introduced to serious mathematics, one must inevitably engage with the concept of mathematical proof and that traditionally means seeing the proof that √2 is irrational, i.e. a real number that’s not a rational number, before almost anything else. Several well-known mathematical constants like 𝜋 and Napier’s constant e provide further examples, but the proofs that these things are irrational tend to be somewhat involved—often requiring fiddly, complicated integrals and the √2 proof being a conceptually awkward descent argument by contradiction, at least if you’re new to this stuff. In the opinion of your author, there are other examples that are even easier to deal with and behave in a much more interesting way. But to introduce these numbers and their properties, we need to have a quick chat about logarithms.

Logarithms

Upon first meeting logarithms, students are often scared and unsettled by these strange-looking and unfamiliar functions, but they are much less mysterious than they seem at first. Moreover, for centuries they were a cornerstone of scientific computation, with even the humble slide rule fundamentally depending on them. Indeed, it was only the advent of cheap and widely available electronic calculators in the 1970s that rendered them useless and even then, by that time, they had already fundamentally embedded themselves into pure mathematics: arguably, the whole of the complex variable theory is nothing more than investigating the consequences of the non-existence of a well-defined and well-behaved logarithm on the complex plane.

So what are these unruly beasts? Recall that “2 raised to the power of 3 is 8” is the statement that 2 multiplied by itself 3 times is 8, i.e. in more compact notation,

Rearranging this slightly, we can phrase this differently as saying “the logarithm of 8 in base 2 is 3”, i.e. in more compact notation,

It’s easy to discern what we mean more generally by logₘ(n). The basic rules that are easy to understand in terms of powers immediately give us many of the standard common properties of logarithms. For example, it’s easy to see that

and rearranging this in terms of logarithms tells us that

Some Irrational Numbers

Suppose m and n are coprime, i.e. have no prime divisors in common. Then consider the number logₘ(n). This number is irrational and the proof of this fact is really simple: to make matters more concrete, let’s focus on the case of m=2 and n=3.

Suppose, for a contradiction, that log₂(3)=x/y for some whole numbers x and y. Then we have that

But this contradicts The Fundamental Theorem of Arithmetic. It really is as short, sweet and simple as that. This argument clearly also works just as well for any coprime m and n. Wow!

Sums

Upon meeting the classic proof that √2 is irrational, a standard exercise is to adapt the proof to show that √2+√3 is also irrational. This involves squaring the number and rearranging the result to turn it into the problem of proving that √6 is irrational (if you’ve not seen this before, I strongly recommend you pause, try this for yourself and then come back). In other words, it’s just like the proof that √2 is irrational but with additional ‘preprocessing steps’ to make it more convoluted.

Our class of ‘logarithmic irrationals’ (if this class of numbers has a better name, then your author is unaware of it), logₘ(n) for any coprime m and n, behaves much more nicely when taking sums.

We claim that, for example, the number log₂(3)+log₂(5) is also irrational. This is almost immediately true since our earlier recollection of the basic properties of logarithms tells us that

which, since 15 is just as coprime to 2 as 3 and 5 are, is already of the correct form to be irrational. No extra work needed. Mic drop…

Transcendental Numbers

Of course we can take logarithms of numbers other than integers too. Might these also give us further examples of irrational numbers?

Recall that a number is transcendental if it is not a root of a polynomial with integer coefficients. Note in particular that no rational number is transcendental: a/b is a root of the polynomial au-b. Nor is √2 transcendental: it’s a root of the polynomial u²-2. Numbers that are transcendental include 𝜋 and e, though the proofs of these facts are even harder than the proofs that they’re irrational.

Given a transcendental number t, it’s almost certainly the case that log₂(t) is also transcendental (though I have no idea how you would prove this); these quite easily give us a fresh supply of irrational numbers and again with a really elegantly straightforward proof. Suppose, for a contradiction, that log₂(t)=x/y for some integers x and y. Then

which immediately implies that t is a root of the polynomial uʸ-2ˣ. Since this is a polynomial with integer coefficients, we have a contradiction. So yes, numbers like log₂(𝜋), log₂(e) and log₃(𝜋) are further examples of irrational numbers and yes, our observations about sums from earlier also immediately adapt to give us, for example, numbers like log₂(𝜋)+log₂(3) also being irrational. Pretty cool, huh?

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