Mathematicians in ancient times were shocked to discover there are numbers which are not the ratio of two integers and legend has it that Hippasus was drowned for suggesting such a thing might exist. These days, we are less perturbed by such irrational notions, and use them every day. We all know that \(\sqrt{2}\;\) is irrational and you’ve almost surely seen a proof before (there are quite a few), but here’s a neat variation you might not have seen which involves a little bit of analysis as well as arithmetic.
Suppose \(\;\sqrt{2}=p/q\;\) for some \(\;p\), \(q\in\mathbb{Z}\;\) with \(q>0\;\) and consider the sequence \(\;a_n=q\left(\sqrt{2}-1\right)^n\).
Notice that \(\;a_0=q\), \(\;a_1=p-q\;\) and \(\;a_n=a_{n-2}-2a_{n-1}\;\) for \(n\geq 2\). In particular, \(\;a_n\in\mathbb{Z}\;\) for all \(\;n\geq 0\).
Since \(\;0<\sqrt{2}-1<1\;\), we have \(\;a_n\rightarrow 0\;\) as \(\;n\rightarrow\infty\;\) and so \(\;0<a_n<1\;\) for sufficiently large \(n\).
Interesting fact: there are no integers strictly between \(0\) and \(1\). This contradiction implies \(p\) and \(q\) can’t exist.
This last seemingly trivial fact turns out to be immensely useful. The same trick (construct an impossible integer between \(0\) and \(1\)) and related ideas can be used to show that many other naturally occurring numbers are irrational, such as \[\sqrt{2}+\sqrt{3},\quad \sqrt[22]{23},\quad e,\quad \pi,\quad \log{2},\quad \sin{3},\quad \zeta(3)=\sum_{n=1}^\infty \frac{1}{n^3},\quad...\] Some of these are quite easy, but others not so much. Euler first demonstrated that \(e\) is irrational in 1737 but it was only in 1978 that Apéry found a wonderfully bizarre proof that \(\zeta(3)\) is irrational. To this day, no-one can say for sure whether \(\zeta(5)=\sum_{n=1}^\infty 1/n^5\) is rational or not.
A key aspect to the ideas involved is: how well can we approximate a given number with rational numbers? This is the subject of Diophantine approximation which is intimately connected to the theory of continued fractions that you maybe recall from Elementary Number Theory. In fact, Euler’s original proof that \(e\) is irrational was a consequence of finding a continued fraction for \(e\) explicitly.
In the 19th century, there was a further discovery that some numbers are not even algebraic. An algebraic number \(x\) is a root of a non-zero polynomial with integer coefficients \[a_nx^n+a_{n-1}x^{n-1}+...+a_1x+a_0=0\quad\text{where $a_i\in\mathbb{Z}$}\] and non-algebraic numbers are called transcendental. Cantor shocked mathematicians again with his diagonal argument, showing in a precise sense that almost all real numbers are transcendental. This argument doesn’t tell us anything about a given number, however, but mathematicians were already beginning to show that many well-known (and less well-known) mathematical constants are transcendental, such as \[\sum_{n=0}^{\infty}\frac{1}{10^{n!}},\quad e,\quad \pi,\quad \log{2},\quad 2^{\sqrt{2}},\quad e^{\pi},\quad \sum_{n=0}^{\infty}\frac{1}{10^{2^n}},\quad 0.12345678910111213141516...,\quad... \] And the general idea often boils down to the same trick as in our \(\sqrt{2}\) proof above (though in much, much more involved ways) - suppose your number is algebraic and construct an impossible integer between \(0\) and \(1\).
In this project, we’ll investigate irrational and transcendental numbers. What properties do they have? How do we show that some of our favourite constants are irrational, or indeed transcendental? We’ll take a tour of this fascinating topic with a view towards discussing various general irrationality and transcendence results such as Dirichlet’s Approximation Theorem, Liouville’s Theorem, Irrationality measure, the Lindemann-Weierstrass Theorem and the solution to Hilbert’s 7th Problem.
Elementary Number Theory II is necessary. There will be a lot of playing around with sequences, series, integrals and inequalities if that’s your kind of thing. Number Theory III, whilst not essential, would be useful - that module concerns algebraic numbers and knowing things about these tells us things about irrational and transcendental numbers.
Many standard number theory textbooks cover elements of irrationality, transcendence and Diophantine approximation and some focus specifically on our intended topic. For instance
Irrationality and Transcendence in Number Theory (library link) by David Angell. Very recently published and covers the topic in an approachable way.
Diophantine Analysis (library link) by Jorn Steuding. Another well-written modern textbook on the topic.
Irrational numbers (library link) by Ivan Niven. A classic in the subject. Quite old-fashioned and very short, but covers a huge amount of ground in 160 small pages.
An Introduction to the Theory of Numbers (library link) by G. H. Hardy and E. M. Wright. One of the sacred texts in number theory, with a significant amount concerning Diophantine approximation.
Roots to research: a vertical development of mathematical problems (library link) by J. Sally and P. Sally. Chapter 4 contains a nice overview of things we could cover.
There are also some books devoted specifically to the deeper topic of transcendental number theory, generally going much further than we ever could. The following take different approaches.
Making transcendence transparent : an intuitive approach to classical transcendental number theory (library link) by E. B. Burger and R. Tubbs. This book is remarkable in the way it goes to great lengths to describe the ideas behind the proofs.
Transcendental number theory (library link) by Alan Baker. This is much more terse and technical. Burger and Tubbs take 70 pages to prove the transcendence of \(e\), \(\pi\) and the Lindemann-Weierstrass theorem, whereas Baker has already done them by page 8.
Auxiliary polynomials in number theory (library link) by David Masser. This covers the more general idea of working with auxiliary polynomials that can be used to construct our impossible integers. It is entertainingly written, contains a wealth of extraordinary calculations, and gets very deep.
If you have any questions, feel free to drop me a message at daniel.evans@durham.ac.uk