Mathematicians in ancient times were shocked to discover there are numbers which are not the ratio of two integers and legend has it that the Greek philosopher 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). 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 \[\sqrt{26}+\sqrt{27},\quad \sqrt[26]{27},\quad \log_{26}{27},\quad e,\quad \sum_{n=0}^{\infty}\frac{1}{10^{n^2}},\quad \pi,\quad \sin{2},\quad \ln{2},\quad \zeta(3)=\sum_{n=1}^\infty \frac{1}{n^3},\quad...\] 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 the number \(\zeta(5)=\displaystyle\sum_{n=1}^\infty \frac{1}{n^5}\) 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. In fact, Euler’s original proof that \(e\) is irrational was a consequence of finding an explicit continued fraction for \(e\).
Whilst \(\sqrt{2}\) is irrational, it is at least the root of a simple polynomial \(x^2-2\). Generally, an algebraic number is a solution of a polynomial equation with integer coefficients \[a_nx^n+a_{n-1}x^{n-1}+...+a_1x+a_0=0\quad\text{where $a_i\in\mathbb{Z}$ and $a_n\neq 0$.}\] In the 19th century, mathematicians were shocked again to discover that some numbers are not even algebraic. Non-algebraic numbers are called transcendental and Cantor showed using his diagonal argument that, in a precise sense, almost all real numbers are transcendental. Unfortunately, this argument doesn’t tell us anything about specific numbers. However, around the same time, mathematicians were already beginning to show that many well-known (and less well-known) mathematical constants are transcendental, such as \[\sum_{n=0}^{\infty}\frac{1}{10^{n!}},\quad e,\quad \pi,\quad \ln{2},\quad 2^{\sqrt{2}},\quad e^{\pi},\quad \sum_{n=0}^{\infty}\frac{1}{10^{2^n}},\quad 0.12345678910111213141516...,\quad... \] And the general idea often boils down to the same trick as in our \(\sqrt{2}\) proof above (though in much more complicated ways). Suppose your number is algebraic and construct an impossible integer between \(0\) and \(1\).
The group project will involve studying basic concepts and initial examples of the topic. By the end of the group project, we should understand the following.
The concepts of rational vs irrational and algebraic vs transcendental numbers.
Techniques for determining irrationality, including
numbers such as \(\sqrt{2}\) and similar irrational surds,
numbers such as \(e=\displaystyle\sum_{n=0}^{\infty}\frac{1}{n!}\) and \(\displaystyle\sum_{n=0}^{\infty}\frac{1}{10^{n^2}}\) via rapidly converging rational approximations,
numbers such as \(\pi\), \(e^r\) and \(\sin{r}\) (for non-zero \(r\in\mathbb{Q}\)) via integrals and Hermite/Niven’s method,
general irrationality criteria such as Dirichlet’s approximation theorem.
The existence of transcendental numbers and first explicit examples such as
Liouville’s constant \(\displaystyle\sum_{n=0}^{\infty}\frac{1}{10^{n!}}\) via rational approximations and Liouville’s Theorem ,
the constant \(e\) via Hermite’s method.
The project will involve learning through reading, with a focus on underlying theory, mathematical rigour, and development of conceptual understanding. Students will work within their group, demonstrating understanding by exploring results and examples, solving relevant problems, and clearly communicating in both written and oral formats.
The individual project will build on knowledge gained in the group project and explore additional advanced topics. Some examples of topics you will be able to investigate are:
The theory of continued fractions, including quadratic irrationalities, the continued fraction for \(e\), and extensions of Dirichlet’s approximation theorem.
Further Diophantine approximation results, including the concept of irrationality measure and extensions of Liouville’s Theorem.
Evaluating the Riemann zeta function \(\zeta(s)=\displaystyle\sum_{n=1}^{\infty}\frac{1}{n^s}\) for positive even integers \(s\) and the proof that \(\zeta(3)\) is irrational using multiple integrals.
Extending Hermite’s method to prove the transcendence of \(\pi\) and perhaps beyond to very general results such as the Lindemann-Weierstrass Theorem.
The project will involve learning through reading, with a focus on underlying theory, mathematical rigour, and development of conceptual understanding. Students will work individually, demonstrating understanding by exploring results and examples, solving relevant problems, and clearly communicating in both written and oral formats.
Calculus I and Analysis I are necessary prerequisites. There will be a lot of playing around with sequences, series, integrals and inequalities if that’s your kind of thing.
Whilst not essential corequisites, Number Theory III and Galois Theory, Groups and Geometry III could be useful. Those modules both study algebraic numbers and knowing things about those 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. Recently published and covering the topic in an approachable way, this will form the principal source for the group project.
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, including failed attempts at proving things.
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.
If you have any questions, feel free to drop me a message at daniel.evans@durham.ac.uk