Mathematical Stories 3 – Irrationality and Infinity

The Ancient Greek Pythagoreans discovered the existence of irrational numbers in the fifth century BCE, to the legendary demise of one of their number, Hippasus, and were perturbed by them since they contradicted the firmly-held belief that everything related to number and geometry came back to natural numbers and their ratios (rational numbers).

The Greeks were rather late to the party, though. It’s thought that Indian mathematicians were thinking about irrational numbers a whole three hundred years earlier – Manava (c. 720 BCE) thought that certain square roots could not be determined – but, as with many things in the history of mathematics, it is not always clear where, or from whom, an idea originated. Much like any academic history, mathematics is a mix of human thought across time and space.

Irrational numbers remained a constant part of mathematics, relatively unstudied (if that’s a word) until the eighteenth century, which saw a renewed interest in their scholarship with the rise of number theory and mathematicians such as Euler. It was Euler who, in the mid eighteenth century, defined the “transcendental numbers” – one of two mutually exclusive subcategories of numbers, the other being the algebraic numbers. Algebraic numbers are those that can be the solution to a polynomial equation with rational coefficients, transcendental numbers are those that cannot.

This means that any number that solves a linear, quadratic, cubic, quartic, … equation (where the coefficients are rational numbers) is algebraic. All rational numbers are algebraic – I could come up with an equation along the lines of 0.7x = 1 for any rational number – and all roots of rational numbers are algebraic (take a moment to convince yourself why) but there are plenty of other irrational (and complex) numbers that are not. These are the transcendental numbers, and include such heavyweights as π and e. People conjectured the transcendence of these numbers but it wasn’t until as late as 1844 that Liouville actually proved that such a category definitely exists. Liouville’s constant, his example of a transcendental number, is the number 0.1100010000000000000000010… which has a 1 in each position after the decimal point that is a factorial (the 1!th position, the 2!th position, etc.) and 0s everywhere else.

What became even more interesting, in the nineteenth century, was work on the infinity of sets of numbers. We had been thinking about infinity since Zeno of Elea posed his paradoxes in the fifth century BCE but infinity had largely remained the domain of the theologians and philosophers. It was Georg Cantor, in the 1870s, who established a formal mathematical principle of infinity and different types of infinity.

The natural and rational numbers, any infinite subset of these numbers, and the algebraic numbers, are countably infinite. This means, for instance, that there are as many even numbers as there are natural numbers – their infinity is the same size. There are also as many rational numbers as there are even numbers. Such a counter-intuitive idea was quite ugly to many in the establishment of the time, sometimes rejected as nonsense. Poincaré called Cantor’s work a “grave disease” on mathematics and some theologians rejected the ideas as anti-Christian, challenging the infinity of God.

Cantor demonstrated that the irrational numbers, and hence the real numbers, were uncountably infinite. You might think of this as a different size of infinity to the rational numbers, we say the two sets have different cardinality. Within this idea comes the mind-blowing result that the size of any interval on the real numbers is the same as the size of the whole set of real numbers – there are as many real numbers between 0 and 1 as there are between 0 and 1000000, or in the whole set of real numbers. You can see why these ideas shook people to the core, but at the same time their hostility most probably shook Cantor, who suffered from depression in the last thirty years of his life.

The irrational numbers and infinity form a thread woven through the timeline of mathematics which leads us to some of the most abstract of mathematical thought and some of the most interesting philosophical discussions. How do you broach infinity with your students?

The ideas here form part of a book I have written for Oxford University Press, called “How to Enhance Your Maths Subject Knowledge: Number and Algebra for Secondary Teachers” which should be available at the end of June and which you can order here or here.

This post follows from two earlier ones, which you can find here:

Mathematical Stories

Mathematical Stories 2 (or Shakespeare never saw a decimal point)

Mathematical Stories 4 – An historical and international endeavour



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