Some people seem to think anything involving numbers must be mathematical, which is a lot more generous than most mathematicians would ask them to be. But numbers stations — an eerie sound from the Cold War era — do have a little bit of real mathematics in them.

Numbers stations are shortwave AM radio frequencies on which one can hear apparently inexplicable broadcasts. Along with a variety of bizarre elements they usually include a human voice reading out long lists of numbers. If you’d like to listen to some but don’t have a shortwave radio receiver, check out the recordings archived by The Conet Project.

Nobody really knows what these stations are or why they exist, but the general assumption is that they’re used by diplomatic, military or intelligence agencies as a way to pass on coded messages.

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Alongside all the historical, political and hauntological resonances of these sounds, two mathematical things spring to mind. One is the use of encryption. I was recently amused to read Simon Jenkins, in his latest missive against maths education, recruiting the number theorist G H Hardy to his side of the argument. Hardy, he says, understood that pure maths has no applications, and therefore is not worth studying.

Hardy was well-known for being quotable, but if you only know the quotes you miss the context. Hardy’s celebration of his subject’s impracticality was a response to the ever-growing use of maths and science in the defence industry; something he disapproved of on ethical grounds. In fact, the field of maths to which he dedicated his life would indeed find its first applications in military encryption of the kind we hear on the numbers stations.

Number theory went on to become one of the foundations of the digital world we now live in, making possible everything from mobile phones to electronic payments. You can only believe it has no applications if you don’t know what it is, or have carelessly forgotten the history of the last century. This would probably not have surprised Hardy, who observed that pure maths is much more useful than so-called “applied” maths because it deals in general techniques rather than solving particular problems.

The second mathematical theme is that what we often hear on a numbers station is an orderly sequence of numbers, and sequences are of fundamental importance in many areas of maths. We meet them a bit in GCSE maths, but only the cases of arithmetic and geometric progressions. Some of the most important sequences are those that, like the one on the left, seem to “settle down” and get closer and closer to a single value as they go on. In this case we say the sequence converges to that value even if it never quite gets there.

Convergent sequences are part of the foundation of calculus, the language of much of science and “pure” mathematics. At least, they perform that role in traditional foundations: there are dissenters who prefer other ways of doing things, as we briefly discuss in our City Lit course on the history and philosophy of calculus. Here’s a video that introduces convergent sequences:

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The sequences broadcast by numbers stations can’t do this — unless they end up being the same number repeated over and over — because they involve whole numbers. Convergent sequences require smaller and smaller differences, which means they needs fractions, or as mathematicians sometimes call them “rational numbers”. In fact, in building calculus from the ground up we usually define a new number system, the so-called “real numbers”, in which every number is constructed as an infinite collection of infinite sequences, each of which converges to that number.

Whether you think the philosophical objections to this approach are justified or not, this is certainly a rather wild idea of what a number might be. The intellectual rigour of the calculus is a high-stakes question, since it’s used in everything from designing aircraft to macroeconomic modelling. Learning to use the tools isn’t good enough: we also need to be able to think critically about them.