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Hello, I'm interested in mathematics. Wrote the Cambridge mathematician, G.H. Hardy, only as a creative art. Hard to revered pure maths and scorned his application to the real world. In great mathematics, he wrote in the same book published in 1941, there is a very high degree of unexpectedness.

Yet what Hardney himself never expected was that in the decade since he wrote those words, pure mathematics would yield such an array of unintended consequences. Not least, through the wartime work of his Cambridge colleague Alan Turing, whose experiments with prime numbers led first to his crucial role in beating Hitler's codes and then to the birth of the computer.

Yet Hardney shouldn't have been surprised. Pure maths was making itself useful long before the struggle against the Nazis. In 1801, centuries-old work on probability helped predict the path of an asteroid, then led on to the birth of population statistics, and the discovery of non-uclidean geometry meanwhile was vital to Einstein's theory of relativity.

To explore mathematics, how it generates all these unintended consequences, I'm joined by Colbert-Roney-Dougall, lecturer in pure mathematics at the University of St. Andrews, John Barrow, Professor of Mathematical Sciences at the University of Cambridge and Professor of Geometry at Gresham College, London, and Marcus Jusottoi, Charles Cimerni, Professor of the Public Understanding of Science and Professor of Mathematics at the University of Oxford.

Colbert-Roney-Dougall, can you start off by giving us an idea of the impact of these unintended consequences? Let's take cubic equations. The work on cubic equations, how did they lead 400 years later rather dramatically to the electric chair?

Okay, so we're going to start in 16th century Italy, and I'd better tell you what a cubic equation is. So a cubic equation is something like, I'm thinking of a number, I'm going to call it x, and I want so that x times x times x is the same thing as x.

x plus 6. Now I've picked that as an example because I can do it in my head and the answer there would be 2, but in general these things are much harder to solve.

Now this had been a long-standing problem for thousands of years, people had known how to solve quadratic equations x times x, but 16th century Italy, they finally discovered a formula for solving cubic equations, a bit like the quadratic formula that you might learn at school, but more complicated.

Now Jorolama Cardano published this formula and ran into difficulties when he was trying to solve certain cubic equations, he knew that the answer was something straightforward like 4.

But part way through his calculations he was getting strange numbers such as the square root of minus 121.

Now a positive number times a positive number is positive and a negative number times a negative number is positive, so there's no such thing as the square root of minus 121.

But he took a deep breath and said, well, let's just pretend that there is, and this was the birth of what's called complex numbers.

If he pretended that there is, then when he finally finished his calculation, all of those funny numbers, the square roots of negative numbers, the complex numbers, cancelled out and he was left with a perfectly valid number.

Now let's jump forward to Victorian times and electricity.

So the first kind of electricity which was widely in use is direct current DC, that's the sort we get out of batteries now.

And Thomas Edison was going around electrifying America and building power stations and connecting it to homes and factories and people were bringing electricity in.

There was a later kind of electricity discovered called alternating current where the direction of the electricity changes direction all the time, hence the alternating.

Now the mathematics of alternating current is far more complicated because of this change of direction, but given Victorian technology, it was possible to transmit alternating current over far greater distances than direct current without wasting it through resistance.

So Tesla and Westinghouse were strongly arguing that America ought to adopt alternating current and so should the rest of the world.

Edison was already wedded to direct current, had built a lot of the things, and so there came a period called the current wars, when Edison basically was running a PR campaign in favor of direct current and was trying to convince the world that alternating current was far more dangerous, which on some level it is.

Now Edison wasn't particularly pro-the-death penalty, but he so wanted to win these current wars, that he got his employees to design, build and later get the state of New York to operate the world's first electric chair, running on alternating current to convince everybody it was dangerous.

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