What's the most useless number?

Chris Smith asked mathematician Bobby Seagull, from the University of Cambridge. Surely all numbers are precious to a mathematician?

Bobby - Asking a mathematician what is the useless number is always the reverse of asking a parent to select their favourite child but if we must, we must give an answer. Let’s take a time machine back to 16th century Italy. Let’s go to Lombardy and let’s meet Gerolamo Cardono. This mathematician was a polymath; he actually did biology, physics, chemistry, philosophy, writing. He even had a dabbling in gambling. And he was looking at solutions to cubic equations.

For our listeners: we have linear equations, like the straight line like my rap: y = mx = c. Then we’ve got our quadratics: that’s x squared - looks like a smiley face. And then the cubics where it’s an x cubed type graph. He was looking at solutions for these. He came across some solutions which were imaginary. An example that he gave was: what happens when you expand (5 + sqrt(-15)) and you multiply that by (5 - sqrt(-15))

If you can mentally picture that, you multiply the 5 and the 5, so a double bracket expansion boys and girls, you get 25.

Chris - Giles has done it already!

Bobby - So you get the 25 there so we get a minus of 5 lots of root minus 15. We get the opposite a plus 5 lots of root minus 15 so they cancel out. At the end we get minus lots of the root of minus 15 squared. So now we’ve got 25, we’ve got a minus, minus 15, so that gives us 25 plus 15 gives us 40. So what Cardono said was, in italian… I’ll do it in English “Thus far does arithmetical subtlety go of which this, the extremes as I have said, so subtle, that it is useless.”  So he thought that the minus squared of 15 imaginary number was useless. But interestingly, over time, imaginary numbers became very useful.

Who’s planning on going on a holiday this summer?

Chris - Pretty much everyone I think.

Bobby - Taking a plane I guess? Actually, air traffic relies on radar, and radar uses complex computations where they distinguish stationary objects and moving ones and for this they use imaginary numbers because it makes the calculations a lot more manageable than if you just had straightforward, standard, real numbers. So there you go, imaginary numbers are real and not as useless as Cardono thought.

Chris - So there are useless numbers but they’re not really useless?

Bobby - Exactly.