## Question

What's the most useless number?

## Answer

**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.

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