Are there more irrational numbers than rational numbers, or more rational numbers than irrational numbers? Well, there are infinitely many of both, so the question doesn't make sense. It turns out, however, that the set of rational numbers is infinite in a very different way from the set of irrational numbers.

Not necessarily, when it comes to money.

How to make a hard problem easy by changing the way you look at it.

A triangle has many centres....

What happens when you shrink infinity to a point? You get a sphere!

Number theory is famous for problems that everyone can understand and that are easy to express, but that are fiendishly difficult to prove. Here are some of our favourites.

Did you learn at school that the angles in a triangle always add up to 180 degrees? If yes then your teacher was wrong. Find out why here.

A quick and easy way of adding using handshakes.

Is there a perfect voting system? In the 1950s the economist Kenneth Arrow asked himself this question and found that the answer is no, at least in the setting he imagined.

Here's a well-known conundrum: suppose I need to buy a book from a shop that costs £7. I haven't got any money, so I borrow £5 from my brother and £5 from my sister. I buy the book and get £3 change. I give £1 back to each my brother and sister and I keep the remaining £1. I now owe each of them £4 and I have £1, giving £9 in total. But I borrowed £10. Where's the missing pound?

We've been dabbling a lot in the mysterious world of quantum physics lately, so to get back down to Earth we thought we'd bring you reminder of good old classical physics.

How to take square roots of negative numbers and invent a whole new family of numbers in the process.