Want facts and want them fast? Our *Maths in a minute* series explores key mathematical concepts in just a few words. From symmetry to Euclid's axioms, and from binary numbers to the prosecutor's fallacy, learn some maths without too much effort.

Maths in a minute: The Riemann sphere

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

Maths in a minute: Number mysteries

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.

Maths in a minute: Not always 180

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.

Maths in a minute: Arrow's theorem

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.

Maths in a minute: The missing pound

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?

Maths in a minute: Newton's laws of motion

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.

Maths in a minute: Complex numbers

Solving equations often involves taking square roots of numbers and if you're not careful you might accidentally take a square root of something that's negative. That isn't allowed of course, but if you hold your breath and just carry on, then you might eventually square the illegal entity again and end up with a negative number that's a perfectly valid solution to your equation.

Maths in a minute: Take it to the limit

Sequences of numbers can have limits. For example, the sequence 1, 1/2, 1/3, 1/4, ... has the limit 0 and the sequence 0, 1/2, 2/3, 3/4, 4/5, ... has the limit 1. But not all number sequences behave so nicely. Can we still discern some sort of limiting behaviour?

Maths in a minute: Countable infinitiesAn infinite set is called *countable* if you can count it. In
other words, it's called countable if you can put its members into
one-to-one correspondence with the natural numbers 1, 2, 3, ... .

Maths in a minute: Clever sums

How would you go about adding up all the integers from 1 to 100? Tap them into a calculator? Write a little computer code? Or look up the general formula for summing integers?

Maths in a minute: Regression to the mean

Sometimes you just can't argue with the evidence. If a large sample of very ill people got better after dancing naked at full moon, then surely the dance works. But hang on a second. Before you jump to conclusions, you need to rule out a statistical phenomenon called regression to the mean.