# Maths in a minute

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: 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.

Maths in a minute: St Paul's dome

The dome of St Paul's, rising elegantly above London since the cathedral was rebuilt late in the seventeenth century, hides an intriguing early example of the interplay between maths and architecture.

Maths in a minute: The prisoner's dilemma

Sometimes people are nasty when it would have been better to be nice.

Maths in a minute - levers

Kneeling in the mud by a country road on a cold drizzly day, I finally appreciated the wonder that is a lever. I was trying to change a flat tyre and even jumping on the end of the wheel wrench wouldn't budge the wheel nuts. But when the AA arrived they undid them with ease, thanks to a wheel wrench that was three times the size of mine. There you have it ... size really does matter!

Maths in a minute — triangles

Ah the humble triangle. This simple shape is one of the first we ever learn. But perhaps you didn't realise just how important triangles are...

Maths in a minute: The fundamental group

Topologists famously think that a doughnut is the same as a coffee cup because one can be deformed into the other without tearing or cutting. In other words, topology doesn't care about exact measurements of quantities like lengths, angles and areas. Instead, it looks only at the overall shape of an object, considering two objects to be the same as long as you can morph one into the other without breaking it. But how do you work with such a slippery concept? One useful tool is what's called the fundamental group of a shape.

Maths in a minute: negative numbers

Negative numbers are easy to imagine if you think of the number line as
a giant thermometer which includes sub-zero temperatures. This makes
addition and subtraction easy, as you just move up or down the number
line by the according amount.

• Want facts and want them fast? Our Maths in a minute series explores key mathematical concepts in just a few words.