Here's a strange fact: if you look up some numbers, for example the
numbers in your tax return, population sizes of Chinese provinces, or
the length of the world's rivers, then most likely around 30% of these
numbers start with the digit 1, around 18% start with the digit 2,
12.5% start with a 3, and so on, all the way to 9 (which only heads up
around 5% of the numbers) - the larger the digit, the fewer numbers in
your list start with it. This fact, known as Benford's law, applies to
so many different kinds of data sets that it's often used to detect
fraud. But why does it work?
Well, if the processes that give rise to your list of numbers do
produce a universal distribution of first digits, then this
distribution should apply no matter what units you use. It should work
no matter if you do your tax return in pounds or in euros, or measure
your rivers in metres or miles - it's universal after all. This means
that the distribution of first digits remains the same when you
multiply your numbers by whatever constant you need to change between
units. And it turns out that the only distribution with this property
of scale invariance is precisely the Benford distribution.
As an example, imagine that your first digits are distributed equally
(roughly the same proportion of numbers begin with the digit 1, 2,
3...) – so NOT according to the Benford distribution. Is this
distribution scale invariant? Let's see what happens when we multiply
by 2. All numbers starting with 5, 6, 7, 8, and 9, when multiplied by
2, give a number starting with 1. By contrast, the only way to end up
with a number beginning with, say, 3, is to start out with a number
starting with 1. In other words, the resulting distribution of first
digits, after multiplying by 2, is skewed towards 1. It's not uniform,
so your original distribution is not scale invariant. It's not too
hard to show that in order to be scale invariant, the first digits
have to be distributed in the way stipulated by the Benford
distribution. It's worth noting though that Benford's law only
applies to data sets that are neither too random, nor too constrained:
alas, it doesn't work for lottery numbers.
To find out more about Benford's law read our article Looking out for number one. This news story explores an application of Benford's law to uncover potentially fraudulent
We were delighted when the BBC approached us back in 2005 to ask if some of the Plus posters we had created for schools could be used as set dressing in an episode of Doctor Who with a storyline revolving around maths classrooms. And accordingly, in April 2006, the Plus posters made their TV debut in the Doctor Who episode School Reunion, alongside tenth Doctor David Tennant. The Doctor has infiltrated a school under cover as a science teacher, and discovers an alien plot to use the enhanced intelligence of the school students to solve the Skasis Paradigm, the key to the fundamental secrets of the universe. Solving the Paradigm requires imagination as well as intelligence - and when you're looking for maths students who are both imaginative and intelligent, of course they'll be reading Plus...
The posters which featured in the episode were launched in 2004, funded by EPSRC, in order to showcase the wide range of careers open to people with a background in maths. As Plus readers know know, the huge range of subjects featured in our articles and career interviews is proof of the amazing range of possibilities open to mathematicians : from avalanche researcher to audio software engineer, almost every field you can think of is represented on the Careers with Maths posters. The posters were intended to spread this message and to be a resource for teachers and career advisors dealing with mathematically-minded students.
All the Plus posters
We are extremely proud of the appearance of the Plus posters on Doctor Who, and very grateful to Charles Trevelyan, Plus's graphic designer. The posters were hugely successful: at one point we were receiving over 1000 requests for them a day! Although the posters are long out of print and no longer being distributed you can still download and print out a copy for yourself. With The Doctor's help, maths fame has now spread throughout the TV-viewing world and beyond, reaching intelligent civilisations of all ages and dimensions. But then, if they're intelligent they probably already got the message. Enjoy the 50th anniversary episode tomorrow night, whichever time and relative dimension in space you find yourself!
In the eighteenth century the city we now know as Kaliningrad was
called Königsberg and it was part of Prussia. Like many other great
cities Königsberg was divided by a river, called the Pregel. It contained two
islands and there were seven bridges linking the various land masses. A
puzzle at the time was to find a walk through
the city that crossed every bridge exactly once. Many people claimed
they had found such a walk but when asked to reproduce it no one was able to. In
1735 the mathematician Leonhard Euler explained why: he showed that such a walk didn't exist.
Euler's solution is surprisingly simple — once you look at the
problem in the right way. The trick is to get rid of all
unnecessary information. It doesn't matter what path the walk takes on
the various land masses. It doesn't matter what shape the
land masses are, or what shape the river is, or what shape the bridges are. So you might as well
represent each land mass by a dot and a bridge by a line. You don't have
to be geographically accurate at all: as long as you don't disturb the
connectivity of the dots, which is connected to which, you can distort
your picture in any way you like without changing the problem.
Once you have represented the problem in this way, its features are
much easier to see. After playing around with it for a while you might
notice the following: when you arrive at a dot via a line (enter a
land mass via the bridge), then unless it is the final dot at which your
walk ends, you need to leave it again, by a different line as those are
the rules of the game. That is, any dot that is not
the starting and end-point of your walk needs to have an even number of
lines coming out of it: for every line along which you enter there has
to be one to leave.
For a walk that crosses every line exactly once to be possible, at most two dots can have an odd number of lines coming out of them. In fact there have to be either two odd dots or none at all. In the former case the two correspond to the starting and end points of the walk and in the latter, the starting and end points are the same. In the Königsberg problem, however, all dots have an odd
number of lines coming out of them, so a walk that crosses every
bridge is impossible.
Euler's result marked the beginning of graph theory, the
study of networks made of dots connected by lines. He was also able to
show that if a graph satisfies the condition above, that the number of
dots with an odd number of lines is either zero or two, then there
will always be a path through it that crosses every line exactly
The result also marked the beginning of topology, which
studies shapes only in terms of their connectivity, without taking
note of distances and angles. The London tube map is a great example
of the topological triumph. By distorting distances and angles it
turns what would otherwise be an unintelligible mess into a map that
every tourist can read effortlessly. You can find out more here.
"That's when I thought, 'this could be the moment we all die'. The de-pressurisation alarm went off and this meant that there really was a hole in the station. I felt the pressure fall in my ears. And then it got cold. Really, really cold."
No, it's not a quote from Gravity, but the words of astronaut Michael Foale recalling an incident in 1997 that nearly cost him his life. Foale's story has all the ingredients for a movie: a damaged space station, no ground communications, no power, and only one chance to get it right. The crew used the soyuz's thrusters — their only means of getting back home — to control the station's chaotic spin. Their main tools were their brains and the basic maths of motion. As Foale put it, "You cannot be an effective mission designer, astronaut, flight controller or engineer if you don't know maths."
Read our interview with Foale for some real-life adventure, space and maths, or watch the Gravity trailer below.
How do you balance a cardboard cut-out of a triangle on a pencil? Trial and error is one way, but maths can save you lots of bending down and picking it up. Take the pencil and a ruler and connect the mid-point of each side to the opposite corner. You'll find that the three lines intersect in a single point, which lies exactly a third of the way from the midpoint of each side to the opposite vertex. That point, called the centroid, is the centre of mass of the triangle. If the triangle is made from uniform material, so it's not more lumpy in some places than in others, then the centroid is the unique point on which you can balance it without it tipping over. Amazingly, the centroid would also be the centre of mass of the triangle if its mass was concentrated only at its corners, and evenly divided between them.
Instead of drawing a line from the mid-point of a given side to the opposite corner, you could also draw the line which passes through the mid-point but forms a right angle with the side the mid-point is on. If you do this for each side you again get three lines, and again these meet at a single point, called the circumcentre of the triangle. If you now draw a circle with the circumcentre as its center passing through one of the triangle's corners, you will find that the other two corners of the triangle lie on the circle too! The circumcentre of the triangle is also the centre of the unique circle that contains the three corners of the triangle. But it doesn't need to lie inside the triangle — in fact, it only does if all the triangle's angles are less than 90 degrees (so the triangle is acute). If one angle is greater than 90 degrees (the triangle is obtuse) then the circumcenter lies outside the triangle, and if one angle is exactly equal to 90 degrees then it is the mid-point of the hypothenuse.
But there's another point that qualifies as a centre of a triangle. You find it by drawing aline
from each corner that is perpendicular to the opposite side. Amazingly, the three lines again meet in a single point, called the orthocentre. As for the circumcentre, the orthocentre lies inside the triangle if the triangle is acute and outside it if it is obtuse. If one of the angles is exactly equal to 90 degrees then the orthocenter will be one of the corners.
And what links all these points together? A straight line! The beautiful fact that the centroid, circumcentre and orthocentre of a triangle all lie on a straight line was first noticed in the 18th century by Leonhard Euler, one of the most prolific mathematicians of all time. That line now carries his name: it's called the Euler line of a triangle. You can play around with the three different centres and the Euler line on Math Open Reference which has beautiful interactive demonstrations, from which we made the images illustrating these articles.