Plus Blog

March 22, 2013

Want to meet some inspirational female mathematicians? Then come to the Florence Nightingale Day, a free day of activities, at Lancaster University on April 17. The day is aimed especially at girls in year 12 at schools in the Morecambe Bay and Preston areas but it is also open to boys. "Everyone talks about getting more women into mathematics but I wanted to do something to actively encourage it," says Nadia Mazza, a mathematics graduate from Lausanne University who is behind the idea. "Year 12 is a critical stage when students make crucial decisions affecting their future career plans. We want to show how appealing it can be to do maths."

Florence Nightingale

A young Florence Nightingale

Speakers at the morning session will include Professor June Barrow-Green from the Open University and pure mathematics researcher Professor Reidun Twarock, from the University of York. Opening the afternoon session will be first generation female mathematician Professor Dona Strauss from the University of Leeds. Then Beth Penrose (University of Nottingham), Fiona Murray (Principal Integrity Engineer – TA Pipelines and Structures at Centrica) and Suduph Imran (a former mathematics teacher at Our Lady's Catholic College in Lancaster who is studying for her Masters in Education) will speak about their jobs and how they use mathematics every day. Beth, Fiona and Suduph are ambassadors from STEMfirst, an organisation which promotes opportunities in science, technology, engineering and maths.

The last speaker will be one of the Plus editors, having a look at the careers of some female mathematicians.

The day will also include a hands-on mathematics contest when attendees will spend time solving tricky problems in small groups under the supervision of coaches, all PhD students from Lancaster University's Department of Mathematics and Statistics. Plus there will be displays featuring opportunities offered to women by a degree in mathematics or statistics, which will stimulate informal discussion between pupils and mathematicians.

For more information and to book places please contact Nadia Mazza at n.mazza@lancaster.ac.uk or James Groves at j.groves@lancaster.ac.uk.

The event is named after Florence Nightingale, best remembered for her work as a nurse during the Crimean War. This amazing woman had an immense love of both subjects and was a pioneer in statistics, especially in the use of visualisation of statistical data. You can find out more about her in this Plus article.

March 21, 2013
Still Life: Five Glass Surfaces on a Tabletop

We're looking for beautiful mathematical images. Still Life: Five Glass Surfaces on a Tabletop by Richard Palais won the 2006 Science and Engineering Visualisation Challenge.

We're looking for inspiring images that illustrate your favourite mathematical ideas. Illustrations, photographs, computer simulations or even clever doodles — anything that's colourful and inspirational. The best fifty images will be used as part of a book fifty to be published by Oxford University Press to coincide with the fiftieth anniversary of the Institute of Mathematics and its Applications (IMA). The book will contain fifty examples of the best writing on mathematics, both popular and technical, aimed at a general audience. We also plan to reuse the best images (fully credited to you) in publicity for the IMA, especially its 50th Anniversary.

The idea is that these images should be able to stand alone, like pictures in an art gallery, with minimal explanation. They should ideally be approximately square or portrait style and sufficiently striking to be readable when reproduced at a size of approximately 10cm2. You need to hold the copyright for the image.

Please submit images, in low resolution at this stage, to ima50@maths.cam.ac.uk by or before 12th May 2013, along with any appropriate explanation or attribution text. Please using the word IMAGE in the header. We encourage you to be creative!

March 19, 2013

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?

The answer is that the £10 are a red herring. There's no reason why the money I owe after the whole transaction and the money I still have should add up to £10. Rather, the money I owe minus the change I got should come to the price of the book, that is £7. Giving a pound back to each my brother and sister just re-distributes the amounts. The money I still owe is reduced to £8 and the money I still have to £1. Rather than having £10-£3=£7, we now have £8-£1=£7. Mystery solved!

March 13, 2013

What a lovely coincidence! Pi day (the 14th of March, written 3.14 in the US) is also Albert Einstein's birthday. How are you going to celebrate? You could join Marcus du Sautoy and over a thousand other people in a mass online experiment to calculate pi or you could join Plus in Cambridge to watch our favourite mathematical movie Travelling Salesman. And to celebrate both the number and the man, here are some favourite articles.


How to add up quickly
One of our favourite authors, Chris Budd, takes a look at some famous infinite series involving pi and presents a trick for making them converge quicker.


Einstein as icon
In 1905 Albert Einstein changed physics forever with his special theory of relativity. Since then his name — and hair do — have become synonymous with genius. John D. Barrow looks at Einstein as a media star.


What is the area of a circle?
You might know the famous formula for an area of a circle, but why does this formula work? Tom Körner's explanation comes with a hefty estimate of pi.


What's so special about special relativity?
Most of us are aware that Einstein proved that everything was relative ... or something like that. But we go no further, believing that we aren't clever enough to understand what he did. Hardeep Aiden sets out to persuade you that they too can understand an idea as elegantly simple as it was original.


Pi not a piece of cake
Every phone number on the planet, all of our names (with the characters suitably encoded), even the works of Shakespeare can be found in the digits of pi — if these digits are truly random that is. So are they?


How does gravity work?
Einstein's theory of general relativity doesn't look at gravity as a force, rather it replaces the concept of force by that of geometry. How does that work?

March 7, 2013

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.

The London velodrome

The London Velodrome's track is designed for maximum speed using Newton's laws of motion.

Newton's first law: An object at rest will remain at rest unless acted upon by an external and unbalanced force. An object in motion will remain in motion unless acted upon by an external and unbalanced force.

This is also called the law of inertia and it doesn't need much explanation. No stationary object will start moving of its own accord without a force being applied. And the reason why in our everyday experience moving objects tend to slow down unless they are being powered by something is due to factors such as friction and air resistance.

Newton's second law: The acceleration a of a body is parallel and proportional to the net force F acting on it. The exact relationship is F=ma, where m is the body's mass.

In this equation both F and a are vectors with a direction and a magnitude.

Newton's third law: When two bodies exert a force on each other the forces are equal in magnitude, but opposite in direction. For every action there is an equal and opposite reaction.

Thus, if you kick a ball with your foot, then the ball exerts an equal and opposite force on your foot.

The three laws of motion were first published in 1687 in Newton's famous work Philosophiae Naturalis Principia Mathematica which translates as Mathematical Principles of Natural Philosophy. Newton's law of universal gravitation and mathematical techniques we'd now call calculus were also published in Principia Mathematica and together with the laws of motion they gave the first comprehensive description of the physical processes we observe in everyday life. It later turned out that the laws don't hold when you look at the world at very small scales (that's where quantum mechanics reigns) or at objects that move at very high speed or when there are very strong gravitational fields. However, Newton's laws still give a very good approximation for the physics we observe in our normal lives.

To read more about Newton's laws and its applications, from understanding the melting Arctic to building the Olympic Velodrome, have a look at our teacher package on classical mechanics.

February 19, 2013

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.

People first noticed this fact in the 15th century. A lot later on, in the 19th century, William Rowan Hamilton noticed that the illegal numbers you come across in this way can always be written as $x+iy$ where $x $ and $y$ are ordinary numbers and $i$ stands for the square root of $-1.$ The number $i$ itself can be represented in this way with $x=0$ and $y=1.$ Numbers of this form are called complex numbers.

You can add two complex numbers like this:

  \[ (x+iy) + (u+iv) = (x+u) + i(y+v). \]    

And you multiply them like this:

  \[ (x+iy)(u+iv) = xu + i(xv+yu) + i^2yv = xu - yv + i(xv+yu). \]    
The complex number with real part 1 and imaginary part 2

The complex number 1+2i.

But how can we visualise these numbers and their addition and multiplication? The $x$ and $y$ components are normal numbers so we can associate to them the point with coordinates $(x,y)$ on the plane, which is where you get to if you walk a distance $x$ in the horizontal direction and a distance $y$ in the vertical direction. So the complex number $(x+u) + i(y+v),$ which is the sum of $(x+iy)$ and $(u+iv),$ corresponds to the point you get to by walking a distance $x+u$ in the horizontal direction and a distance $y+v$ in the vertical direction. Makes sense.

What about multiplication? Think of the numbers that lie on your horizontal axis with coordinates $(x,0).$ Multiplying them by $-1$ flips them over to the other side of the point $(0,0)$: $(1,0)$ goes to $(-1,0),$ $(2,0)$ goes to $(-2,0),$ and so on. In fact, you can think of multiplication by $-1$ as a rotation: you rotate the whole plane through 180 degrees about the point $(0,0).$

Multiplying by i.

What about multiplication by $i,$ the square root of $-1$? Multiplying twice by $i$ is the same as multiplying by $-1.$ So if the latter corresponds to a rotation through 180 degrees, the former should correspond to rotation by 90 degrees. And this works. Try multiplying any complex number, say $2+i5,$ by $i$ and you will see that the result corresponds to the point you get to by rotating through 90 degrees (counter-clockwise) about $(0,0).$

And what about multiplying not just by $i$ but by a more difficult complex number $u+iv?$ Well, multiplying by an ordinary positive number corresponds to stretching or shrinking the plane: multiplication by 2 takes a point $(x,y)$ to $(2x,2y)$ which is further away from $(0,0)$ (that’s stretching) and multiplication by 1/2 takes it to $(x/2,y/2)$ which is closer to $(0,0)$ (shrinking).

The effect of multiplying by two

Multiplying by 2 is stretching.

It turns out that multiplication by a complex number $u+iv$ corresponds to a combination of rotation and shrinking/stretching. For example, multiplication by $-1+1.732i$ is rotation through 120 degrees followed by stretching by a factor of 2. So complex numbers are not just weird figments of the imagination designed to help you solve equations, they’ve got a geometric existence in their own right.


You can find out more about complex numbers and things you can do with them in the Plus articles Curious quaternions, Unveiling the Mandelbrot set, Non-Euclidean geometry and Indra's pearls and Maths goes to the movies.

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