Plus Blog

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.

February 17, 2013

Science advisors to government are an embattled lot. Remember the l'Aquila earthquake debacle or David Nutt's stance on drugs that cost him his job. Bridging the gap between politics and science isn't easy. Politicians like clear messages but science, and the reality it tries to describe, is rarely clear-cut.

Buckyball

Full marks for Obama.

So how do you advise a politician about science, its uncertainties and about risk? What better person to ask than John P. Holdren, Assistant to President Obama for Science and Technology. As he told a packed auditorium at the annual AAAS meeting in Boston, Holdren is actually quite happy with his own boss. Obama, he says, always wants to know the level of confidence scientists have in a specific result. And Holdren's first memo for Obama, which stuck to the traditional two pages, came back with "where's the rest?" scrawled over it. That's reassuring! Anne Glover, Chief Scientific Advisor to the European Commission, on the other hand, has come across leading European politicians who prefer to do without science advisors altogether, since "scientists never agree".

Glover says that it's important to emphasise consent rather than disagreement. Scientists may be unsure or disagree about the details of something, say the exact relationship between CO2 emissions and global climate, but they may be certain about the big picture — that climate change is happening. Holdren points out that it's important to know where uncertainties are coming from — can they be sorted out with a bit more time and effort, or are they down to deeper gaps in or understanding or to processes we just can't pin down with better accuracy. Advisors should make sure uncertainties aren't exaggerated or understated, or simply ignored because they are too difficult to deal with.

Both Glover and Holdren agree that it's important to speak plainly to politicians. Advisors should use examples and visualisations of uncertainties (see this Plus article for some ideas) and when there's a range of possible outcomes of something, say an epidemic, use scenarios to examine the possibilities. Holdren advises to look for a policy that remains robust in the face of all of them. It's important to be prudent, since new evidence may always come along. But when there is a large, coherent and consistent body of evidence, as there is with climate change, it's safe to talk in terms of certainty.

For more on risk and uncertainty see our understanding uncertainty section.

February 16, 2013

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.

Limits

But not all number sequences behave so nicely. For example, the sequence 1/2, 1/3, 2/3, 1/4, 3/4, 1/5, 4/5, ... keeps jumping up and down, rather than getting closer and closer to one particular number. We can, however, discern some sort of limiting behaviour as we move along the sequence: the numbers never become larger than 1 or smaller than 0. And what's more, moving far enough along the sequence, you can find numbers that get as close as you like to both 1 and 0. So both 0 and 1 have some right to be considered limits of the sequence — and indeed they are: 1 is the limit superior and 0 is the limit inferior, so-called for obvious reasons.

But can you define these limits superior and inferior for a general sequence $(a_ n) = a_1, a_2, a_3, ...$ , for example the one shown in the picture? Here’s how to do it for the limit superior. First look at the whole sequence and find its least upper bound: that’s the smallest number that’s bigger than all the numbers in the sequence. Then chop off the first number in the sequence, $a_1,$ and again find the least upper bound for the new sequence. This might be smaller than the previous least upper bound (if that was equal to $a_1$), but not bigger. Then chop off the first two numbers and again find the least upper bound.

Keep going, chopping off the first three, four, five, etc numbers, to get a sequence of least upper bounds (indicated by the red curve in the picture). In this sequence every number is either equal to or smaller than the number before. The limit superior is defined to be the limit of these least upper bounds. It always exists: since the sequence of least upper bounds is either constant or decreasing, it will either approach minus infinity or some other finite limit. The limit superior could also be equal to plus infinity, if there are numbers in the sequence that get arbitrarily large.

The limit inferior is defined in a similar way, only that you look at the sequence of greatest lower bounds and then take the limit of that.

You can read more about the limits inferior and superior in the Plus article The Abel Prize 2012.

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