## 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! |
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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
How to add up quickly Einstein as icon
What is the area of a circle? What's so special about special relativity? Pi not a piece of cake How does gravity work? |
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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's track is designed for maximum speed using Newton's laws of motion.
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.
In this equation both
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 |
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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 where and are ordinary numbers and stands for the square root of The number itself can be represented in this way with and Numbers of this form are called complex numbers. You can add two complex numbers like this: And you multiply them like this: The complex number But how can we visualise these numbers and their addition and multiplication? The and components are normal numbers so we can associate to them the point with coordinates on the plane, which is where you get to if you walk a distance in the horizontal direction and a distance in the vertical direction. So the complex number which is the sum of and corresponds to the point you get to by walking a distance in the horizontal direction and a distance in the vertical direction. Makes sense. What about multiplication? Think of the numbers that lie on your horizontal axis with coordinates Multiplying them by flips them over to the other side of the point : goes to goes to and so on. In fact, you can think of multiplication by as a rotation: you rotate the whole plane through 180 degrees about the point Multiplying by What about multiplication by the square root of ? Multiplying twice by is the same as multiplying by 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 by and you will see that the result corresponds to the point you get to by rotating through 90 degrees (counter-clockwise) about And what about multiplying not just by but by a more difficult complex number Well, multiplying by an ordinary positive number corresponds to stretching or shrinking the plane: multiplication by 2 takes a point to which is further away from (that’s stretching) and multiplication by 1/2 takes it to which is closer to (shrinking). Multiplying by 2 is stretching. It turns out that multiplication by a complex number corresponds to a combination of rotation and shrinking/stretching. For example, multiplication by 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 |
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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. 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 CO Both Glover and Holdren agree that it's important to speak plainly to politicians. Advisors should use examples and visualisations of uncertainties (see this For more on risk and uncertainty see our understanding uncertainty section. |
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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.
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 But can you define these limits superior and inferior for a general sequence , 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 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. |