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 1+2i.
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 i.
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.
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
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 thisPlus 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.
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 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 , 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, 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 ), 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
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
An 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, ... . For
example, a bag with infinitely many apples would be a countable infinity
because (given an infinite amount of time) you can label the apples 1, 2,
Two countably infinite sets A and B are considered to have the same "size"
(or cardinality) because you can pair each element in A with
one and only one element in B so that no elements in either set are left
over. This idea seems to make sense, but it has some funny consequences.
For example, the even numbers are a countable infinity because you can
link the number 2 to the number 1, the number 4 to 2, the number 6 to 3
and so on. So if you consider the totality of even numbers (not just a
finite collection) then there are just as many of them as natural
numbers, even though intuitively you'd think there should only be half
Something similar goes for the rational numbers (all the numbers you can
write as fractions). You can list them as follows: first write down all
the fractions whose denominator and numerator add up to 2, then list all
the ones where the sum comes to 3, then 4, etc. This is an unfailing
recipe to list all the rationals, and once they are listed you can label
them by the natural numbers 1, 2, 3, ... . So there are just as many
rationals as natural numbers, which again seems a bit odd because you'd
think that there should be a lot more of them.
It was Galileo who first noticed these funny results and they put him
off thinking about infinity. Later on the mathematician Georg Cantor
revisited the idea. In fact, Cantor came up with a whole hierarchy of
infinities, one "bigger" than the other, of which the countable infinity
is the smallest. His ideas were controversial at first, but have now
become an accepted part of pure mathematics.
Yesterday we opened the Plus New York office, amidst snow covered streets at the foot of the Empire State Building!
The day started with a trip to MoMath, the recently opened maths museum in central New York. It was filled with a fascinating array of interactive exhibits demonstrating the beauty and playfulness of mathematics. And as one of the volunteers told us, playfulness is what it's all about. There were musical spheres demonstrating the maths of music, a fractal machine with cameras creating fractals from their surroundings, and a chance to discover the paths of mathematical rolling stones. It was full of children and the young at heart discovering the joy of maths for themselves.
We also discovered an illuminated buckyball in the park just across from our hotel and the arithmetic of relationships in the High Line park. Maths is everywhere in NYC!
The Institute for Advanced Study in Princeton.
Today we had a very early start, taking the train from New York Penn Station to Princeton to visit the Institute for Advanced Studies. We were very lucky to speak with Freeman Dyson and Edward Witten about quantum field theory (QFT), the mathematical framework that has made much of the advancements in physics possible in the last century. This is the main reason for our trip to the States and we are looking forward to more interviews this week with other luminaries of theoretical physics to continue our series telling the story of QFT. You can read our first articles here. We'd like to thank Jeremy Butterfield and Nazim Boutta, our gurus in QFT for all their help in preparing for the trip!
After a Manhattan or two tonight we're heading to Boston tomorrow to continue our quantum adventure!
On 14 March at 1.59pm GMT, Marcus du Sautoy will host Pi Day Live, an interactive exploration of the number which has fascinated mathematicians throughout the ages. He wants to rediscover pi using ancient and intriguing techniques, and he needs your help!
Everyone at Pi Day Live will be using marbles, pins, maps and other household items to discover pi using methods that range from 3500 to around 250 years old. It’s not all low-tech, though, as they will be using the web to gather everyone’s results live, combining them to find out if they can collectively calculate a more accurate approximation of pi. Will it be possible to derive pi to one, two, three or more, decimal places? Can we do better than the ancient Greeks or have we lost the ability to rediscover this amazing number without using computers?
Mathematicians (and the American House of Representatives) have christened 14 March Pi Day because the date, when written in the US date format, is 3.14. Add the 1.59pm time of the Pi Day Live experiment and you get 3.14159, or pi at around the accuracy Archimedes calculated it over 2000 years ago using simple geometry.
Pi has obsessed generations of mathematicians for millennia because it is integral to one of the most important and elegant geometric objects in nature, the circle. Attempting to calculate an accurate value for this never-ending transcendental number has been one of the big themes running throughout the history of mathematics.
Even though you only need to know pi to 39 decimal places to calculate a circumference the size of the observable universe to the precision comparable to the size of a hydrogen atom, mathematicians have pushed the limits of computing technology to calculate the number to over one trillion digits. How close can Pi Day Live get to this accuracy using ancient techniques?
You can connect with Marcus and Pi Day Live via an Online Lecture Theatre or by watching online on the ‘Big Screen’. If your computer can run YouTube videos then you have what you need to get involved. The event will be recorded and will be available on YouTube afterwards for anyone who can’t take part on the day. Just go to Pi Day Live website to find out more. And you can get live updates and all the pi facts you could ever want on Twitter at and Facebook.