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.
How would you go about adding up all the integers from 1 to 100? Tap them into a calculator? Write a little computer code? Or look up the general formula for summing integers?
Legend has it that the task of summing those numbers was given to the young Carl Friedrich Gauss by his teacher at primary school, as a punishment for misbehaving. Gauss didn't have a calculator or computer, no one did at that time, but he came up with the correct answer within seconds. Here's how he did it.
Notice that you can sum the numbers in pairs, starting at either end. First you add 1 and 100 to get 101. Next it's 2 and 99, giving 101 again. The same for 3 and 98. Continuing like this, the last pair you get is 50 and 51 and they give 101 again. Altogether there are 50 pairs all adding to 101, so the answer is 50 x 101 = 5050. Easy — if you're Gauss.
Usain Bolt celebrates his victory over 100m and new world record at the Beijing Olympics. Image: Jmex60.
Sometimes you just can't argue with the evidence. If a large sample of
very ill people got better after dancing naked at full moon, then surely
the dance works. Less contentiously, if the country's best-performing
schools produce worse results over time, then surely something is wrong
with the education system.
But hang on a second. Before you jump to conclusions, you need to rule
out a statistical phenomenon called regression to the mean. The idea is
that if you choose a set of measurements because they are quite extreme,
and then do the same measurements a little while later, the result is
likely to be less extreme.
Think of Usain Bolt. If you measured his performance over 100m the day
after he ran a world record, the time you'd get would probably not be
another world record, but something slower. This is because his world
record performance on the previous day was not entirely down to his
physical ability but also to all sorts of other factors - his mood, the
condition of the track, the passion of the crowd - which to all intents
and purposes are random. On his next run some or all of these factors
are probably absent, so his performance will be closer to his personal
average, or mean.
Similarly, if you select a group of very ill people to test a drug (or
dance) on, then on your next measurement they are likely to feel
better simply because their ill-being has regressed to the mean (never
mind the placebo effect). You can't automatically assume it was because
of the drug. And if you select a group of schools because of their
outstanding performance, you're likely to see worse results next time
around. You can't necessarily blame the government.
Regression to the mean was first noticed by a cousin of Charles Darwin,
Sir Francis Galton, in the 19th century. You can read about him and his
discovery in the maths magazine The
Commutator. And for an example of how measurements of school
performance can potentially give misleading results, read this
article by Daniel Read of Warwick Business School.