——By Ian Colley Mathematics Teacher, Suzhou High School
Welcome to the Wonderful World of Mathematics! Or WWW.OM for short. I
want to share with you some of the reasons why I love mathematics so
much. Mathematics for me can be funny, surprising, challenging,
incredible and a linear combination of all these things.
How can it be funny?
Well, it is no secret comedians often include a maths joke or two in
their repertoire, which goes down very well even with people who are not
keen on the subject. Here is a classic maths joke by the famous
Liverpudlian comedian Ken Dodd:"The secretary of State for Education in
England was so worried about the standard of mathematics amongst 15 year
olds that he conducted a survey across the whole country targeting this
age group. In his ndings he concluded that 4 out of every 3 people
could not do fractions! "
Here are 2 more: John,
who was mathematically challenged, went to the Pizza Hut and ordered a
Large Deluxe Supreme with a thin crispy crust and extra anchovy
toppings. The server asked him if he wanted it cut into 6 or 12 slices.
His reply was "You had better make that 6; I could never manage 12."
A man takes his son Fred to see his favorite comedian Mr. Numbers
perform, and when the show starts, Mr. Numbers quickly gets into his
routine. "Welcome everybody, 1 (everyone laughs loudly), 7(some laugh a
little), 26((everyone laughs moderately), 35((everyone laughs a
lot)......726((everyone laughs hysterically and some are laughing so
much that there are tears rolling down their cheeks). Fred is confused
and asks his dad what is going on. He replies "It's simple son,
Mr.Numbers has a loyal fan base who come and see the whole joke he
labels them with a number. The listener then remembers the joke with
that number and then laughs after he has told it to himself. It is
pretty clever, don't you think, Fred? And the best part is that he can
get through so many jokes in a 2 hour performance."
"That's amazing" says Fred, "but I don't understand why everyone was laughing so much when he said 726?"
"That's a good question Fred, and the answer is because we haven't heard that joke before!"
There are lots of these jokes about maths, but the above three are my
favorites. Because I think they are hilarious and are simple to
understand. With mathematics it is important to me to always try to make
it as simple as possible.
Why is mathematics surprising?
Consider the following question: What is the probability that among n
people there are at least 2 who have the same birthday? We assume that
they "choose" their birthdays independently of one another so that the
result is as if they had drawn n balls marked from 1 to 365(ignoring
leap years)from a bag with replacement. This means that after a ball is
drawn out and its number is noted. All these numbers are equally likely
and the total number is . Now we must count these cases in which some
of the balls have the same number. This sounds complicated, but it is
easy to gur out the "opposite event", namely when all n balls are
different. Hence the required probability
What comes as a big surprise is the numerical fact that this
probability exceeds 12 as soon as n 23. What would you have guessed?
When this result was shown to me whilst in my first year at University
by my fantastic Statistic teacher Professor Nick Bingham I was
absolutely astonished! What made things even better was the fact that an
that day in my MIPR class ( year Probability) there were 23 people
attending and 2 people had the same birthday!
Another surprising result which is probably my favorite is the fact that the vector
is perpendicular to the plane ax + by + cz = D. Why is it my favourite
result? It's because it is so simple to write down n when you have the
Cartesian equation of a plane.
Finally on this section another
result which makes my top 3 on surprising result is on a topic called
'Flow through a network'. This is part of an applied branch of
mathematics called Decision Mathematics. Consider a system of pipes
where water can ow down them an different rates of x
from a single source s to a single sink T. .What is the maximum ow of
water per second that can pass from s to T ? The answer is obtained from
the marvelous theorem ( rst proved in 1955 by Ford and Fulkerson)which
states: In any capacitated network with a single source s and a single
sink T., (The value of a maximal ow) = (The capacity of a minimum cut)
For further details see 'Heinemann Modular Mathematics for Edexcel AS
and A Level - Decision Mathematics 1' by John Hebborn. It is a
surprising result but pure genius in its simplicity to solve ow
problems. With this MAX FLOW - MIN CUT theorem, problems which look
complicated on a rst inspection can quite easily be solved. I also had
the privilege of being taught at University by the author - Dr. J. E.
Hebborn, who is without doubt the greatest mathematician and teacher I
have ever had.
Challenging mathematician problems:
There are many challenging questions but the one that stands out in
more recent times is the story of a riddle that baf ed the world's
greatest minds for 358 years. The riddle in question is Fermat's Last
Theorem which states that there are no whole number solutions to the
following equation:
where n is any whole number greater than 2.
This is a simple problem to understand but to prove it is an entirely
different matter. In May 1995, the English mathematician Andrew Wiles
nally proved it after working for 7 years in isolation. It required him
to use the work of many other branches of mathematics including The
Taniyama-Shimura conjecture and elliptical curve theory. In fact he rst
announced his proof in May 1993 at the Issac Newton Institute at
Cambridge University but a problem was discovered in his work by the
Wolfskell committee 3 months later. This committee consisted of 6 expert
mathematicians who were assigned to proof read his manuscript of over
100 pages. This awful hiccup for him set him back for a further 2 year.
Luckily he was able to change his counting system and nally solved the
riddle! It was a truly fantastic moment for someone now in his early
forties. He rst became fascinated by Fermat's Last Theorem when he
accidentally came across it whilst he was visiting his local library as a
ten year old. It was also great that Mr. Wiles had done something
astonishing when he was in his early forties, a large piece of stellar
mathematical breakthroughs are almost always done by young
mathematicians, and certainly no older than 31 or 32. You could say it's
a young man's game.
Of course there are plenty of problems out
there which keep the most brilliant of mathematical minds busy and one
area which contains some very hard problems to prove are that of
combinatorics. There is a great story of the late great Hungarian
mathematician Paul Erdos who used to help PHD students with their
theses. One very talented American student Challenging mathematical
problems: response took him by surprise. "The problem you have chosen is
too difficult for you - try something more accessible." The student
reluctantly agreed but Mr. Erdos was absolutely correct because 50 years
on, the problem the student asked for help on has still not been
solved!
Which mathematics do I had incredible?
For this last section there is not enough room to write down all of
my incredible mathematical result, but here is a small sample:
1)
The use of probability generating functions to find the mean and
variance of some probability models. It is such a clever technique and
you can apply it to more complicated problems with n independent and
identically distributed variables.
2) The Central Limit Theorem in Statistics.
3) The C + i S technique for summing trigonometric series.
4) The cross product of 2 vector you have crossed - quite incredible!
It also has so many applications in vector questions would be so
article' I have certainly enjoyed writing it.
I will nish with one more joke:
A tramp is on the side of the road one morning and spots a smartly dresses man in a suit on his way to work.
He asks the man in the suit "Hey mister, could you lend me ? 100 till pay day?"
The man in the suit replies, "When is pay day?"
The tramp replies, "How should I know, you are the one with the job!"
Remember: A busy mathematician is a happy one!