
Facts
or Did You Know?
Mathematics is full of fascinating
facts and I can only give a small flavour of them here. I hope one or
two of them will make you think "Wow!"
1. Language
 George Bernard Shaw said
Britain and America are "two nations separated by a common
language", but did you know that this happens even in
mathematics which is supposed to be a language all of its own. The
differences aren't confined to spelling (as in centre/center).

British 
American 


maths 
math 
I don't know how this came about but, unlike the next example, noone
I know in the UK uses the US form


soluble 
solvable 
A specialized term in group theory




I came across this one quite recently. The definitions are completely reversed: 

trapezium
trapezoid 
trapezoid
trapezium 
Quadrilateral with one pair of sides parallel
Quadrilateral with no sides parallel


rightangled triangle 
right triangle 


sine rule 
law of sines 
A trigonometric formula for triangles


formulae 
formulas 
The US version of the plural of formula is taking over rapidly in Britain


billion = 10^{12} 
billion = 10^{9} 
In Britain this battle has been lost many years ago. A British
billion used to mean a million million but its use for finance has
ensured that the US thousand million has taken over. There were
similar differences for trillion etc 
 There must be many other differences. Do you know of any?
2. BanachTarski Theorem
 This must be one of the strangest theorems ever proved. In 1924
Banach and Tarski showed that you could cut up a solid sphere into
six pieces and then reassemble them into two solid spheres of
exactly the same size. In fact, the sphere could be cut and
reassembled into any shape or size object whatsoever. However, the
shape of the pieces stretch your understanding of volume and area.
 For more on this and even weirder consequences of the theorem try
 Wreckreational
Math
 Mudd
Math Fun Facts
 and an outline of the proof can be found at The
BanachTarski theorem (but don't expect to be able to understand
the proof!).
3. Ham Sandwich and Hairy Ball Theorems
 Topologists are mathematicians who study a type of geometry which
looks at the properties of objects that are unaffected by continuous
distortions (such as stretching). They love using interesting names
for their results, to make what are quite difficult and abstract
ideas more appealing. So they'll talk about the Ham Sandwich Theorem
rather than planes in 3dimensional space, or the Hairy ball Theorem
instead of tangent vectors on spheres which is much more boring.
 Ham Sandwich Theorem
 Regardless of the distribution of ham and cheese in a sandwich,
you can use one slice to divide the sandwich into two parts
containing equal parts of bread, ham and cheese.
 Hairy Ball Theorem
 (This description comes from The Penguin Dictionary of Curious
and Interesting Geometry by David Wells. Internet
Bookshop, Amazon)
 Imagine that you are combing a tennis ball which is hairy rather
than fluffy. You attempt to comb it so that all the hairs are lying
flat on the surface and so that they change direction smoothly over
the whole surface, but the Hairy Ball Theorem says that you will fail.
 Since the earth is a ball, and the wind at any point has a
direction, as if the air were being combed over the Earth's surface,
it follows that there is always a cyclone somewhere.
4. Calculators Don't Use Series For Trig Functions
 This will be a surprise to most maths teachers (including me). We
tended to assume that series like

 were used to calculate sin of any angle. But it turns out that
calculators 'know' just a few values and calculate the rest from
these. For an excellent explanation download Calculator
Algorithms. This is a pdf file so you will need Acrobat
Reader
to view it. Calculator Algorithms
features on the Peanut Software
site which has other goodies Software
5. A Fascinating Integral

 You could define a mathematician to be someone who finds this
result delightful. Everyone knows about
being an approximation to,
but to see it appear like this is a surprise.
 I have given more details in A
Fascinating Integral and in the original T_{E}X
format as written in Scientific
Notebook.
6.
= 3.2 By Law
 In 1897 the General Assembly of the State of Indiana in the USA
tried to pass legislation that appears to say that
is to be 3.2, though the Bill does not make it very clear. On top of
that they had the nerve to try to get everyone else to pay royalties
for this 'discovery'.
 The Bill was referred to the House Committee on Canals, which was
also referred to as the Committee on Swamp Lands! By chance a
professor of mathematics happened to be present during a debate and
heard an exteacher saying "The case is perfectly simple. If we
pass this bill which establishes a new and correct value for ,
the author offers to our state without cost the use of his discovery
and its free publication in our school text books, while everyone
else must pay him a royalty." Fortunately, the professor was
able to teach the senators about mathematics and the Bill was stopped
becoming an object for ridicule.
 For more on this fascinating story, and and analysis of the
'mathematics' in the Bill see Legal/pi
indiana and Legislating
Pi
7. ,
The ShoveHalfpenny Board and Buffon's Needle
 In the eighteenth century the French naturalist Comte de Buffon
showed that you could estimate
by experiment using a needle and a shovehalfpenny board, which
consists of a grid of parallel lines.
 Suppose the lines are distant a from one another, and the
needle has length l with l < a (so the needle
can only cross one line at a time). Then you can conduct an
experiment by dropping the needle randomly onto the board and
counting the number of times the needle crosses any line. Suppose you
do this n times, where n is large, and c of
those times the needle crosses a line then
 You can try it for yourself, but you will need lots of trials in
order to get a remotely reasonable value for .
If your browser can handle Java then Buffon
Needle Applet Page will do the work for you.
 Searching for Buffon's Needle will give hundreds of sites. A
proof of the result is given at Ask
Dr. Math
8. Useless Fact
 116! + 1 is a prime number
 See Penguin Dictionary of
Curious and Interesting Numbers
9. The Great Internet Mersenne Prime Search
 A Mersenne
prime is a prime number of the form 2^{p}  1. For this
number to be prime p itself must be prime. There are 38 known
Mersenne primes, the largest being the number 2^{6972593} 1
(as at 6^{th} February 2000).
 You can join the search for these primes by downloading software
to run on your computer and join Roland Clarkson (a 19yearold
student who found the 37th Mersenne prime) in getting your name in
the newspapers. All the details are at Mersenne
Prime Search
10. The Birthday Problem
 If a class has 23 students in it then the probability that at
least two of the students share a birthday is about 0.5. Surprised?
If there are 50 students in a class then it's virtually certain that
two will share the same birthday. This seems to go against common
sense but is absolutely correct. For a good explanation and
simulation see The
Birthday Problem.
 Probability can catch even the best mathematicians out. A fair
amount of controversy was created in 1990 by The
Monty Hall Problem also known as The
Goat and the Car.

