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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, no-one 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

 

right-angled 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 = 1012

billion = 109

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. Banach-Tarski 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 Banach-Tarski 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 3-dimensional 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

sine series

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 Get Acrobat Reader to view it. Calculator Algorithms features on the Peanut Software site which has other goodies Software

5. A Fascinating Integral

Integral

You could define a mathematician to be someone who finds this result delightful. Everyone knows about22/7 being an approximation topi, but to see it appear like this is a surprise.

I have given more details in A Fascinating Integral and in the original TEX format as written in Scientific Notebook.

6. Pi = 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 pi 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 ex-teacher saying "The case is perfectly simple. If we pass this bill which establishes a new and correct value for pi, 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. Pi, The Shove-Halfpenny Board and Buffon's Needle

In the eighteenth century the French naturalist Comte de Buffon showed that you could estimate pi by experiment using a needle and a shove-halfpenny 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

Estimate for pi

You can try it for yourself, but you will need lots of trials in order to get a remotely reasonable value for pi. 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 2p - 1. For this number to be prime p itself must be prime. There are 38 known Mersenne primes, the largest being the number 26972593- 1 (as at 6th February 2000).

You can join the search for these primes by downloading software to run on your computer and join Roland Clarkson (a 19-year-old 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.

 

 

Steve Mayer

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