4.12 MATHEMATICAL STORIES
The revised Junior Certificate Mathematics syllabus (page 4) states that students "should be aware of the history of mathematics and hence of its past, present and future role as part of our culture". A number of mathematical stories now follow which can be used to enliven a topic or as an attention grabber at the start of a new topic.
INTRODUCING IRRATIONAL NUMBERS- THE MURDER STORY
Pre-requisite knowledge: Theorem of Pythagoras.
Excerpt from Fermat's Last Theorem by Simon Singh (Fourth Estate, London, 1997, p.55)
For Pythagoras, the beauty of mathematics was the idea that rational numbers (whole numbers and fractions) could explain all natural phenomena. This guiding philosophy blinded Pythagoras to the existence of irrational numbers and may have led to the execution of one of his students. One story claims that a young student by the name of Hippasus was idly toying with the number , attempting to find the equivalent fraction.
Eventually he came to realise that no such fraction existed.... Hippasus must have been overjoyed by his discovery, but his master was not.... Pythagoras was unwilling to accept that he was wrong, but at the same time he was unable to destroy Hippasus' argument by the power of logic. To his eternal shame he sentenced Hippasus to death by drowning.
NOTE
Squareroot 2 is related to the theorem of Pythagoras.
A SPHERICAL STORY!
Archimedes was one of the all-time great mathematicians, especially when one realises that all he had access to was a pencil, a ruler and a compass. He is credited with many inventions including giant catapults, solar ray guns, and giant lever systems. However he was very proud of the following equation:
V = p r3
Archimedes proved that a sphere is exactly two-thirds as big as the smallest cylinder into which it will fit. So, a solid ball which would only just fit inside a tin can (with the lid on) takes up exactly two thirds of the space inside the can. There is a small sign consisting of a sphere in a cylinder on his gravestone in memory of this discovery. When the Romans invaded his town one night, the Roman general Marcellus had especially ordered that the 75-year-old Archimedes should be spared. However, a soldier found him doodling in the sand and Archimedes upset him by saying, "Do not disturb my diagrams", so the soldier killed him.
IMPORTANCE OF USING THE CORRECT UNITS - AN EXPENSIVE LESSON
The $125 million Mars Climate Orbiter, launched on 11 December 1998, was lost when it came too close to planet Mars. It had probably come too deep into the atmosphere of Mars, because it had been erroneously navigated on a trajectory bringing it down to only 50 km above the surface, and was very likely destroyed - its safe altitude would have been about 80 km. The error was due to a confusion of metric and imperial units between different collaborating teams.
Arthur Stephenson, chairman of the Mars Climate Orbiter Mission Failure Investigation Board said:
The "root cause" of the loss of the spacecraft was the failed translation of English (imperial) units into metric units in a segment of ground-based, navigation-related mission software....
For further details of this story see http://www.seds.org/~spider/spider/Mars/ms98mco.html http://www.astronomynow.com/mars/mco/991110findings/index.html
PRIME NUMBERS -NOW THAT'S BIG!
On 1 June 1999, the team of Nayan Hajratwala, George Woltman, Scott Kurowski et. al. discovered a new record prime number: 26972593-1. This is the 38th known Mersenne prime (there may be smaller ones as not all previous exponents have been checked). This is the fourth record produced by GIMPS (the Great Internet Mersenne Prime Search) in four years! This prime number has 2 098 960 digits (it would take over 400 typed pages to write it down!!) and is the largest known prime number in the world - until someone discovers the next one, that is. The discoverers got a $50,000 award for finding it!
NOTE
Aprime is any integer greater than 1 which has only 1 and itself for positive divisors. The first few primes are 2, 3, 5, 7 and 11. Mersenne primes are those which are a power of two, minus one. For example, the first few are 22-1=3, 23-1=7, 25-1=31, 27-1=127. These first few were known to the ancient Greeks several hundred years before Christ.
For more information see http://www.utm.edu/research/primes/largest.html#largest
Finally, mathematics teachers in search of suitable stories to weave into their mathematics classes will find useful resources on the following web pages.
- History of Mathematics Archive: http://www-groups.dcs.st-and.ac.uk/~history/index.html
- Female Mathematicians: http://www-history.mcs.st-and.ac.uk/history/Indexes/Women.html
- Earliest known uses of some of the words of mathematics: http://members.aol.com/jeff570/mathword.html
- Earliest uses of various mathematical symbols: http://members.aol.com/jeff570/mathsym.html