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Mathematics = FUN. There's a lot more to it than doing sums, and at the top-end it merges into philosophy. Besides being very USEFUL, mathematics is also interesting, intriguing, fascinating, and involves FUN with advanced rational thinking.
Mathematics as a science is unusual in that you seldom see mathematicians performing experiments.
Mathematics is a science of absolute fact which exists independently of the Universe, though the Universe may be a function of mathematics!
How Logarithms work - the secret behind the conjuring trick revealed!
PI = 3.14159...
log-e = 2.71828
% = parts per hundred
How to work out radio frequencies and wavelengths
Working out the Light-Year
Quadratic Equations - that formula for solving them
Mathematica - software from www.wolfram.com
How to work out credit APR
Geometry of space and how it relates to phenomena in physics such as area and volume
Diamond 16 Puzzle
Solvers create a variety of symmetric designs that illustrate concepts of finite mathematics. A link to related theory is provided. - was http://finitegeometry.org/sc/16/puzzle/index.html
Fine Structure Constant - a 20th Century Mystery
52 Cards; how many permutations?
Maths by Somos and The Elliptic Realm
Quite Big Numbers
Really Big Numbers
Operators: Plus, Multiply, Power, etc - speculative notions
What generally happens with mathematics is that someone thinks up something for fun and works out the maths to do with it, such as "what if there was such a thing as the square-root of minus one?", and then a bit later the maths of it becomes very useful when something technological needs that species of maths (in the case of square-root(-1), it's just exactly the way AC electric phases work). I wonder what 4-dimensional manifolds will come in useful for?!
Yes I know it's a bit scant! Add more stuff to this! Please e-mail me with constructive suggestions, links to interesting sites, and contributions, e-mail
It has been speculated that the Maths Tower at Manchester University (as seen on this photo page) is prevented from falling down by the application of mathematics. It is said to have been inadvertently built on a giant underground boulder, and every time the tower looks like tipping a microscopic amount, calculations are done and the weights on the roof are moved to compensate.