A demonstration of Leonhard Euler's genius:
e - a most mysterious number in mathematics
From computer science to statistics, the number e is everywhere in the mathematical sciences.
e and pi are both examples of transcendental numbers, a type of number whose baffling
complexity is the very antithesis of the plain everyday integers 0, 1, 2, 3, 4 and so on.
Whereas the integers are easy for humans to understand, transcendental numbers are infinitely harder to pin down.
Definition of a transcendental number
e and pi are examples of special irrationals known as transcendental numbers. A transcendental number is entirely unrelated to the integers by any sequence of ordinary arithmetical operations. You can multiply it by itself as many times as you wish, combine these powers and divide and multiply by integres in whatever fashion you want, but you will never arrive back in the familiar territory of the integers. This is the definition of a transcendental number.
Even numbers like the square root of 2 are tame compared with transcendentals. By definition if you multiply the square root of 2 by itself the result is 2 - an ordinary number, or an integer. So we get back to the integers after just one step. But e and pi are different, no ordinary calculation will connect them back to ordinary numbers.
In one awe-inspiring equation Euler
tied e to the four other fundamental numerical entities 0, 1, pi,
and i. e as the symbol for the base of natural logarithms; i for the square root of -1, and pi for the ratio of circumference
to diameter in a circle - all three transcendental numbers ( that is non-terminating ) which means they are not precisely defined, but - almost magically - in Euler's equation they unite as a whole precisely defined !
Euler's formula became central to our understanding of number and exponentiation, and is celebrated for the beautiful way it unites the five fundamental constants of mathematics.
After demonstrating a proof of this equation a famous mathematician told the audience:
"Gentlemen it is absolutely paradoxical; we cannot understand it, and we don't know what it means. But we have proved it, and therefore we know it is the thruth." The physicist Richard Feynman described it as "the most remarkable formula in mathematics". What an incredible achievement : to mathematically proof something that we cannot yet understand !
For more explanation go to : The number e
|( The above text is excerpted from the NewScientist, July 21-27, 2007 )|