15 May, 2006

Transcendental numbers

The first time I came across this kind of number was in a book called Contact, by Carl Sagan. It said that pi, which is quite familiar to all of us, was from a class of numbers called Transcendental numbers. I did a bit of digging and came across the fact that there are infinitely more transcendental numbers than there are integers, or even rational numbers. The same place told me that e too, is one of them. Also, the natural log of any positive rational number other than one. What is funny, is that, it is not known whether π+e is one.

There are many definitions of these numbers, but the coolest one, again from the book itself, was that there is no expression which is of finite length, which can accurately express such a number.
Waitaminute, so what about 22/7? Well , that goes out of the window. 22/7 actually is wrong after the first couple of decimals. It is used proabably because it is a good approximation and is easy to remember. For an accurate formula, try this one -

Or this...

I remember how I used to cringe every time the 7 remained in the denominator, but in retrospect, given the bad boys above, 22/7 has been a blessing all these years. (All these formulae are from here, btw, Wiki rocks!)

Two things amazed me while reading these pages on Wikipedia - one, by the level of detail and breadth of knowledge and the various problems, both solved and unsolved. (Look at the Wiki link for a list of such problems);Reminded me of Fermat's last theorem - which Fermat mentioned as a note in the margin, and also added that he had a very elegant proof in mind, but the space there (in the margin) was not enough. That added to the mystique of the theorem, especially as no one could solve it for over 350 years. It was solved in 1995 (357 years later) and the size of the proof - 200 pages - slightly more voluminous than as hinted by Fermat.

The second thing was the Wiki - Almost all things I have looked up, I have found very rich and well researched information on the Wiki. It truly is a modern miracle. Over 4 million pages and 55 million edits, it is the 17th most visited website as measured by Alexa.

Not so much an oddity anymore, but pretty much a mainstream website.

More ramblings soon...

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