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As for the square root of two, ever tried using taylor polynomials? I get an odd sense of satisfaction from taking the value of e, paper, pencil, and finding the result of a number raised to a fractional power and getting a result that agrees with my TI-83 to about four decimal places :) It takes a while if you do it by hand, though.
MsMittens already recommended Singh's books before; haven't had a chance to read them yet though :(
Anyways, here's what Fermat's Enigma was all about:
We all know Pythagoras' theorem. There are lots of trios of integers that qualify for Pythagoras' equasion (a² + b² = c², eg. 3,4 and 5). There are even more non-integers that qualify for the equasion (1, 2 and SRQ{5},...). Let's focus on the integers though...
You also could try this with a³ + b³ = c³: search for integers a, b and c that qualify for a³ + b³ = c³.
Pierre Fermat stated in the 1730's that you can't find integers a, b and c so that a³ + b³ = c³, or more generally for every n-th power bigger than 2.
He claimed to have found the ultimate evidence for his theorem, but it was never found.
In 1993, Andrew Wiles finally presented the complete evidence... Most specialists don't believe that Fermat ever found the evidence himself (based on the limited mathematical knowledge in the 1700's).
This is what differentiates computers from human beings and what makes Fermat's theorem particularly interesting: you can run computers to find billions times trillions of integers that do not qualify for a³ + b³ = c³; that still wouldn't proove the theorem (sure, the chance it IS true will grow, but the computer can never be sure about it - it's enough to find three integers that DO qualify for the equasion, and your 'computer-proof' is worthless...). It took a human being to proove that searching for those true integers is a waste of time...
blow: For precision.
Weirdos... :D
I thought pi was just 22/7... :(
no pi isnt quite 22/7...... what IS the real formulia for pi? 22/7 is what they tell you when your 10 to plug into your calculator incase you forget 3.14, it agrees closly but veers off after the first few digits. what is the actual formulia? any one know?
The circumference of a circle divided by its diameter = pi
SunDots - I'll post the source code in this thread as soon as I can find it. It was a few years ago, and I have to check my backups for the file itself. As soon as I do, it'll be up. :-)
AJ
22/7 is just like "3.1416", they are approximate values that people use because they are too lazy to use a closer approximation such as 3.141592 :D
As for ye olde 'proof by computer', the essential problem is that you can't program a computer to think 'outside the box' or to derive 'obvious' things. That, and mathematics is still an uncertain science (albeit greatly more rule-based than some less cut-and-dried sciences.)
For instance, take a circle. Programming a computer to find the points on a circle would cause it to never find the number of points, but the limit would be at infinity. Yet if you ask a person, they might say that it either has no points or infinite points. And in a way, both answers are correct. On one hand, an infinite number of points would cause a 'smooth' circle/polygon.
Othe other hand, if a straight line is drawn between points, a circle has no straight lines, only a continuous curve, therefore it has no points...
Bleh. Have I sidetracked things? :)