What's the most overcomplicated proof of a simple proposition you've seen or can think of?
An outline of the proof or a link to it is perfectly acceptable. The proposition should be simple (say early college level at most), but the proof can be as deep as you like.
Answering another question made me realize how poor my mental stock of completely overdone proofs is, hence this question.
2011-11-04T04:33:21Z
Here's a random and not-very-good example.
Theorem: 7 is prime.
Proof: Checking every possible group table for groups of order 7, one finds each valid group to have an order 7 element, i.e. to be cyclic. If 7 were composite, we would find a non-cyclic group, since for n=ab, <a> x <b> has order ab = n. Thus 7, using a proof by contradiction, an extensive computer search, and some basic group theory, is prime.
2011-11-04T08:55:06Z
Thank you for the answers so far. They've reminded me of 3 more, though they're above the "early college" level I set: computing the fundamental group of the circle, showing the 2-sphere is simply connected, and proving the isoperimetric inequality. Each of these can be cast in early college level terminology, so they (and similar statements) are also fine.
?2011-11-07T15:42:53Z
Favorite Answer
There is an old book titled "Mathematic Made Difficult" by Carl. E Linderholm that is loaded with such stuff. Wikipedia lists the start of his proof that 2 is a prime number. It's a wonderful book written for mathematicians. A bit humorous yet it takes a serious look at things we tend to take for granted. Like many of my books that I love and have read dozens of times, I loaned it to someone and never saw it again.
Personally, I have always loved the derivation of the equation of an ellipse from the two focus definition. There is an elegance to the way it almost explodes in size and then shrinks down to a simple equation.
There's no shortage of "overcomplicated proofs" in mathematics, where some of the simplest propositions seem to require mind-bogglingly complicated proofs. A classic that comes to mind is the Jordan Curve Theorem, which states that every non-self-interesting continuous loop on the plane divides it into two regions. See link for Jordan's proof. Another is proving that 1 + 1 = 2, which actually ended in failure for Russell and Whitehead in their herculean treatise "Principia Mathematica" (the failure being due to Kurt Godel's incompleteness theorems). Just recently, I think I "proved" that given a non-trivial triangle on a plane and a line not passing through any of its vertices, then the line cannot pass through all three of its sides. See link for my attempt, and Duke's comments on this subject.
Edit: Josh, glad I helped you jog your memory. Like my nephew has said to me more than once, "sometimes I realize that I'm smarter than I think I am".
The proof that a negative number multiplied by a negative number makes a positive product is quite complicated when done rigorously. You must start from the definition of a negative number. To avoid lots of brackets, I will use x' for negative x.
First define negative x by the equation x + x' = 0
Now prove that positive and negative multiply to negative. 0 = 0*y = (x + x')*y = xy + x'y = 0
But by definition xy + (xy)' = 0 ---->(x')*y = (xy)' or the product of a negative multiplied by a positive is negative. Something very similar can be done for the two numbers in reverse order.
Now consider 0 = (x + x')*y' = x*y' + x' *y' ----> (xy)' + (x')*y' = 0 from above. But by definition (xy)' + xy = 0 ----> (x')*(y') = xy The product of two negative numbers is positive.
since you alreadly have answers... can you help me with geometry? PLEASSSE! IM STUCK IN PROOFS! http://answers.yahoo.com/question/index;_ylt=AiglPIy_IISFSs0wAs8J7ebsy6IX;_ylv=3?qid=20111105121349AAwQRia