 Let's have fun with unanswerable questions. Questions we can prove are unanswerable! Do you ever write about anything that has to do with WORDS as opposed to MATH? My dad is a math person and my mom is a word person, so I like both.... *crickets chirping*..... Ok, how 'bout those unanswerable questions? The coolest math fact ever is Thurston's geometrization conjecture/theorem. elspi:

do you mean that the fact that i called the coolest follows from Thurston's theorem? or that you have a different opinion on the coolest?

either way, i admit that i don't know nearly enough math to understand the explanation of that theorem on wikipedia. coolest math fact ever:
e^(pi*i)+1 = 0
proof time! I'm a big fan of pathological functions myself. Fun little examples, like the function that's defined everywhere, but continuous at exactly one point (f(x)=x on the rationals, f(x)=0 on the irrationals).

For more advanced functional fun, there's the fact that most (!) continuous functions are not differentiable at any point, and the fact that there exist space-filling curves. If an integer, n, is divisible by neither 2 nor 5, that integer *must* divide a number of the form 10^m-1 (where m is a positive integer). I have no idea how to prove this except that, given that 1/n, a rational number, is expressible as a repeating decimal, it must be the case that 1/n = b/(10^m-1), where b is the integer value of the repeating sequence in the decimal, and m is the number of digits in that sequence. (Yes, I thought this up when contemplating 0.999...=1).

I really think there should be a cleaner proof. Twinkle, that statement is special case of Euler's Theorem. Wikipedia has a nice proof, but it involves some group theory. for example you wanna buy something worth \$97.00, but you don't have money, so you borrow from you 2 friends \$50.00 each for a total of \$100.00. you still have change of \$3.00, you gave \$1.00 dollar to each of your friend whom you borrowed money from, that gives you 1 dollar left and a debt of 49 dollar from both of you friends. if you add, 49+49=98. plus the 1 dollar left with you. 99 in total. where did the other dollar goes? of the three dollars, you subtracted two of them (50-1=49) but then added the last (98+1=99). you should subtract that last dollar, 98-1=97, and 97 was the cost of the item.

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