paradox

Arrow’s Impossibility Theorem

Elections are democracy in action. People go to polls and express their preferences, and somehow we must aggregate the preferences of many individuals to make a joint decision. So the choice of voting method is very important. Is there an ideal voting method? According a...

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All Numbers are Interesting

There are clearly many interesting whole numbers. For instance, 2 is the only even prime number, 3 is the first odd prime, 6 is a perfect number (the sum of the proper divisors is the number itself), etc. But did you know that all whole...

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Derivative Paradox

Here’s a fun (but untrue) fact. You know from calculus that the derivative of x2 is 2x. But what’s wrong with the following calculation? x2 = x + x + … + x (repeated x times) so by taking the derivative of both sides we get (x2)’ = 1...

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One Equals Zero: Integral Form

Consider the following integral: INTEGRAL (1/x) dx Perform integration by parts: let u = 1/x , dv = dx du = -1/x2 dx , v = x Then obtain: INTEGRAL (1/x) dx = (1/x)*x – INTEGRAL x (-1/x2) dx = 1 + INTEGRAL (1/x) dx which implies that 0...

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One Equals Zero!

The following is a “proof” that one equals zero. Consider two non-zero numbers x and y such that x = y. Then x2 = xy. Subtract the same thing from both sides: x2 – y2 = xy – y2. Dividing by (x-y), obtain x + y = y. Since x = y, we see...

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