Tim Reynolds
Premium Member
Oooh someone posted a math riddle. When does 3 + 4 = 111?
When you're working mod 104
When does 9+4=1.
When you're working mod 12
How about the flaw in this "proof"?
assume x > 0 (no tricky divide by zero nonsense)
x^2 = x^2 ; identity, also it is true on the interval x > 0
x^2 = x*x ; (expand the exponent)
x^2 = x + x + x + ... + x ; (x instances of the variable x) ; expand the meaning of multiplication
d/dx (x^2) = d/dx (x + x + x + ... + x) ; (x instances of the variable x) ; differentiate both sides
2*x = 1 + 1 + 1 + ... + 1 ; (x instances of the number 1) ; evaluate the derivative, remembering that the derivative is a linear operator.
2*x = x ; (add together the x instances of the number 1)
2 = 1 ; divide by x, which is ok here because I differentiate on the interval of x > 0, and because of this restriction diving by x has a meaning and is not undefined
2 must really equal 1 :rolleyes:
Chris
(serious this time) x^2 = x + x + x + ... + x is only valid for positive integer values of x. To take the derivative of the right side, it needs to be defined at all real values of x (well, to be precise, it needs to be defined in some neighborhood about whatever x_0 you're differentiating at). Since Z+ is not open in R, you can't take the derivative like that.
But 2=1 when you're working mod 1, so it all works out in the end.