I was shown the following proof a few days back and it left me puzzled:
1 = 1
also,
-1 = -1
Now lets divide both sides by 1. Thus:
-1/1 = -1/1
thus,
1/-1 = -1/1
Now lets take the square root on both sides (i.e. raise to 1/2. When i use the word "sqrt", what i mean is raising it to (1/2))
sqrt(1/-1) = sqrt(-1/1)
sqrt(1) /sqrt(-1) = sqrt(-1)/sqrt(1)
Lets call
x = sqrt(1) (which means, x = (1)^(1/2))
y = sqrt(-1), (which means, y = (-1)^(1/2))
then we have:
x/y = y/x
or,
x^2 = y^2, and thus
-1 = 1
Clearly this is absurd, so what went wrong ?
We can see that we are substituting x = 1^(1/2), which is supposed to mean that x^2 = 1, but x^2 = 1 means that x could as well have been - 1 i.e. - (1^(1/2)). Similarly, if we substitute y as sqrt (-1), then y^2 = -1, so y = +/- sqrt(-1). Essentially, if we call sqrt(1) as x, we must also understand that -x is also sqrt(1), and as a result the trouble with this solution is that after a stage it become "ambiguous".
On similar lines, if we define 'j', the imaginary number as +sqrt(-1), we face the following problems:
A) j^2 = j * j = sqrt(-1) * sqrt (-1) = ((-1)^(1/2))^2 = -1
B) j^2 = j * j = sqrt(-1) * sqrt (-1) = sqrt (-1 * -1) = sqrt(1) = +/- 1
i.e. ambiguous, and this is the reason we define j^2 = -1 (and j = +/- ((-1)^(1/2))). In fact Wikipedia is also careful about defining the imaginary number in this way (http://en.wikipedia.org/wiki/Imaginary_number) i.e. basically that we note that by this definition, j = +/- sqrt(-1) (and not j = + sqrt(-1)), so when someone uses e^(jtheta), it means e^((+/- ((-1)^(1/2)) * theta) -- kinda sucks -- but we have notation, to help us get used to this mess.
Thus we must be careful in defining the imaginary number as a "imaginary" square with area -1.
in a land that uses square coins, and if we owe -5 dollars to someone, we really owe 5 "imaginary" square coins. Someone can always argue that these coins are "imaginary" and impossible to "feel" by the senses, so there is no way one can give such a coin. However, we DEFINE a way to end this debt, by saying that giving a "real" coin, eliminates the imaginary coins' debt.
Other ways of looking at the imaginary number is as a "rotation" in the complex plane (using x+iy), but we sometimes loosely use j = sqrt(-1) and in my opinion it is not right i.e. has the potential to cause quite a bit of confusion.
