Untitled Page check out this picture !
i want to know what was the wrong step that guides us to -1=1 , thnx
Untitled Page check out this picture !
i want to know what was the wrong step that guides us to -1=1 , thnx
the problem, as other have pointed out, is this step:
$\displaystyle \sqrt{-1}\sqrt{-1} = \sqrt{(-1)(-1)}$
-1 actually has TWO square roots, i and -i. unfortunately, there is no way to tell "which is the positive one".
when you write out what this actually means you get:
$\displaystyle (\pm i)(\pm i) = \pm 1$ which is far less controversial.
if one writes out these complex numbers in polar form:
$\displaystyle (\cos\left(\frac{\pi}{2}\right) + i\sin\left(\frac{\pi}{2}\right))^2 = \cos(\pi) + i\sin(\pi) = \cos\left(\frac{2\pi}{2}\right) + i\sin\left(\frac{2\pi}{2}\right)$
that is, the "proper" square root of 1 to take on the RHS is the negative one, which leads to -1 = -1, which is hardly surprising.
put another way, "angles" aren't uniquely defined by a number, we have to restrict the range of these numbers. this can lead to confusing results, when talking about "half-angles"
(half of 360 degrees is 180 degrees, but half of 0 is 0, and these two angles (0 degrees and 360 degrees) represent "the same direction").