Original post by math42So we want to show that the equation |z| = z + 2i has no solutions, where z is a complex number.
To do this, we assume that it does have a solution, and see where this leads us. If it leads us to something which is clearly wrong/makes no sense, our initial assumption was wrong. This is a "proof by contradiction", which you may have seen.
Well, start by writing z = x + iy, where x,y are real (we know by definition that all complex numbers can be written this way). The equation for |z| is then |z| = root(x^2 + y^2) - it is the length of the line joining z to 0 in the complex plane.
So we have root(x^2 + y^2) = x + iy + 2i = x + i(y + 2).
The real and imaginary parts on each side have to match up. The left hand side is real, the right hand side has the imaginary bit i(y + 2). This must be 0 to match up with the left hand side, so y + 2 = 0, i.e. y = -2.
Then we get root(x^2 + y^2) = x. But y = -2, and (-2)^2 = 4, so root(x^2 + 4) = x. Can you see why this doesn't make sense?