I must find both such complex solutions and express them in Euler form and usual form.
So it's been a while since I've touched the imaginary/real plane. However, from what I remember, $z = a + bi$. If $|z| = 1$ we can assign specific values to a and b.. what would these values be?.. And how would we use the result to find specific $z$ such that $|z| = |z-1|$?
Hints:
Write $\;z=x+iy\;,\;x,y\;\Bbb R\;$ , so that
$$\begin{cases}|z|=1\iff x^2+y^2=1\\{}\\|z|=|z-1|\iff x^2+y^2=(x-1)^2+y^2\end{cases}\;\;\;\implies$$
$$2x=1\iff x=\frac12\;\;$$
So now just substitute in the first equation above and...
Hint: Think two-dimensionally. Our $z$ is equidistant from $(0,0)$ and $(1,0)$. Geometry shows that $z$ must lie on the perpendicular bisector of the line segment that joins $(0,0)$ to $(1,0)$. In complex number terms, the real part of $z$ is $\frac{1}{2}$.
One way to think of this geometrically is that the function $f(z) = |z - p|$ gives you the distance of a complex number $z$ from the point $p$. If we have the equality $$ |z - 0| = |z| = |z-1| $$ this tells you that we are looking for points $z$ that are equidistant from both the origin, 0, and the point $1 \in \mathbb{C}$.
It isn't too hard to see that this is the line of points $x + iy \in \mathbb{C}$ such that $x = \frac{1}{2}$ (draw a picture).
From this, it follows that you're looking for the intersection of this line with the circle $|z| = 1$, from which you should be able to determine your solution.
I think the mistake in your reasoning is here:
If |z|=1 we can assign specific values to a and b. what would these values be?
In fact, knowing that |z|=1 does not allow you to determine values of a and b. All that |z|=1 tells you is that z is exactly one unit from the origin -- in other words, it lies on a circle of radius 1, centered at 0.
Likewise, the condition that |z|=|z-1| means that z is equidistant from the two points 0 and 1, which means it lies on a line through 1/2 and perpendicular to the real axis.
Finishing touch: In how many points does that line cross the circle?
So if I were to draw out the point z, it would lie on a unit circle (radius 1) on a complex plane. Subtracting 1 from that point would displace it by just enough so that it is now lying on the circle but in a different quadrant. This gives us the value of a.. which is 1/2. We then substitute 1/2 into the equation (sqrt(a+bi)^2) = 1 to find the value of b. With these coordinates we can determine the angle theta as well.
Is that it?
– user122661 May 02 '14 at 19:32Note that $|z|=1$ if and only if $z=e^{it}$ for some real $t$. So we have to solve $1=|e^{it}-1|$. But that's easy: $$1=|e^{it}-1| \iff 1 =|e^{it}-1|^2$$ and $$|e^{it}-1|^2 =|(\cos t -1)+i\sin t|^2 =(\cos t - 1)^2 + \sin^2 t$$ $$= \cos^2 t -2\cos t + 1 + \sin^2 t = 2 - 2\cos t$$ This means we have $$1 = 2 - 2\cos t \iff \cos t = \frac12$$ which means that $$\sin t = \pm \frac{\sqrt 3}{2}$$ and so $$z = e^{it} = \cos t +i\sin t = \boxed{\frac12 \pm i \frac{\sqrt 3}{2}}$$
