Solving simple complex equations I'm starting to read a book on complex analysis, and I'm having some troubles envolving simple equations with complex numbers. How can I solve equations envolving these numbers- what methods and strategies do you recommend? When it envolves aspects like $\bar z$, $|z|$ and $Arg(z)$, what should I do? Please take this equation as an example:$$|z|-z=1+2i$$
 A: For your specific question, remember that $z$ can be written as $z=x+iy$ and $|z|=\sqrt{x^2+y^2}$ is a real number. Also, $a+bi=x+yi$
 if and only if $a=x$ and $b=y$ at the same time.

There are some handy tricks when dealing with complex number equations. Many of them will involve replacing $z$ with one of these:


*

*$z=x+iy$

*$z=|z|(\cos\theta+i\sin\theta)$

*$z=|z|e^{i\theta}$


where $\theta=Arg(z)$. Then you will be able to apply some formulas and identities and to get your result. 
Sometimes it may be useful to think about it as a point in the complex plane. Then the trigonometric representation will make more sense. 
Other times, when dealing with more analytical expressions (such as complex polynomials), it may be better to think in terms of the exponential.
Sometimes it will suffice to compare the real and imaginary terms (as in your exercise).
In the end, it all comes down to practice and familiarization. Do many exercises, work out some proofs in your books and try to work with them in other subjects, like Linear Algebra. Once you do, you'll have one of the most powerful tools of mathematics and physics at your disposal.
A: Let $z = x + iy$. Rewrite your equation as $|z| = 1 + 2i + z$. Since $|z|$ is a real number, we conclude that the imaginary part of $z$ must be $-2i$. So the equation we're really trying to solve is
$$\sqrt{x^2 + 4} = 1 + x$$ This is equivalent to $$x^2 + 4 = 1 + 2x + x^2$$ i.e.,  $x = \frac{3}{2}$.
So the only solution is $z = \frac{3}{2} - 2i$.
