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I'm learning Complex Numbers by myself. I've already understood basics concepts, but now I have problems with plots/graphs.

For example, I don't know, how to plot:
(1) $|z-1|+|z+1|=4$

(2) $|z+1| - \Im z \leq 1$

(3) $\Re\left(\frac{1-z}{1+z}\right) = 1$

($z \in \mathbb{C}$)

These examples are from some problem set I'm learning from (it's for first year college students)

I would really appreciate your help. I don't want to get answers for these particular problems. I want to understand, how can I plot similar things.

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Take z=x+iy and redice the equations in complex variables to an equation in $x,y\in\mathcal{R}$ Notice that all the operations in your question return real arguments (Re, modulus, Im, etc) – Please Delete Account Mar 20 '11 at 20:53

some of them are easier to view in $\mathbb{R}^2$, but some can be thought about directly. for instance 1) is the locus of points $z$ in the plane the sum of whose distances from the points 1 and -1 (i.e. (1,0) and (-1,0)) is 4. this is a description of an ellipse (you may recall).

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To graph these equations, I would just write out $z = x + iy$, and then use a standard graphing device, recognizing that we can identify $\mathbb{C}$ with the plane $\mathbb{R}^2$. For instance, for the first equation, you could write:

$$ |x+iy - 1| + |x + iy + 1| = 4 \Longrightarrow \sqrt{(x-1)^2 + y^2} + \sqrt{(x+1)^2 + y^2} = 4, $$

and then plot the equation as usual.

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Shouldn't there be square roots involved with your norm formula? – Dan Donnelly Mar 20 '11 at 21:25
Why, yes there should be. I apologize for not being sufficiently careful. – JavaMan Mar 20 '11 at 22:20

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