What are the steps to sketching of complex number equations?

Question

Eg. Sketch on the complex plane the set of complex numbers z that satisfies the conditions and determine the cartesian equation of each:
1) {z: Re(z)+Im(z)=6}
2) {z:Iz-4iI<3
3) {3
Thanks!

Answer 1

In the complex plane, the real part is on the x axis and the imaginary part is on the y axis.

So for number 1, just substitute x for Re(z) and y for Im(z) and you get the equation x + y = 6. I'll assume you know how to graph that.

So if x is the real part and y is the imaginary part, then the magnitude of z, |z| = √(x² + y²). So for the others, expect to draw a circle.

z - 4i is the same as taking z and shifting it down 4 (subtracting 4 from y).

So |z - 4i| = | x + (y - 4)i | = √(x² + (y - 4)²)

|z - 4i| < 3
√(x² + (y - 4)²) < 3
(x² + (y - 4)²) < 9

So this will be a circle, centered at (0,4), with a radius of 3. The < sign means that you will color in the circle, since all points inside of it satisfy the inequality.

The last one is just

3 < √(x² + y²) < 5
9 < (x² + y²) < 25

So that's TWO circles, centered at the origin, one with radius 3 and one with radius 5, and you color in everything between them to get a "ring".

Answer 2

(Remember the y axis is imaginary and x is real) For these equations, just let z = x+yi and go from there. for example for 2, x^2+(y-4)^2<3 would be your graph. that's a filled circle at (0,4) with radius 3. For the first, Re(z) = x and Im(z) = y. So x+y=6 is the line to graph.

Answer 3

a) write it interior the widely used format for a quadratic: a x^2 + b x + c = 0 x^2 + 0 x + 16 = 0 the position a = a million, b=0 and c=16 Then use the quadratic formula: x = [ -b +/- SQRT( b^2 - 4ac ) ] / 2a b) ? something lacking from the question? c) both numbers are: ok and (ok+17) their product (p) is: p = ok(ok+17) The minimum value will happen the position dp/dk = 0 have relaxing