I Sketching Complex Numbers in the Complex Plane

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1. Oct 26, 2016

MickeyBlue

I've just had my first batch of lectures on complex numbers (a very new idea to me). Algebraic operations and the idea behind conjugates are straightforward enough, as these seem to boil down to vectors.

My problem is sketching. I have trouble defining the real and imaginary parts, and I don't understand how some subsets translate into conic sections.

Does anyone have any tips or advice on complex number sketching? (And, specifically, why {z∈C : Rez =|z−2|} is a parabola?)

2. Oct 26, 2016

Math_QED

What do you mean? Sketching a number like $z = 2 + 3i$ in the complex plane?

3. Oct 26, 2016

Staff: Mentor

Let z = x + iy
What is Re(z)?

Geometrically |z - 2| represents the distance between an arbitrary complex number and the number 2 (a purely real complex number). How do you calculate the magnitude of a complex number?

4. Oct 26, 2016

MickeyBlue

x is Re(z). The magnitude is the square root of the sum of the real part squared, and the imaginary part squared. Does this mean that Re(z) = I x + iy - 2I?

5. Oct 26, 2016

Staff: Mentor

Yes. Now can you simplify the right side by finding the magnitude? Note that x + iy - 2 = x - 2 + iy.

6. Oct 27, 2016

MickeyBlue

Oh, I think I see now. Re(z) = x = I x + iy - 2I. This can be simplified to:

x = √(x-2)2 + y2

x2 = (x-2)2 + y2

0 = y2 - 4x +4

x = ¼y2 + 1

Thank you so much. I must not have taken note of the modulus.