Undergrad Sketching Complex Numbers in the Complex Plane

[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?)

 

[member 587159]

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?)

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

 

[Mark44]

And, specifically, why {z∈C : Rez =|z−2|} is a parabola?

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?

 

[MickeyBlue]

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?

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?

 

[Mark44]

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?

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

 

[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.