# Describing points in the complex plane

When describing sets of points in the complex plane such as $$|2z-i| =4$$ would it be correct to describe the set exactly as we would in the $$x,y$$ plane?

For instance, Letting $$z=x+iy$$ for some $$x,y \in \mathbb{R}$$

yields the equation $$x^2+(y-\frac{1}{2})^2=4$$

And would it be correct to say this is the circle in the complex plane described by $$x^2+(y-\frac{1}{2})^2=4$$

• Do you know Euler's formula? It provides for an identification of $\Bbb R^2$ and $\Bbb C$. It's $e^{i\theta}=\cos\theta+i\sin\theta$. Then every $(x,y)$ can be identified with $(r,\theta)$ via $z=re^{i\theta}$.' The coordinates $(r,\theta)$ are called "polar coordinates". – Chris Custer Jan 22 at 3:36

Would it be correct to describe the set exactly as we would in the x,y plane?

If there is no more structure given to $$\mathbb{R}$$ and $$\mathbb{C}$$ else than the norms, I will say yes, since we have the isomorphism $$iso:\mathbb{C}\ni z=x+iy\mapsto (x,y)\in\mathbb{R}^2$$ (bijection that conserving norm).

And would it be correct to say this ($$|2z−i|=4$$) is the circle in the complex plane described by $$x^2+(y−\frac{1}{2})^2=4$$

Yes, in sense that $$iso(\{z\in\mathbb{Z}:|2z-i|=4\})=\{(x,y)\in\mathbb{R}^2:x^2+(y-\frac{1}{2})^2=4\}.$$

• Is the $i$ in front of the first set a mistake? – 68e1515 Jan 22 at 2:48
• @68e1515 Sorry, it may be misleading notation. I fixed it. It means the image of the given set by isomorphism from $\mathbb{C}$ to $\mathbb{R}^2$ can be uniquely specified as a set of $\mathbb{R}^2$. – An Jin Jan 22 at 2:54

I would say, rather, that this is the circle in the complex plane analogous to $$\displaystyle x^2+\left(y-\frac{1}{2}\right)^2=4$$.

The reason is, $$\left |z-\frac{i}{2}\right|=2$$ is well understood to mean a circle with radius $$1$$ and center $$z=\frac{1}{2}$$, and that the complex plane, as you note, uses $$z$$ instead of $$x$$ and $$y$$.

• What if the equation in the complex plane is $|z-1|=|z+i|$, can we say this is the line $y=-x$ in the complex plane?Or would we again say this is the line analogous to $y=-x$? – 68e1515 Jan 22 at 2:39
• You should have, after dividing by$2$, the equation $|z-i/2|=2$. This is of course the circle of radius $2$ centered at $i/2$. – Chris Custer Jan 22 at 2:48