Understanding/Visualizing Complex Numbers

Question

So I've always imagined complex numbers as a point in a 2D plane. Is this a problem? Is there a fundamental difference between a point on a 2D plane and a complex number? Will I have issues with more advance concepts regarding complex numbers due to such a visualization? How should I otherwise imagine complex numbers?

Answer 1

As topological spaces, $\mathbb{C} \cong \mathbb{R}^2$ so it is perfectly reasonable (and useful) to identify a complex number with a ordered pair of real numbers.

Answer 2

There is nothing wrong with that. The thing is that complex numbers, as a set, can be thought as the points on a plane. But you have more than a set. Namely, you have a set with a structure defined on it. You can add and multiply numbers and certain rules are satisfied. This is what they call a field. This structure is actually more fundamental than the set itself. You can actually replace the set by a different "substrate".

For example, the vector space $\mathbb{R}^2$ can be identified with the set of points on a plane too. But it is a completely different animal, it has a different structure. You cannot multiply vectors, for example.

Answer 3

I would say there's no harm in viewing them this way since, as vector spaces over $\mathbb{R}$, $\mathbb{R}^2$ and $\mathbb{C}$ are isomorphic. Regarding visualization, it might help your studies down the line if you think of a slightly modified geometric representation of complex numbers: every complex number can be written as $z=re^{i\theta}$. In this case $r$ is the absolute value, and $\theta$ describes the angle between the positive real axis and the number represented as a vector. So if you multiply two complex numbers $z_1=r_1e^{i\theta_1}$ and $z_2=r_2e^{i\theta_2}$, this corresponds to multiplying their absolute values and adding their phases and you'll end up with a complex number of the form $z_3=r_1r_2e^{i(\theta_1+\theta_2)}$. Without adopting this view further in your studies, you might not be able to appreciate what's happening geometrically.

Visualizing complex numbers as so as opposed to merely points in a set has helped me tremendously when thinking about their applications in AC circuits with apparent power, frequency responses, filtering, and sinusoidal voltage/current sources since their behavior is intrinsically described using complex numbers.