Why can we identify complex numbers as points on a plane?

Question

Modern mathematicians seem to define the complex number $a+bi$ as the ordered pair $(a,b)$, with the usual rules for complex addition and multiplication. I'm reading a book on the history of the complex numbers and it mentions that Wessel was the first to associate complex numbers with points on a plane, with the imaginary axis perpendicular to the real axis. It also says that others like Gauss had similar ideas at around the same time. I'm failing to see the intuition though. What is the justification or motivation for identifying complex numbers with points on a plane, with the imaginary axis perpendicular to the real one (without resorting to modern ideas of vector spaces, since Wessel and Gauss didn't have this machinery)?

Answer 1

Edit: The 90 degrees thug just indicates the independence between the real and imaginary parts.

The insight in using a complex plane is that by doing so, you can forget about the imaginary unit per se, and instead visualize it along an axis. Up and down being positive and negative imaginary part. Left and right being negative and positive real part. Every complex number can be expressed in the form $a+bi$. So, every complex number corresponds to a single point on the plane.

Well at the very least, it's a helpful way to break the complex number down into real, manageable quantities. Recall that the whole idea of the imaginary unit was uncomfortable, the least to say , for the mathematicians of the day. Italso has a geometrical value to see it in a pseudo-Cartesian light. Take for example finding the nth roots of a complex number. The roots are spaced out equally around the origin in the plane. Similarly, unit circle mathematics, and Euler's identity for $e^{i \theta}$ can be intuitively applied in a plane. You can basically use much of the analysis that had been developed for the Cartesian plane, including the work on vectors, and consequently phrase algebraic complex problems in the language of analysis that those mathematicians were familiar.