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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)?

nilcit
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  • You fail to see why the complex number $a+bi$ was identified with the ordered pair $(a,b),$ or you fail to see why the ordered pair $(a,b)$ was identified with a point in the plane, or both? – bof Apr 13 '16 at 03:02
  • I guess mainly I'm confused why $a+bi$ was identified with $(a,b)$. I'm OK with an ordered pair being a point on a plane. It doesn't seem obvious to me that the $a+bi$ can be thought of as a point on a plane, with the imaginary axis perpendicular to the real one (why not at 60 degrees or something?) – nilcit Apr 13 '16 at 03:05
  • Well, for one thing, the polar notation $e^{i\theta}$ directly relates to polar coordinates as if the reals are the "x" and the imaginary part is the "y". – Jared Apr 13 '16 at 03:06
  • I'm not sure how exactly this question differs from your other, or really what sort of thing would be included in a good answer, in your opinion. – pjs36 Apr 13 '16 at 03:14
  • @pjs36, it's basically the same question, I thought for a moment I understood it when I asked it previously, but I've fallen into confusion again, so I thought I'd try again, and see what other people have to say – nilcit Apr 13 '16 at 03:16

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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.

KR136
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    yeah, I definitely get why it's an extremely useful interpretation if it's valid. But I'm having trouble firmly convincing myself that it is a valid interpretation. – nilcit Apr 13 '16 at 03:07
  • @nilcit I think it is arbitrary to a certain extent. Because the imaginary/real parts are independent it is just another way of representing the number. It's valid because of the uniqueness of every complex number, when fully reduced. Every distinct point has a distinct real part and a distinct imaginary part. The logic behind it is that both the $a$ and $b$ in $a+bi$ can be treated as variables which range over many values. – KR136 Apr 13 '16 at 03:11
  • Sorry, I'm still confused about the 90 degrees thing. Without the notion of vector spaces, basis, etc.. How would it be justified that the real and imaginary parts are 'independent'? If I claimed we should identify $a+bi$ as a point on a plane with the imaginary axis at 60 degree to the real one, what would be the objection? – nilcit Apr 13 '16 at 03:13
  • You could do it that way. Do you really think that would work better? – bof Apr 13 '16 at 03:17
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    The issue with 60 degrees is that if you don't transform the entire coordinate grid accordingly, then there will be a point on the imaginary axis that has a real value greater than zero. Think about it the y axis were tilted on a normal coordinate plane. Then, there would be some points on the y axis that are "above" a point on the x axis and hence have an x-coordinate. The definition of being on the imaginary axis is that the real part is zero. If it were tilted them there would be a real part. In other words, the real part is strictly real and the imaginary part is strictly imaginary. – KR136 Apr 13 '16 at 03:18
  • Wait, then why does @bof think I could do it that way? – nilcit Apr 13 '16 at 03:23
  • If you transform the entire coordinate grid, then it makes sense. But you need to actually, preserve the grid relative to the transformation of the axes. Cf. eigenvalues/eigenvectors – KR136 Apr 13 '16 at 03:26
  • If you like literally tilt every line on the coordinate grid along with the axes, then you could theoretically make it work. But it would need to be homeomorphic-type translation of every point. – KR136 Apr 13 '16 at 03:28