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I've read a bit about the history of the complex numbers, and many seem to credit Caspar Wessel with the idea of associating the complex numbers as points on a 2-dimensional plane (or at least the first to explicitly publish the idea). But what was the motivation behind this? Why did people make this leap, and what were their justifications for this?

edit: just to be clear, my question is only about the justification of identifying complex numbers as points on a plane, NOT about justifying complex numbers themselves.

nilcit
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    People like to make things more complex, it is in our nature. – copper.hat Mar 27 '16 at 02:48
  • But introducing the complex plane actually simplified things enormously, didn't it? I'm just wondering exactly how people came up with the idea, and their justifications for it. – nilcit Mar 27 '16 at 02:53
  • People have always been interested in the solutions to polynomials. We needed somehow to extend $\mathbb{R}$ to include the solution to equations like $x^2 + 1 = 0$, whose solution requires the ability to take the square root of $-1$. It turns out that the extension of $\mathbb{R}$ to $\mathbb{C}$ contains ALL of the solutions to any polynomial with coefficients in $\mathbb{C}$. – Edward Evans Mar 27 '16 at 02:54
  • The ironies of life... – copper.hat Mar 27 '16 at 02:54
  • I would assume simply because humans like pictures. Complex numbers $a + bi$ are basically just ordered pairs $(a, b)$ of real numbers, and we know how to put those on a plane, thanks to Descartes. (I'm assuming this question isn't about the birth of complex numbers, but literally just putting them on a plane) – pjs36 Mar 27 '16 at 03:19
  • @pjs36, you're correct, my question was only about putting them on to the plane, not the birth of them. How did people at that time realize that $a + bi$ is "basically" just an ordered pair? Without modern ideas of vector spaces and field extensions, how did people "realize" this in the 17/1800s? – nilcit Mar 27 '16 at 04:56
  • @nilcit Weird question, most of the times I introduce the complex numbers as $\Bbb R^2$, with regular vector $+$, and a "modified" $\cdot_{\Bbb C}$ (product) operation (i.e $\Bbb C= (\Bbb R^2,+,\cdot_{\Bbb C})$) to new students, only later, I show the notation $a+bi$ (after proving $(0,1)^2=(-1,0)$). – YoTengoUnLCD Mar 27 '16 at 05:02
  • Yeah, I pretty much think of them that way too. But I'm curious how people in the early 1800s, who didn't have access to the modern math that we do, came up with the idea. Historically, the $a + bi$ stuff came first, then the ordered pair business. The modern way does it the other way round, and it's much clearer, but I'm just curious how the original pioneers thought about it. – nilcit Mar 27 '16 at 05:03
  • @nilcit Wouldn't the function $\exp$ be enough motivation (representing complex numbers by the pairs $(r,\theta)$ makes them much easier to deal with in a lot of cases, and it's only natural to associate this with the same point in $\Bbb R^2$)? – YoTengoUnLCD Mar 27 '16 at 05:05
  • @yoTengoUnLCD Again though, the exp function would (historically) come after Wessel decided to identify complex numbers as points on a plane. – nilcit Mar 27 '16 at 05:08
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    @nilcit Fair enough! I don't know anywhere near as much math history to keep up any (useful) discussion. Good luck! – YoTengoUnLCD Mar 27 '16 at 05:12

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Complex numbers first arose in the works of Cardano and Bombelli in the 16th C who showed that formal calculation with square roots of negative numbers with the assumptions that:

(1) $(a+ib)+(c+id)=(a+c)+i(b+d)$

(2) $(a+ib)(c+id)=ac+i(bc+ad)+i^2bd=(ac-bd)+i(bc+ad)$

where $i$ is a formal symbol such that we may replace $\sqrt {-1}$ by $i$ led to the solution of many geometric problems. An example was the value of $x$ at the intersection of the graphs of $y=x^3$ and $y=15x+4$. This however led to uncomfortable questions like what kind of an animal is an entity $a+ib$? Merely defining it as a formal expression did not seem sufficient and it was a cause of great suspicion whether manipulation with such formal symbols was mathematically justified.

Wessel gave a formal mathematical meaning to the complex numbers by defining it as an ordered pair. His work was followed up by Argand and Gauss. This once and for all dispelled all doubts regarding calculations with complex numbers in the computations of solutions of cubic equations and other related problems.

Further edit: Wessel's motivation may not have been directly to give a meaning to complex numbers. He had a background in cartography and surveying and was naturally motivated to problems involving direction. He gave geometric meaning to "algebraic operations" on directed line segments, in other words he defined addition and multiplication geometrically for such directed line segments. Addition of course was done by the familiar parallelogram law, while for multiplication he defined an abstract unit $1$ and gave the now familiar similar triangle based definition. This also led him to define another directed line segment by rotating $1$ by $\pi/2$ and he observed that under his multiplication scheme this became $\sqrt{-1}$. Next Wessel established algebraic properties of these directed line segments, such as associativity, commutativity etc. Since these properties were exactly the same as those of complex numbers it was no great leap to make and conclude that the directed line segments were precisely one way to define complex numbers. As far as I am aware Wessel did not explicitly claim this though. However it is implicit in his work and Wessel was probably aware of it. Later mathematicians refined this idea and since an ordered pair such as in Cartesian or polar coordinated characterizes a directed line segment so the idea that a complex number is an ordered pair was born.

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    They're not "negative square roots"; rather they are square roots of negative numbers. $\qquad$ – Michael Hardy Mar 27 '16 at 04:13
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    @MichaelHardy: Corrected. Thanks. –  Mar 27 '16 at 04:16
  • So my question is exactly what prompted Wessel to define them as ordered pairs, and what was his justification for doing so? – nilcit Mar 27 '16 at 04:53
  • @nilcit: See my updated answer. –  Mar 27 '16 at 05:12
  • @Shahab THANK YOU! You're the first to address the question I was originally asking. You mentioned that he "observed that under his multiplication scheme this became $\sqrt{-1}$". Do you happen to know what his multiplication scheme was? I'm curious what he could've possibly have been doing that seemed to coincidently give him complex multiplication. – nilcit Mar 27 '16 at 05:19
  • @nilcit: I believe in modern language it must be equivalent to the fact that the triangles with one vertex $0$ and the other $1,z_1$ and $z_2,z_1z_2$ respectively are similar. I think you can see the explanation in any standard complex analysis book for that. –  Mar 27 '16 at 05:29
  • Yeah, I had meant to ask, what did you mean by the "familiar similar triangle based definition?". I don't think I know what that means. Is there a phrase or something I could Google to look up what you're talking about? – nilcit Mar 27 '16 at 05:31
  • Let us continue this discussion in chat. –  Mar 27 '16 at 05:40