The history of how the concept of complex numbers developed is convoluted.

On physics.stackexchange questions about complex numbers keep recurring. It seems to me this indicates that when authors of physics textbooks introduce complex numbers there is a need (in the students) that is not met.

Let me make a comparison.
When non-euclidean geometry was introduced (spherical geometry and hyperbolic geometry) the development to robust branches of geometry was fairly quick.

In the case of complex numbers: from the first inklings to comfortable understanding took centuries.

There is a theorem in the theory of equations that states that the field of complex numbers is algebraically closed.

Quoting the wikipedia article about the fundamental theorem of algebra:
Every non-zero, single-variable, degree n polynomial with complex coefficients has, counted with multiplicity, exactly n complex roots.

This property of the field of complex numbers is sometimes presented as the most important property of complex numbers. I agree with that statement, but it took mathematicians centuries to come to that realization.

It appears to me that for centuries complex numbers had quite a nebulous status. I'm using the name 'complex number', but of course throughout those centuries, throughout the stages of understanding, there were various names for the concept.

Histories of complex numbers describe that Descartes coined the name 'imaginary number'.

I tried to find whether Isaac Newton had ever discussed the imaginary unit. The contributions of Isaac Newton to mathematics are wide and varied, but it appears Newton never discussed the concept of the imaginary unit, or complex numbers, in whatever form. (Arguably that's a deafening silence.)

Next in the history of mathematics is the explorations of the exponential function, and the sine and cosine function, by Euler.

As we know, the exponential function is its own derivative. The sine function is its own derivative too, only it goes through a 4-step cycle.

$$\frac{d(\sin(\alpha))}{d\alpha} = \cos(\alpha) $$

$$\frac{d(\cos(\alpha))}{d\alpha} = -\sin(\alpha) $$

$$\frac{d(-\sin(\alpha))}{d\alpha} = -\cos(\alpha) $$

$$\frac{d(-\cos(\alpha))}{d\alpha} = \sin(\alpha) $$

As we know, the imaginary unit $i$ provides an efficient way to express the connection between the exponential function and the sine function

$$ e^{ix} = \cos(x) + i \sin(x) $$

At this point in time this formula, Euler's formula, is not yet particularly important. But not long after that physicists start recognizing that many phenomena involve oscillation, and many phenomena involve propagation of transversal waves.

It seems to me that it is only around here that things start to come together for complex numbers.

It appears to me that until then complex numbers were, if used at all, used in an ad hoc manner. Starting with the work of d'Alembert and Cauchy I get a sense of systemization.

I'm curious whether people will agree with this view.

The history of how the concept of complex numbers developed is among the most convoluted in the history of mathematics. Compared to other concepts complex numbers remained nebulous for a very long time.
Do you agree? Do you disagree?

  • $\begingroup$ It is not quite clear what are you asking about. I suggest that you read "How to ask questions" on this site. $\endgroup$ – Alexandre Eremenko Aug 5 at 1:26
  • $\begingroup$ @AlexandreEremenko Yeah, being a long time user I was aware that the form of my question is not along the stackexchange guidelines (as in, for example, asking for opinion). The underlying question is: in physics education, how should complex number notation be introduced? In the history of mathematics complex numbers were used to an extent, but at the same time the concept was generally not trusted. This lasted centuries. The physics student, when first encountering complex number notation, feels that same instinctive discomfort. I feel that discomfort should be acknowledged as understandable $\endgroup$ – Cleonis Aug 5 at 3:34
  • $\begingroup$ I think "the physics student" should be replaced with "some physics students", and I doubt that the historico-educational analogy applies even to all those. The historical discomfort over complex numbers dissipated when a geometric model was found, in exact parallel with hyperbolic geometry. Spherical geometry, where it was plain, was studied since antiquity. Modern discomfort has more to do with the misguided idea that real numbers are "more real" inculcated by earlier education. $\endgroup$ – Conifold Aug 5 at 5:48

This is a reply to some of the points you have made rather than an answer to the very broad question you have posed, which is essentially "yes, it is a very convoluted and tortured history".

A reliable source on the subject of the history of complex numbers is the mathematical historian Leo Corry.

According to Corry and contra your source, Descartes used the term “false” rather than imaginary, as Cardano had also done much earlier, though Descartes did refer to them as existing “only in our imagination.” While Descarte seemed to understand the basic idea behind the fundamental theorem, namely that if $a$ is a root of the polynomial $f(x)$, then $f(x)$ can be divided exactly by the factor $(x-a)$, this had already been stated formally by Albert Girart in his 1629 work L’invention en algebra.

Corry writes that Newton may have been the first to formulate a version of the fundamental theorem of algebra, to be found in his early lecture notes from Cambridge. It asserts that

[T]he number of roots of a polynomial equation cannot surpass the highest order of the unknown in the equation, but these roots may be positive, negative, or “impossible” (rather than “imaginary”). And what he meant by impossible he explained by reference to the equation $$x^2 - 2ax +b^2 = 0$$

Here, we obtain two roots, namely $$a+\sqrt{a^2-b^2}$$ and $$ a - \sqrt{a^2-b^2}.$$

Now, when $a^2$ is greater than $b^2$ - Newton wrote - the roots are “real”. In the opposite case [...] the root “impossible”. But, interestingly, Newton nevertheless went on to stress that both expressions are roots of the polynomial for the simple reason that when they are introduced into the equation in place of the unknown, the equation is satisfied because “their factors eliminate each other.” In other words, a square root of a negative number is an impossibility and hence does not represent a number in the proper sense of the word, but expressions containing such impossible entities are legitimate roots of an equation and allow for an appealing formulation of the fundamental theorem of algebra, as Newton conceived of it.

Gauss proved the fundamental theorem in 1799 and appears to have been working with a geometric interpretation of complex numbers. However, efforts to better understand the nature of imaginary (and indeed negative) numbers continued. Success started with the work of mathematicians such as Caspar Wessel (largely ignored), Jean-Robert Argand, Jacques Français, and several others including a brilliant interpretation by Adrien Quentin Buée.

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  • $\begingroup$ Thank you, particularly for the corrections regarding Newton. Newton did much work in the field of general methods for finding roots of equations, so inevitably he is going to run into some precursor concept of complex numbers. Another source mentioned that de Moivre at some point stated that as early as 1676 Newton already had an expression equivalent to de Moivre's theorem $\endgroup$ – Cleonis Aug 4 at 0:42
  • $\begingroup$ Anyway, the way my thinking goes now is that today's usage in physics of complex numbers is pretty much excusively in the context of Euler's formula. And Euler's formula isn't about complex numbers; its about the deep relation between the exponential function and the sine function. $\endgroup$ – Cleonis Aug 4 at 0:47
  • $\begingroup$ @Cleonis It is a noteworthy historical point is that Roger Cotes pubished Euler's formula in the equivalent form $ln(cos\theta + isin\theta) = i\theta$ in 1714, some 34 years before Euler, although the $ln$ function is multi-valued over $\mathbb C$. $\endgroup$ – Nick Aug 4 at 3:31
  • $\begingroup$ I chose to emphasize Euler's formula because that representation embodies that oscillations and propagating transfersal waves can fluently be represented with the exponential function. That's the physicists' bread and butter. Still, it would be interesting to learn how Cotes arrived at that form, as that would provide information on how at that time the-unit-that-is-the-square-root-of-minus-one was understood. $\endgroup$ – Cleonis Aug 4 at 6:54
  • $\begingroup$ In general a hard lesson in history is: don't rely on secondary sources, all to often the story is highly embellished. (Case in point, I succumbed to the temptaton of embellishing myself, recently.) In an answer to a Quora question about Cotes and Euler Dean Rubine went to the primary source. iImplicitly it's kind of there, but certainly not as explicit as the modern notation suggests. $\endgroup$ – Cleonis Aug 4 at 7:44

This is not a self-answer, I'm using this space to present my views as they have evolved to this point.

I want to recapitulate the following:

The exponential function is it's own derivative:

$$ \frac{d(e^x)}{dx} = e^x $$

The hyperbolic sine and the hyperbolic cosine are each others derivative; a 2-step cycle.

$$ \sinh = \tfrac{1}{2}e^x - \tfrac{1}{2}e^{-x} $$

$$ \cosh = \tfrac{1}{2}e^x + \tfrac{1}{2}e^{-x} $$

That is, we can think of the $\cosh$ and $\sinh$ functions as constructed out of the exponential function in such way that you get a sign flip resulting in a 2-step cycle.

For the trigonometric functions the taking-the-derivative cycle is a 4-step cycle, with sign flips.

It doesn't matter how that 4-step cycle is implemented, but the more efficient the better of course.

$$ e^{ix} = \cos(x) + i \sin(x) $$

I have now convinced myself that this formula is exclusively about the connection between the exponential function and the sine function.

Using complex number notation is purely a matter of implementation here; it so happens that it fits the bill.

In physics the exponential-function-in-4step-cycle is ubiquitous of course. the exponential-function-in-4step-cycle allows efficient manipulation of quantities such as frequency, amplitude, phase.

If I were to write a physics textbook I would treat Euler's formula as a standalone formula, emphasizing that it is about the connection between the exponential function and the sine function.

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