"Veritasium" (YT) - "The Big Misconception About Electricity" ?

[adx]

The question then remains whether this fundamental nature of the number (and complex plane) has direct relevance to phase of sine waves, or whether phasor analysis merely purloins the property of the complex plane as a "handy trick"?

If it is only a "handy trick", it is already useful and worthy of our attention as engineers. We want shortcuts to solutions for our engineering problems.

I don't think I suggested any different, except maybe the notation is confusing.

It's just that we are taught something that turned out not to be necessary or relevant, in a sense. Complex numbers are an interesting story, but not knowing what sqrt(-1) 'is', trips me up if I'm told "it is completely fundamental" rather than "based on a true story of i, but events or characters might have changed, so don't fret" - instead I am left to myself to work out that mathematics is better ignored in engineering. That's about where I put a  .

Yes, if it "adds" nothing practical or needs to be applied abstractly by some engineers who might then not know what they are doing as clearly.

Of course it adds practicality, otherwise it wouldn't be taught. Not only that, it adds insight, which is essential for engineering.

Not for everyone it doesn't. I wasn't describing a theory of mind, but a reality of minds and situations. There are people and situations for which multiplication might not work for them as well as addition, and might result in much less understanding. Addition might even be more practical in a computer, some sort of realtime situation. Ok so maybe these people are not cut out to be engineers, but there are much more advanced mathematical topics in engineering that are even more optional (as the power triangle example above shows). I'm not suggesting it isn't taught, just that it can't be assumed to be useful. I think Maxwell's equations succumb to the same problem, of looking too 'theoretical and optional', when they are not (the concepts are fundamental, even if the analytical solutions are too much for many engineers to really get).

Mathematical models as-taught for engineering added mess and confusion, not insight for me. You could assume that makes me completely stupid. I could be pretentious and say all engineering (and physics) concepts are simple to me and I find the math a distraction. With the reality somewhere within that space. For all the millions of engineers now in the world, there have to be some differences between them.

Because you don't have to work with AC circuits, filters, control systems, or RF, and you see no use for it in your daily tasks, it doesn't mean that mathematical concepts like complex numbers should be abolished from engineering.

For working with ADCs, for example, a different set of theorems and math tricks are required.

I could conversely say that the Nyquist theorem is a waste of time, if I my job as an engineer didn't involve sampling analog signals. Or that the Viterbi algorithm, without which CDMA, GSM, WiFi, speech recognition and a whole bunch of other technologies wouldn't be possible, that I had to study while in engineering college, is rubbish if my job as engineer had nothing to do with telecom.

I do work with AC circuits, filters, control systems, even RF. Most of it has become unavoidably cookie-cutter or specialised because of chips and advances in technology.  About the only thing which uses as-taught maths of those is analogue filters, where even for me the cookie-cutter solutions do not always, well, cut it. You've seen the power triangle reference, and can see how complex numbers could be optional, so why are they needed? Tradition? I guess I am saying abolish it, but I know it wouldn't be practical because it is such a strong tradition, and I know it is at least partly motivated by my prejudice against mathematical notation in engineering, which I know not everybody shares.

The Nyquist sampling theorem is easy to describe in words and see why in a simulation (perhaps in Excel, to impress what I mean there), so isn't the kind of thing I'd want to say is a waste of time. It's more the pages of mathematical descriptions which I can only assume professors must know half their students don't even begin to comprehend properly. Then there's stuff I think is outright misleading, like there being complex numbers in an FFT - real values (again Hermitian) go in, so in the output why do the sines get a j while the coss get nothing, when (despite complaints) imaginary numbers are very different in character from reals and there is nothing like reactance in the phasor to even suggest some imaginaryness (despite complaints) to one of the axes? j definitely has a supposed meaning as one of the roots of x^2=-1 (and how do we know the one we pick as positive is the -+ one or the +- one?), 1 is a natural number with a clear positive. I don't think I would ever call it "rubbish" though, it's a minor annoyance. But to fresh students with a weakness (or perhaps a strength) in maths, it can be extremely (and I have to assume unnecessarily) confusing.

Anyway, I can't see the point in getting too worked up over (or taking too seriously) theory and learning at university. Your mention of Viterbi decoding reminded me of a seminar thing (with sausages) I went to recently (10 years ago!) about LTE and one of the presentations went right over my head with words like "Bayesian" this and that. A quick search for that now turns up things like:

... The coexistence problem is modeled as a decentralized partially-observable Markov decision process (Dec-POMDP) and Bayesian inference is adopted for policy learning with nonparametric prior to accommodate the uncertainty of policy for different agents. A fairness measure is introduced in the reward function to encourage fair sharing between agents. Variational inference for posterior model approximation is considered to make the algorithm computationally efficient. ...

Students need to choose their poison and pick their battles.

Oh noes, too long again, I meant to reply to other stuff.

[HuronKing]

Back now after a weekend trip. A few comments.

Only time for a partial reply for now:

First, my concern over sqrt(-1) in electrical engineering, penfold has it right: "and the j is an operator rather than a quantity ... it is stretching it a bit far to say that it is a physical quantity".

Did I say it was a physical quantity? Please show me (I've tried to find where I might've implied that but I don't see it). j is not an Ohm. But it is a representation of phase-shift in Ohms and a damn good one. Is that not physically relevant?

Tricky semantics. What I and I assume penfold were referring to was somewhere between a physical unit and representation as a tool. You said sqrt(-1) "has immense physical significance, just as 'zero' and 'negative' have immense physical significance" which I took to be that middle meaning. Saying j is physically relevant is different from saying sqrt(-1) is, to me. The latter being a very abstract mathematical concept, but j being defined as a practical tool by Steinmetz (yes, with overlap). sqrt(-1) is the first whole positive imaginary number (if there is such a thing) hence a quantity (of sorts), j is a rotation operator as defined by SandyCox in (a, b)(c, d)  = (ac-bd, ad+bc) (with j as b or d). They happen to be algebraically identical.

That difference is only in your mind - at least as far as us engineers are actually concerned.

For example, the vector is an abstract mathematical concept. In fact, no one thought they were very useful or had much relevance until Heaviside showed the world what it could do (remember the 4 equations of Maxwell are really the Maxwell-Heaviside Equations). Seriously, Heaviside had to FIGHT to get vectors accepted. I recommend you read The History of Vector Analysis. Here is a short timeline synopsis of the book but the book itself is loaded with a colorful stories of what seems so 'obvious' to us [simple vectors] had to be hard-won:
http://worrydream.com/refs/Crowe-HistoryOfVectorAnalysis.pdf

But here we are using vectors all the time. A vector has magnitude AND direction... and that direction property necessarily is subject to a property of rotation (because I need a reference direction for the concept of 'direction' to even make sense), which is connected directly to solutions of x^2+1 = 0.

There is no coincidence that Heaviside vectorizing electromagnetism led Steinmetz to the realization that complex analysis of phasors is another, much simpler, way of solving power problems. I don't care about Descartes' idiotic 'imaginary' and 'real' naming convention that we've chosen to stick with. He never solved a circuit. 

I don't think it is any sort of tautology to say mathematical concepts are not real, if one then goes on and asserts that some part has physical relevance. Not all engineers are naturals at maths and can easily identify where that link appears (ie goes from nothing to something without explanation). Some people here seem to be struggling with it too - perhaps from over-familiarity.

You might as well be arguing that multiplication has no 'physical relevance' to engineering because you could just add the numbers up... like, yes? What is your point? Should we count on our fingers and toes because applying math makes us feel dumb? 

Yes, if it "adds" nothing practical or needs to be applied abstractly by some engineers who might then not know what they are doing as clearly.

Such an engineer wouldn't even know how to apply the abstraction. Honestly, they need to 'git gud.' If not, those engineers should be replaced with engineers who can solve it using the abstractions. I've provided copious amounts of examples of problems that were incredibly difficult or even sometimes completely inscrutable to solve without complex phasor analysis.

If someone wants to solve 100x100 by adding up 100 100 times... their billable hours will be higher than mine who can solve it in 2 seconds with my 'handy trick too-hard abstraction.'  I know who the employer is going to hire.

And if such an engineer is never going to apply to solve big addition problems that need multiplication because the abstraction is too hard... fine. Good for them. But they'll never land a man on the Moon counting on their fingers and toes. 

For something like sqrt(-1), I don't know where it gets real.

Get... Descartes... out.... of... your...head... Why won't you listen to Gauss?

I'm not citing waffle-y texts at you. I'm citing actual engineering practices. You can take them or leave them.
https://www.electronics-tutorials.ws/accircuits/power-triangle.html

Although I've clarified more since, this is exactly what I don't have a problem with. j is defined only in the annotations on the diagrams as a 90 degree shift pictorially and as reactance. j doesn't appear in any of the body text or its formulae. The only hint as to what j might be (as a symbol) is mention of "which is the vector sum of the resistance and reactance".

This is what I mean by things like "to the point they realise sqrt(-1) has no physical relevance, with j being the unit vector that I say it is".

And you straight up ignored the Keysight Impedance Measurement manual.

I give up. 

I'm sticking this series here again just because:

[bpiphany]

In this, of all threads, we can of course not miss pushing this video =D

[SiliconWizard]

I also mentioned epsilon numbers, and for something "closer" to complex numbers, you have dual numbers. https://en.wikipedia.org/wiki/Dual_number
Another "handy trick".

Now the question remains. What really makes "real numbers" more real than complex numbers? Are rational numbers more real than irrational numbers? Are transcendental numbers less real? Or are they just a handy trick? Is infinity in R the same as infinity in N? Is infinity even "real"?

Do you think dual numbers are less real than complex numbers?

Is there some kind of hierarchy of reality that makes sense outside of just being another handy trick?

[TimFox]

A mathematical approach or method applied to engineering is only "not real" if it predicts results that do not agree with practical outcomes, like the frequency response of an RIAA R-C network measured with simple equipment.

[adx]

One I needed to explain...

This is all a bit silly - it started with a gentile troll about i vs j, then we're now back to arguments over half-arsed engineering.

You reduced engineers to mere solder monkeys (no Cartesian coordinates, no vectors, no y, no functions).



What do you expect?

Silliness? I was responding to "And of course we should remove the study of Cartesian coordinate system from electronics engineering because y doesn't appear on the screen of any oscilloscope ...". (Plus how did you find a monkey wearing Dave's T-shirt?)

But I thought I better explain what I mean by stuff like "Yes, stop studying Cartesian coordinates and silly unit vector formulas. Forget about y. Ignore "functions". Teach the oscilloscope display for what it is.".

It is a little story. I was suffering a non-authentic crisis of confidence wondering if my ignorance is more functional than I assumed, so when this appeared...

https://archive.org/details/ThePhysicsOfVibrationsAndWavesH.J.Pain/page/n15/mode/2up

...I paged through it to check if it made sense. I even looked at some of the equations. Going "yep", "know it", "know that too".

Oh, can't find it, might have been a different reference. That kind of ruins my little story. Anyway, undeterred by it losing all context, it was something along the lines of the wave function operator with an omega t and x somewhere, and it said something like "this is valid for any function of x and t" - I stopped and thought "what function? itself? javascript?". A less erudite version of penfold's "whaaa?! a 1024-point DFT is just a 1024-dimension vector... with 1024 components... that represents the 1024-dimension signal vector... nooo, how can this be, it's frequency components!", in post number well lost that too after pasting it. It was there seconds ago.

Anyway it took me a few seconds to bend my brain around what that meant - by function they mean signal, waveform, shape, deflection of string, wiggles on scope. Not something to be 'solved' or 'refactored' or 'pondered in math101' or whatever the mathematicians do with equations. Yes, it's a concise description, but it doesn't represent what happens until you in effect solve it in your mind. It's kind of backwards. If engineering is applied physics and math, then you wouldn't expect an average engineer to work out bandgaps in a new semiconductor, so why the need for mathematical chops many won't understand and few will ever use in their entire careers? Many concepts in engineering are presented / taught / described in this overly abstract way - stuffy, boffiney, hard to access.

Of course you could say any engineer worth their salt should suck it up and learn to think like a mathematician - which is natural for some. But it's still obvious there is a divide between academia and what is used in the 'real' world, and one which by and large academia seems unaware of (or unwilling to accept). It reminds me of this type of mindset from 'experts':

Exactly what is so complicated about:
   …
   x = ((PORTB & _BV(PB3) == _BV(PB3)); //x gets state of  bit 3

It's hardly the most complicated C ever is it?!?

... instead of x = PORTB.3 (which can be done in some other languages and nonstandard dialects of C). It is harder, it's not a fault of students or beginners that they find it so.

Yes, I'm not saying stop teaching the concepts of Cartesian coordinates of course (or erase all abstract mathematics) - but this implication that it's best for engineers to crowd round the textbook in candlelight to learn the ways of classical and renaissance mathematicians, is just too much. Claptrap (or perhaps Klaptrapp).

[adx]

[...]
Why does the algebraic 'what if' solution to x^2+1=0 have direct mathematical relevance to phasors?

I had half-baked a response to that earlier actually (hopefully that doesn't get taken as evidence of non-causality), I was pondering my own initial reaction to complex numbers from high-school maths. I think that 'what if?' solution is typical of most people's first exposure to complex numbers, demonstrating that there are still "some roots" to an apparently 1d problem. I initially just accepted that it was 'nice' and plodded along.

I'm sure I remember you saying that in an earlier post - now I'm worried for causality!

I can't remember my reaction to complex numbers in high school - though I wrote a little story lastnight then wondered if it would be a good idea to post my tattered academic achievements. Can't do much more harm .

I can't get completely past that what if. I (still) reluctantly accept bsfeechannel's "So it is clear that x is in another dimension." argument, which would seem to give complex numbers the genuine fundamental relevance to phasors that I sought.

By about first or second year EEE maths, when functions of a complex variable were introduced formally with power series (of a complex variable), residues, etc, it shed a little more light on things, at least to demonstrate that the function of x, for which we'd only ever assumed to be a function of a real value (and yet had complex roots... go figure) could actually be a function of a complex 'z=x+jy' which more naturally has a complex root, where f(z)=z²+1 is now a surface plot with height defined for values of x and y... only the height is complex but only goes completely to zero at +j and -j (i.e. y=+1,-1). If you were to draw cross-sections of the surface plot (as x²+1 is the cross-section at y=0) and the same function will look slightly different... you can even plot a cross-section at an angle where both x and y are varying... or any arbitrary function that links x and y in response to an arbitrary parameter (I'm too tired to wonder if that was relevant... could be Euler's formula with 'phase' as a parameter... really not sure where I'm heading with that).

I'm yet to really work that out (or plot it in Octave), but as you say a function of a complex thing more naturally has a complex root. But when it comes to an extra dimension being generated out of 'nothing', it's drop anchor and haul back until I get to the what if place. Not that I think such a thing would be impossible (or icky), but because of that "despite the algebra of the square being the source of the number in the first place" chicken and egg situation (similar to the DNA argument a few posts back, where the machinery of its own encoding is needed to make it work). It's made of algebra (what if), so what if any modern impression of the 'reality' (particularly the neat 2D Cartesian uses) of imaginary numbers is no more than a product of our overactive imaginations? (Of course these uses would stay valid, but so would any arbitrary 2D system with the properties useful for phasors.)

If so, an illusion might in part be fostered by the implication of an orthogonal dimension by the x^2. The failure of that to solve if negative is a whole new dimension on top of that (for the output). Per bsfeechannel's words "x can be neither a positive nor a negative number, because such numbers give you a positive area. So it is clear that x is in another dimension." (which is for x rather than x^2, so I can see my argument doesn't really work). The area usually works for +ve x^2, then for -ve, something otherworldly happens to x and less so to the area. Or one of the dimensions (depending on the root) flips midway. Or it is real; squaring generates negative numbers (or the negation operator) from nothing in the same way I am complaining about about i. Then using those negative numbers (or really just the operator) and the same squaring operation, two imaginary roots are generated (or rotated into being).

Whatever the explanation (real or otherwise), a neat 2D phasor view does seem quite leapey faithey to me. Hence that anchor. Fortunately, I am not a mathematician, so it doesn't really matter.

I half had something about pole-zero responses and phase wrapping around, but better end that one there!

[adx]

That difference is only in your mind - at least as far as us engineers are actually concerned.

I know - that is the basis of my argument about pedagogy. Otherwise I wouldn't care (and didn't for decades). I was going to say in response to SiliconWizard's idea about "what reality is" is that I think that I think therefore I am, therefore what I think is my reality. If that makes sense (which I'm hoping it doesn't, because you'd be doing better than me). Use me as a model of the most obtuse students, who may not understand a thing until it is rammed in directly. If I were a student now (odd mix of tenses) I would have feigned belief long ago. The use of a difficult to accept concept as a foundation pretty directly implies difficulty in accepting things that are built on it (either an unresolved struggle, or an unquestioning acceptance; the opposite of what we want science to be). It's really odd: We are saying to them that a number when multiplied by itself results in -1 is a 90 degree phase shift. Or if you want to understand a 90 degree phase shift, just 'work out' the square root of -1 and you'll be golden. It's mind-shatteringly difficult. Never mind the name, I think the hint given by the word "imaginary" is useful in learning to ignore it, and just use the rules. Anyway, I'm off on the wrong axis again.

I obviously don't know much about the history of vectors, even after reading that pdf, it might just be that I don't know what vectors are. I better leave that alone, after a quick look Wikipedia revealed no surprises.

But here we are using vectors all the time. A vector has magnitude AND direction... and that direction property necessarily is subject to a property of rotation (because I need a reference direction for the concept of 'direction' to even make sense), which is connected directly to solutions of x^2+1 = 0.

Ok, despite what I wrote above I am a lot closer to accepting that makes some sense: If I am going straight ahead, negative velocity might not be real for me at that moment but applying it will result in my distance units decreasing rather than increasing, so the 'mark' of what I will call "physical" numbers (positive reals) versus the imaginary nature of the negative numbers (because they are generated from those by a minus operator) has physical significance. In the same way, if I want to describe "sideways" as some unholy mix of positive and negative (or negative and positive), the lack of operator still points the way forward (as my reality, there is no delta to new freedoms like going in reverse). Similar for my FFT; the 'mark' of the real is how the DC component (oops I nearly said term) comes about, although positive or negative reals count and no DC offset need exist (which rains on my new parade a little). Still, if 180 degrees phase is produced by negation, then what's to say an extracorporeal mix of minus and positive can't produce all phases quantifiable? (Which is your point, I know.)

But "what's to say" isn't a proof. And we are clear in our claim that 90° = sqrt(-1), or rotation is "connected directly to solutions of x^2+1 = 0" - it's an extraordinary claim, unscientific in its boldness coming from historical ideas of something no one ever really worked out (to my knowledge). (In this sense perhaps mathematics is to engineering what the pre-science medicine is to modern medicine - full of ideas (many good) but isn't science?)

And generation of a whole new dimension? To the point where complex numbers are thought of as inherently "one number" while the same detail presented as an ordered pair is two (is a vector a number?). Forward / reverse is connected directly to solutions of x^2 = 1 as I mentioned lastnight. x^2 is effectively a statement of area which 'invokes' an extra dimension of our own making. Is it any surprise that doing strange things to that area can result in something which appears to have excess dimensionality? It could be fundamental, or it could be we set ourselves up for a trick and believe this illusion means more than it does.

And that's possibly all I need to say on it without knowing more. I have learned why complex numbers have fundamental physical relevance, but also why they might not.

I don't think it is any sort of tautology to say mathematical concepts are not real, if one then goes on and asserts that some part has physical relevance. Not all engineers are naturals at maths and can easily identify where that link appears (ie goes from nothing to something without explanation). Some people here seem to be struggling with it too - perhaps from over-familiarity.

You might as well be arguing that multiplication has no 'physical relevance' to engineering because you could just add the numbers up... like, yes? What is your point? Should we count on our fingers and toes because applying math makes us feel dumb? 

Yes, if it "adds" nothing practical or needs to be applied abstractly by some engineers who might then not know what they are doing as clearly.

Such an engineer wouldn't even know how to apply the abstraction. Honestly, they need to 'git gud.' If not, those engineers should be replaced with engineers who can solve it using the abstractions. I've provided copious amounts of examples of problems that were incredibly difficult or even sometimes completely inscrutable to solve without complex phasor analysis.

If someone wants to solve 100x100 by adding up 100 100 times... their billable hours will be higher than mine who can solve it in 2 seconds with my 'handy trick too-hard abstraction.'  I know who the employer is going to hire.

And if such an engineer is never going to apply to solve big addition problems that need multiplication because the abstraction is too hard... fine. Good for them. But they'll never land a man on the Moon counting on their fingers and toes. 

I didn't say they shouldn't learn multiplication. Just "if" addition works better. I gave the example (in a different reply) of measuring out a liquid rather than calculating the volume out. This is a practical result which just works better in many situations. Or a computer solution arrived at by filling pixels with colour then going over that counting them (say some sort of floor plan calculator). I also know some clients who would be frightened by multiplication (yes) and might not pay me if I went against their wishes and used 'complicated maths'. If wanting to put someone on the moon (then back here especially) it's probably better to replace both the consultant and client.

It's all abstraction anyway. I didn't bring up the example of not using multiplication, I just wanted to show how general the idea is (that it is always possible that the mathematics is too abstract).

Get... Descartes... out.... of... your...head... Why won't you listen to Gauss?

Bah, my answer to this part was to be a quote (I thought from Gauss) saying the true nature of the imaginary numbers remains elusive. Can't find it anywhere.

Anyway, you see from above why I think imaginary remains a good name. It stands as a warning that we (at least I) don't know for sure, and as humans we tend to get ideas into our heads and believe them without adequate evidence. I like to use qualified language in that case. I'm not saying Gauss was wrong, but I think there is a chance he was wrong.

And you straight up ignored the Keysight Impedance Measurement manual.

I give up. 

I looked at that, and saw "complex quantity" and "imaginary part" at the start. I had a bit of a laugh at "imaginary components" (always fun). Searching the pdf it does talk about complex and imaginary a fair bit in places, being 140 pages long and devoted to LCR measurement. I'm not suggesting that these terms don't appear anywhere reputable - I know full well what they mean in engineering.

For another example of absent imaginary, look at:

https://en.wikipedia.org/wiki/In-phase_and_quadrature_components
(I and Q suggested by TimFox on page 67 - I had half-penned a reply)

Not one mention of complex or imaginary.

I'm happier with that approach, but it doesn't mean I think complex phasors are "wrong" (they never stopped working), and now I understand sqrt(-1) better I might even begin to like the idea.

[TimFox]

Yes, the voltages indicated as I and Q on a two-phase lock-in amplifier are "real values" in the common mathematical sense of the word.
However, when I use these values to calculate something useful, such as the frequency response of an amplifier or an impedance as a function of frequency, being of sound mind I do the simple complex algebra in Excel, setting the imaginary part of the voltage to "Q" and the real part of the voltage to "I".  Both values are functions of frequency going into the algebraic calculations.

[HuronKing]

But "what's to say" isn't a proof. And we are clear in our claim that 90° = sqrt(-1), or rotation is "connected directly to solutions of x^2+1 = 0" - it's an extraordinary claim, unscientific in its boldness coming from historical ideas of something no one ever really worked out (to my knowledge). (In this sense perhaps mathematics is to engineering what the pre-science medicine is to modern medicine - full of ideas (many good) but isn't science?)

Gauss and others worked it all out for us. In fact, some of the most brilliant minds in human history turned their attention towards this. It's the basis of the Fundamental Theorem of Algebra. It's not so mysterious, really.

It could be fundamental, or it could be we set ourselves up for a trick and believe this illusion means more than it does.

I'm content that it's not an illusion since the mathematics has tremendous predictive power in physics and engineering.

And that's possibly all I need to say on it without knowing more. I have learned why complex numbers have fundamental physical relevance, but also why they might not.

This is progress. 

Bah, my answer to this part was to be a quote (I thought from Gauss) saying the true nature of the imaginary numbers remains elusive. Can't find it anywhere.

You're probably referring to the 'shadow of shadows' quote which should be weighted in its context. Gauss was tackling Euler's Identity in his doctoral dissertation to prove the Fundamental Theorem of Algebra.

Anyway, you see from above why I think imaginary remains a good name. It stands as a warning that we (at least I) don't know for sure, and as humans we tend to get ideas into our heads and believe them without adequate evidence. I like to use qualified language in that case. I'm not saying Gauss was wrong, but I think there is a chance he was wrong.

As a teacher it is the WORST name to give it.

Me: "Okay class, now that we've learned about real numbers, let's now learn about imaginary numbers."
Student: "Wait, why are we learning fake math?"
Me: "No, it's real math."
Student: "But you said it's imaginary."
Me: "Not really, the better name is complex numbers."
Student: "Oh God no! Why do we need to learn complicated math?"
Me: "It's not complicated. It's complex."
Student: "Yea! That's what I said. Math is stupid. You're making me learn complex imaginary math that I'll never use. Blegh."

There is nothing qualified about the language calling it 'imaginary.' It is straight up just repeating Descartes' lack of, heh, imagination in foreseeing where numbers in the complex plane could be used for helping humanity. Our understanding of complex numbers has advanced significantly since Descartes.
If I can make an analogy, we don't call particles of light "corpuscles" even though Newton conceived of the first particle-models of light. We call them photons, because calling them corpuscles would carry with it a lot of baggage from Newton's other arcane ideas.

Earlier someone mentioned that being mad about the 'imaginary' convention is like being mad about our plus-minus red/black current convention. I soft disagree with that. No one has any trouble learning electricity with the historical convention, the math all works out the same, and a simple sign reversal is all that's required to talk about the direction of charge flow for current.

Whereas students think there is something actually meaningful about the name 'imaginary' number. Or even, as you're suggesting, that there is a chance Gauss was wrong. There isn't - at least in as much as ANY portion of mathematics has meaning.

For another example of absent imaginary, look at:

https://en.wikipedia.org/wiki/In-phase_and_quadrature_components
(I and Q suggested by TimFox on page 67 - I had half-penned a reply)

Not one mention of complex or imaginary.

Lulz - the suggested additional reading is Charles Steinmetz' Theory and Calculation of Electrical Apparatus where that icky j appears on page 2:
https://www.google.com/books/edition/Theory_and_Calculations_of_Electrical_Ap/UjEKAAAAIAAJ?hl=en&gbpv=0

Maybe take your investigations beyond Wikipedia? 
https://www.dsprelated.com/showarticle/192.php

I'm amused by this site also taking the great pains to explain how the j is unfortunately named and glossed over too quickly when it is taught. In any case, You can thank Euler for making sines and cosines equivalent to j rotations.

I'm happier with that approach, but it doesn't mean I think complex phasors are "wrong" (they never stopped working), and now I understand sqrt(-1) better I might even begin to like the idea.

If you want to clunk around with sines and cosines you can - and sometimes its better. Other times it isn't. Being comfortable with both makes you a better engineer.

When I used to be a private tutor, I always told my students that mathematics is like long hair.
You can wear it up.
You can wear it down.
You can color it.
You can cut it... and it'll grow back.
You can part it in the middle, on the side, wear it as bangs, or tie it into pigtails and ponytails.

But at the end of the day... it's the same hair, just dressed up differently.

And some social occasions require the hair to look a certain way. And sometimes the way it looks doesn't matter - but it's function matters (like putting the hair up so its out of the way). And other times the way it looks is ALL that matters regardless of how impractical it is.

Sometimes, someone comes along with a new way of styling hair. Maybe that styling method sucks or looks really ugly... until fashion changes or you find a really good reason to do hair that way.

If you're a hair stylist, you can be a boring technician who only knows 3 haircuts and 3 ways to comb hair. And you can have a perfectly successful career as a stylist. But that's all you'll ever be capable of doing.
Or, you can be a stylist who embraces new fashions, learns new ways of constructing and deconstructing the hair with the tools of the trade (scissors, clippers, steamers, gels, dyes, shampoos, etc etc). You'll then be sought out for your talents at solving any kind of hair problem and get paid lots of money to do it. And you might even find that emotionally fulfilling.

Or all of that is too hard and that's not the kind of work you want to do. You don't care about getting girls ready for prom or dressing hair for weddings. You're satisfied giving buzzcuts to marines. That's fine and admirable and, yes, you don't need to know anything about curling hair to do the buzz-cutting job.

But thank goodness there are skilled stylists who can make a young lady's dreams come true with a gorgeous effortless looking hairdo. 

[adx]

First clean up some unreplieds...

Does the 'value' sqrt(-1) have innate physical relevance for anything like phasors (or even quantum mechanical wavefunctions)? In other words, would these engineering uses suffer some fatal breakdown if they were replaced by two 'ordinary' numbers without some extra special property added? I genuinely didn't know as a student, although I slowly learned they are simply 'hack vectors' and more akin to polar to Cartesian conversion than some mysterious fact of mathematics. (But whether mathematics has more of a reality of its own is a different and much more interesting question.)

Complex numbers ARE ordinary numbers. In point of fact, what the heck IS an 'ordinary' number? That's not a formal definition. What is that?

Real numbers from my context. Avoiding formal definitions was part of my point. I don't care if mathematicians (or you) say complex numbers are a single number or not, I can define any number as an algebraic construction, but going on to say the box set of Star Wars is "an ordinary number" would be silly.

I too read the bit about Gauss suggesting "lateral" and thought that might have helped set the pedagogical direction for engineering uses, but I have no problem with the word "imaginary" or the reason it was originally used, especially if this lateralness is not truly innate (ie, an illusion).

Lateral is an expression of the rotation of the quantity. It is as 'physical' as multiplication is 'physical' as the sine function is 'physical.'

You lost me there, despite knowing what you mean. I would say sines are more physical as a feature of 2D geometry and oscillations. Multiplication is more of an abstract tool relating to quantities - but could be physical. Lateral is geometry. My question is about sqrt(-1), not the complex plane.

"Waffley texts" I meant anything that is used as or perhaps is an "argument from authority" fallacy (per Wikipedia), eg Steinmetz says so so it must be true. Steinmetz says it is a handy trick, so if I read that right, it is an answer to my question that sqrt(-1) has no direct / special / innate physical relevance (because it is a handy trick).

To hell with that. I never appeal to authority. The only reason I or anyone else gives a damn about Charles Steinmetz and Edith Clarke is that they taught engineers all over the world how to use complex numbers to solve problems that stumped EVERYONE ELSE in the engineering industry until they came along. The proof is in their work and the results their analysis produced - nothing else. I've linked their works and plenty of other things to learn about it. The rest is up to you.

That was in response to bsfeechannel being particularly appeal to authority adjacent. Your story about Charles Steinmetz and Edith Clarke is nice, but them being good at what they did has nothing to do with my question as far as I can see.

[adx]

But "what's to say" isn't a proof. And we are clear in our claim that 90° = sqrt(-1), or rotation is "connected directly to solutions of x^2+1 = 0" - it's an extraordinary claim, unscientific in its boldness coming from historical ideas of something no one ever really worked out (to my knowledge). (In this sense perhaps mathematics is to engineering what the pre-science medicine is to modern medicine - full of ideas (many good) but isn't science?)

Gauss and others worked it all out for us. In fact, some of the most brilliant minds in human history turned their attention towards this. It's the basis of the Fundamental Theorem of Algebra. It's not so mysterious, really.

Well I guess I just don't believe. If someone like me insists on being an ignoramus who won't or can't understand (I can't be expected to tell the difference), and you are limited to 'appeal to authority adjacent' claims because there is no trivial proof, then in the absence of launching into full time study I can just remain skeptical. It's not a carload of students trying to get to the top of a hill (then not drive off it). I can still use I and Q, and I can pretend j doesn't mean anything beyond how it gets used.

It could be fundamental, or it could be we set ourselves up for a trick and believe this illusion means more than it does.

I'm content that it's not an illusion since the mathematics has tremendous predictive power in physics and engineering.

A working illusion will also have "tremendous predictive power", so I'm not content.

I don't know what that predictive power is anyway. If you mean frequency domain analysis, a 'frequency' must describe amplitude and phase, it’s direct and obvious. It doesn't need some [inflationary language trigger warning] ridiculous number system to describe it, just 2 reals, or even an unsigned magnitude and direction. That direction's 0 has the reference direction you needed.

I (sort of) regret leaving out ", or both" from my claim above (the false dichotomy sounded more dramatic).

I had some ideas, but I don't think it will help.

It just seems awfully convenient that when multiplying by -1 gives an infinite frequency oscillator ( (-1)^n creates problems ), that it is possible to define an in-between situation (literally i*i=-1) of pathologically orthogonal numbers to create a quadrature oscillation (circularly polarised) to do the job and represent any phase. It's like we half made it up for the purpose that any sane person would call a vector. The other half seems fundamental. Of course complex numbers can be visualised on a plane, because we designed them that way (Cartesian coordinates). The question is whether imaginary numbers deserve to be "an axis", or just happen to work that way because we think they should. Imaginary numbers are dreamed up from fanciful mathematical impossibilities (x^2 is non-physical for -ve x: we can't have negative length). It seems to have more in common with rotation in 3D than any 2D geometrical construction. Sus picious. Might be best to make a tinfoil hat, or if I already have one, make a roll of foil from it so I can make another one later on (metal fatigue and infinite patience permitting).

And that's possibly all I need to say on it without knowing more. I have learned why complex numbers have fundamental physical relevance, but also why they might not.

This is progress. 

Except that leaves me in a clearer version of where I started out - knowing how it's used and vaguely why, but not accepting it.

Bah, my answer to this part was to be a quote (I thought from Gauss) saying the true nature of the imaginary numbers remains elusive. Can't find it anywhere.

You're probably referring to the 'shadow of shadows' quote which should be weighted in its context. Gauss was tackling Euler's Identity in his doctoral dissertation to prove the Fundamental Theorem of Algebra.

Maybe, but I thought it had "elusive" in it. The shadows of shadows thing reminds me of my "banana numbers" which I had planned to invoke in roo-tons esque style to explain the cancellation of sqrt(-1) in that epic math duel video which was a better explanation better than Wikipedia. Best left undescribed even if only to avoid long sentences.

Me: "Okay class, now that we've learned about real numbers, let's now learn about imaginary numbers."
Student: "Wait, why are we learning fake math?"
Me: "No, it's real math."
Student: "But you said it's imaginary."
Me: "Not really, the better name is complex numbers."
Student: "Oh God no! Why do we need to learn complicated math?"
Me: "It's not complicated. It's complex."
Student: "Yea! That's what I said. Math is stupid. You're making me learn complex imaginary math that I'll never use. Blegh."

How did you get that recording of me?! The chances are remote - the one and only maths class I ever bothered turning up to, only to be hurfed out after 5 mins from suffering a hypnic jerk and throwing my bic fluoro pink pen clear across the lecture theatre, delivering that unmistakable clatter as it hit the wood of the sound diffusor panel wall. Not my finest moment*, still, where's that recycled roll of foil.

(* Was actually some boring as all hell circuit analyis class of utter theory (S domain stuff?), and I didn't get hurfed out, just awoke to a couple of hundred incredulous eyes each beaming forth an accusing stare, once I had worked out what was happening from my complex plane induced (therefore involuntary) nap, I had to suffer through the indignity of trying to find my pink pen in that same silence only punctured with things like "do you have a spare pe... oh, ok, where did it go?" shuffle klonk klonk "found it" clatter "oops" clatter CLATTER "sorry dropped it again" muffled scream "sorry I thought that was the foot of the chair" shuffle shuffle shuffle "sorry" and so on until the lecturer asked me "are you quite finished?", to which I could only answer "I think so". Glad all that embarrassment is behind me. Maths wasn't in the big lecture theatres, so the pen couldn't have gone very far.)

Must admit, if the complex plane hadn't been invented, I would be going "ooh ooh you can plot it like this". I just can't accept something that doesn't make sense to me and never seemed to have any practical relevance as that dsprelated article bemoans (quote below).

Lulz - the suggested additional reading is Charles Steinmetz' Theory and Calculation of Electrical Apparatus where that icky j appears on page 2:
https://www.google.com/books/edition/Theory_and_Calculations_of_Electrical_Ap/UjEKAAAAIAAJ?hl=en&gbpv=0

Hardly dents my argument (or really evidence) that some references and areas of engineering use phasors without sqrt(-1), or even j. I'm not saying j isn't widely used.

Whereas "Unfortunately DSP textbooks often define the symbol j and then, with justified haste, swiftly carry on with all the ways that the j operator can be used to analyze sinusoidal signals. Readers soon forget about the question: What does j = √-1 actually mean?" is my point (even to some extent the "unfortunately")...

Maybe take your investigations beyond Wikipedia? 
https://www.dsprelated.com/showarticle/192.php

I'm amused by this site also taking the great pains to explain how the j is unfortunately named and glossed over too quickly when it is taught. In any case, You can thank Euler for making sines and cosines equivalent to j rotations.

I saw that article when looking for guidance on imaginary numbers. Quite neat and a good explanation, but on sqrt(-1) is again 'appeal to authority adjacent' (especially with all that stuff about Herr Euler, Gauss' brilliant introduction of the complex plane and comparison to Einstein - welcome analogies and hyperbole in this context, but not proof (also incorrect)). It describes how it behaves, light on what it actually means. e^(j(pi)/2) = jsin(pi/2) = j doesn't show j = pi/2 as an argument of sin obviously, but a means to scale the 'fake' vector j. It could be because the derivative of e^ix is ie^ix and that's how oscillators (and circles) are made (thereby defining the behaviour of i - fakely, or it could be the 'proof of concept' that I sought). As an engineer and mathematics weakling I am more interested in how things work rather than bathing in the glory and beauty of unquestioningly received wisdoms (pretty much the entire theme of your point) so I can't fall for the Klaptrapp conspiracy and it's time to despool that roll of foil once again: Never trust the math.

Well that's very wearying so I thought a nice conciliatory response would wind down arguments - pity how some things just don't work out .

You could well be right, I just don't know. Like you say, I need to go off and sort that out myself.

Better I go off and wash the tortuous mess that is my hair.

[HuronKing]

I'm going to keep my replies short.

Well I guess I just don't believe. If someone like me insists on being an ignoramus who won't or can't understand (I can't be expected to tell the difference), and you are limited to 'appeal to authority adjacent' claims because there is no trivial proof, then in the absence of launching into full time study I can just remain skeptical. It's not a carload of students trying to get to the top of a hill (then not drive off it). I can still use I and Q, and I can pretend j doesn't mean anything beyond how it gets used.

Again, I never appeal to authority. I've provided ample resources to read from Steinmetz and Clarke down to YouTube level basic introductions. You've got the full gambit of resources at all levels of rigor available. Something something horse to water.

The question is whether imaginary numbers deserve to be "an axis", or just happen to work that way because we think they should. Imaginary numbers are dreamed up from fanciful mathematical impossibilities (x^2 is non-physical for -ve x: we can't have negative length).

You have 3 apples, and want to take away 5 apples. So you have negative 2 apples.

NEGATIVE 2 APPLES? WHAT IS THIS SORCERY? This is just mathematical claptrap invented to compensate for made up problems and invent solutions.

How can you have negative apples? IMPOSSIBLE!!!! 

These excuses make you sound like a pre-medieval mathematician.

Hardly dents my argument (or really evidence) that some references and areas of engineering use phasors without sqrt(-1), or even j. I'm not saying j isn't widely used.

How is that even an 'argument' to have? Like, yes? What is even the point of what you're trying to say now?

I saw that article when looking for guidance on imaginary numbers. Quite neat and a good explanation, but on sqrt(-1) is again 'appeal to authority adjacent' (especially with all that stuff about Herr Euler, Gauss' brilliant introduction of the complex plane and comparison to Einstein - welcome analogies and hyperbole in this context, but not proof (also incorrect)).

The article isn't trying to rigorously prove it - just explain what it is, what it means, and get on with it to do some engineering work. I even showed you Steinmetz's books that introduced all this! If you're saying it's too hard to read the proofs and the cursory introductions are just appealing to authority... That's on you, man and I've done all I can.
 

[SandyCox]

Take the set F of all 50 Hz sinusoidal waveforms. Each element f of F is of the form:
f= A cos(100*pi*t + phi),

The phasor transform maps f onto the complex number A angle(phi).

In fact, the phasor transform is an isomorphism between the field F and the field of Complex number. This means the only difference between the two is a change of notation. So the two are exactly the same.

ADX must agree that F has meaning physical?

[penfold]

[...]
In fact, the phasor transform is an isomorphism between the field F and the field of Complex number. This means the only difference between the two is a change of notation. So the two are exactly the same.
[...]

I like that wording... I think additionally because that transform can be performed in the same "domain" as the measured value, i.e. using analog circuits responding only to voltages or currents that produce the real and imaginary components that it can therefore be of a similar physical significance as using a compass and straight-edge to measure something geometric.

In a general sense what it is that irks me a little about complex numbers and physical significance is that there aren't many measurements I can think of that are genuinely unequivocally negative in their measurement (without considering a direction or rate of change relative to something else) of which one must take a square root of.

[...]
You have 3 apples, and want to take away 5 apples. So you have negative 2 apples.

NEGATIVE 2 APPLES? WHAT IS THIS SORCERY? This is just mathematical claptrap invented to compensate for made up problems and invent solutions.

How can you have negative apples? IMPOSSIBLE!!!! 

These excuses make you sound like a pre-medieval mathematician.
[...]

I'm not sure that kind of response really helps. I mean, literally, how can I have negative apples?! Is that the number of apples that I must possess before I own zero? Where will these apples come from and to where will they go? In the sense of lengths, the negative implies a direction whereby we would still be counting a positive number of lengths in the backward direction... but I cannot own negative apples, I could owe a positive number of apples to a specific person perhaps. But the negative sign contains very little of the necessary information... hence negative numbers don't appear that often in accountancy.

[hamster_nz]

I'm really surprised that nobody has mentioned changes of basis vectors, changes of coordinate systems, translations, transforms, and of ways of changing your 'perspective' and 'view' on the same underlying system. Lagrangian mechanics vs Hamiltonian mechanics vs Newton's laws of motion, and so on.

In some cases sqrt(-1) has a physical meaning - for example, in a physical system it could be energy 'transformed' into a value that you can't measure in the units you are working with - you might be measuring displacement/distance and imaginary quantity might be energy stored in a spring, or in system's momentum. In electrical system the 'imaginary' unit might be current, if you are working with voltages.

And then quite often, the simplest coordinate system to analyze a system isn't the most obvious one, or maybe the coordinate system doesn't have any physical interpretation at all.

Why are people devolving to discussions of "negative apples"? next it will be "I've got zero Ferraris in the garage! How is that possible? I've never had a Ferrari in the garage".

And yes you can measure a negative length, you just need to be careful about defining your basis vectors.

Or what I am left wondering is this: Why are people acting like 11 year old nerds? Has the world gone crazy?

[adx]

Take the set F of all 50 Hz sinusoidal waveforms. Each element f of F is of the form:
f= A cos(100*pi*t + phi),

The phasor transform maps f onto the complex number A angle(phi).

In fact, the phasor transform is an isomorphism between the field F and the field of Complex number. This means the only difference between the two is a change of notation. So the two are exactly the same.

ADX must agree that F has meaning physical?

Yes, I agree with all of that. I'm hoping it helps illustrate my point - if they are the same, then what purpose is served by drawing on a 'contested' (at least students have a lot of problems with it) concept, being sqrt(-1)? Or why require a definition j*j = -1 to generate a dimension that was right there in front of us all along? Is that dimension innately and clearly defined by j*j = -1, or have people extended it by axiom, at least to some extent? If so, is that mapping (phasor to complex number) merely (or partly) synthetic, and if so, is it right to say that this mapping (in itself) is physical?

It is clear to me that people interpret this in different ways, so it is potentially impossible to convey.

[HuronKing]

I'm not sure that kind of response really helps. I mean, literally, how can I have negative apples?! Is that the number of apples that I must possess before I own zero? Where will these apples come from and to where will they go? In the sense of lengths, the negative implies a direction whereby we would still be counting a positive number of lengths in the backward direction... but I cannot own negative apples, I could owe a positive number of apples to a specific person perhaps. But the negative sign contains very little of the necessary information... hence negative numbers don't appear that often in accountancy.

I'm frustrated that after pages and pages of conversation on this point we return RIGHT BACK TO SQUARE ONE. This assignment of impossibility to a definition without even considering all the assumptions that are made about the consideration of the impossibility. Really, that objection is the same thing pre-Medieval mathematicians did say about negative numbers.

Whereas the complex numbers (what Gauss called lateral numbers) are just another type of representation of numbers we deal with all the time.

How can I have negative current? Negative power consumption? Negative dollars in my bank account? Heck, I can even have negative areas in solutions to integrals.

These all assume implicitly that there is a DIRECTION to the quantity. As you say, a negative length implies a direction to the quantity. I didn't say anything about owning negative apples... but someone is owed some apples - that we both see. 

So, if I can assign a forwards and backwards direction to a quantity (positive and negative)... why is it 'OMGZ IMPOSSIBLE MAN!' to assign... rotational direction to the quantity? Rotation isn't just forwards and backwards, but all the places in between. That's all sqrt(-1) means. I know you know that - but after many replies and seeing adx's latest comment (where he asks yet again what the point is of sqrt[-1]) I just shrug now. 

[SandyCox]

This discussion is really more about mathematical philosophy than engineering. For engineers, the subject of mathematical philosophy doesn't put bread on the table.

Here's another interesting one:

Lets say that we have a rectangle with width w and height h. Does its area remain the same if we turn the rectangle through 90 degrees? If it does, then we have "proven" commutativity for the real numbers, i.e. w*h = h*w.

I tend not to worry too much about these type of questions and rather focus on making my designs work. I need to be able to factor polynomials over the complex numbers to design filters and control loops. I need the poles and zeroes of transfer functions. I need the Fourier transform for the frequency domain perspective. I need theorems from Complex analysis. So I tend not to worry about the philosophical meaning of j. It works and that's good enough for me.

[HuronKing]

Early on in this discussion the relevance of sqrt(-1) in a quantum wave function got a passing mention.

What's fun is that Paul Dirac was initially puzzled by the appearance of apparently 'negative energy states' in his relativistic solutions involving the Klein-Gordon equation. He ploughed ahead anyway and ended up mathematically discovering positrons (later experimentally verified):
https://quantummechanics.ucsd.edu/ph130a/130_notes/node478.html
https://quantummechanics.ucsd.edu/ph130a/130_notes/node490.html
https://quantummechanics.ucsd.edu/ph130a/130_notes/node504.html

Thanks @TimFox for sharing this website in another thread. There's good stuff here. 

[TimFox]

This discussion is really more about mathematical philosophy than engineering. For engineers, the subject of mathematical philosophy doesn't put bread on the table.

Here's another interesting one:

Lets say that we have a rectangle with width w and height h. Does its area remain the same if we turn the rectangle through 90 degrees? If it does, then we have "proven" commutativity for the real numbers, i.e. w*h = h*w.

I tend not to worry too much about these type of questions and rather focus on making my designs work. I need to be able to factor polynomials over the complex numbers to design filters and control loops. I need the poles and zeroes of transfer functions. I need the Fourier transform for the frequency domain perspective. I need theorems from Complex analysis. So I tend not to worry about the philosophical meaning of j. It works and that's good enough for me.

At a more elementary level of usefulness, I have mentioned using a two-phase lock-in amplifier to measure "I" and "Q" components of a signal coherent to the reference signal.  From these two voltages (bipolar real numbers), I can treat the values as either (1.  Real and imaginary components) or (2.  Computed magnitude and phase angle).  Elementary complex algebra gives me the relationship between these two representations of the two values.  When adding values, it is simpler to use the real and imaginary components.  When multiplying values, it is simpler to use magnitude and phase.  Doing the computation in either way should give the same result, for example multiplying the complex current times the complex impedance to get the complex voltage.

[adx]

Not implying anything by not having time to reply to everything or one right now, but I'll try these 2:

How can I have negative current? Negative power consumption? Negative dollars in my bank account? Heck, I can even have negative areas in solutions to integrals.

It has effects far beyond what you might assume - I have only thought that through properly (or with any real interest) recently, and in part because of one of penfold's earlier posts (about the chickens) and your earlier power supply example. You might not have negative power consumption if it defined as consumption. Money is a fiction so you don't have anything beyond some person or corporate's idea. You can have a virtual negative area. Many concepts break, change or become otherwise ill-defined. It's not just "direction".

So, if I can assign a forwards and backwards direction to a quantity (positive and negative)... why is it 'OMGZ IMPOSSIBLE MAN!' to assign... rotational direction to the quantity? Rotation isn't just forwards and backwards, but all the places in between. That's all sqrt(-1) means. I know you know that - but after many replies and seeing adx's latest comment (where he asks yet again what the point is of sqrt[-1]) I just shrug now. 

Rotation isn't just a direction, but a quantity on its own. By your argument negating a natural number (say) is still a quantity - you can have more or less of it. Quantities can be offset. Many quantities are relative measurements, where zero has no special meaning. Angle is another quantity. When you multiply by sqrt(-1) to claim it is rotated (or kind of 'subnegated'), you start breaking more features like with some of the negatives, and you imply a whole new degree of freedom. That's why you need 2 real numbers in a complex number to promote to 2D. You seem to think sqrt(-1) is a fundamental property of all numbers so it is always there. To me no, you need to explicitly build a vector. If you go on arbitrarily adding parameters and degrees of freedom, you end up with the box set of Star Wars, and calling that a quantity.

[HuronKing]

You seem to think sqrt(-1) is a fundamental property of all numbers so it is always there. To me no, you need to explicitly build a vector. If you go on arbitrarily adding parameters and degrees of freedom, you end up with the box set of Star Wars, and calling that a quantity.

BECAUSE IT IS ALWAYS THERE. IT IS A FUNDAMENTAL PROPERTY OF ALL NUMBERS.
https://en.wikipedia.org/wiki/Complex_number

Complex numbers, as much as reals, and perhaps even more, find a unity with nature that is truly remarkable. It is as though Nature herself is as impressed by the scope and consistency of the complex-number system as we are ourselves, and has entrusted to these numbers the precise operations of her world at its minutest scales."

— R. Penrose

Get educated, man - complex numbers make up ALL THE NUMBERS:
https://www.mathsisfun.com/sets/number-types.html

[adx]

Wo, no. I see what's going on here. I'm going to go off and not think about it.

[TimFox]

18th and 19th century mathematics is something that happened to other people.