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How imaginary numbers were invented [video] (youtube.com)
187 points by peter_d_sherman 22 days ago | hide | past | favorite | 145 comments

The innovation of "imaginary" numbers is that we can concretely reason about multiple dimensions in a unified manner (unlike classical geometry, which relies on geometric primitives). `i` is not some magic, "imaginary" value, it is an invented syntax that means something like "rotate by tau/4 radians" or "orthogonal to the default vector". It turns out that physical problems are much more accurately modeled in 2 dimensions than in 1 dimension, and even more so in higher dimensions.

> Freeman Dyson: “Schrödinger put the square root of minus one into the equation, and suddenly it made sense … the Schrödinger equation describes correctly everything we know about the behavior of atoms. It is the basis of all of chemistry and most of physics. And that square root of minus one means that nature works with complex numbers and not with real numbers.”

This quote is emblematic of the mysticism that some mathematicians and academics cannot resist using to advance their careers as public intellectuals. Reality is certainly not based solely in "real numbers" (one dimension), but nor is it based solely in "complex numbers" (two dimensions). The idea that mathematics is "the language of the universe" that can be precisely "discovered" by brilliant minds is a ridiculous notion that only serves the status of the mathematical elite. Mathematics is fundamentally about designing models and abstractions that help us reason about real phenomena with minimal cognitive resources. Everyone does it, and anyone can do it. Disclaimer: I have a degree in mathematics.

"All models are wrong, but some are useful" [1]

[1] - https://en.wikipedia.org/wiki/All_models_are_wrong

I cannot agree with calling complex numbers just two dimensional.

Function of a complex variable is very different from a function of two variables. You can say these are two different departments of mathematics.

Real numbers are not algebraically complete, but extending it with 'i' makes it complete. Adding another dimension to go to 'two dimensions' does not do anything like this.

Mathematicians are fascinated with complex numbers because it is THE extension of real numbers that completes them in very important sense but it comes with so many unexpected and fascinating properties.

Quantum phase is not two-dimensional, it is complex and it amazes me much much more than two-dimensionality would.

The complex numbers are 2-dimensional, but their 2 dimensions are not the 2 dimensions of a normal 2-dimensional geometric plane, they are 2 other dimensions.

The 2 dimensions of a geometric plane correspond to the 2 orthogonal translations of the plane.

The 2 dimensions of a complex number do not correspond to translations, but to scalings and rotations of the geometric plane.

The multiplication of the complex numbers corresponds to the composition of scalings and plane rotations, which are invertible operations and that is why the set of complex numbers is a commutative field, unlike the set of points of a geometric plane, which does not have such an algebraic structure.

The set of complex numbers can be viewed as a plane, but it must be kept in mind that this plane is a distinct entity from a geometric plane.

(The Cartesian product of a geometric plane with a complex plane forms a geometric algebra with 4 dimensions.)

> The 2 dimensions of a complex number do not correspond to translations, but to scalings and rotations of the geometric plane.

Similar story for other 2D number systems:

For the dual numbers, they express scalings and "Galilean" rotations (i.e. shears).

For the double numbers, they express scalings and "Minkowski" rotations (i.e. Lorentz boosts).

Unfortunately, some of the nice theory of the complex numbers doesn't generalise easily to the dual numbers or double numbers. I'm thinking specifically of complex analysis which is very, very nice, and much nicer than real analysis. But I think these planar number systems have their own intriguing character: For instance, see "screw theory" and "automatic differentiation" for two distinctive applications of the dual numbers.

> The 2 dimensions of a complex number do not correspond to translations, but to scalings and rotations of the geometric plane.

I disagree. When multiplying complex numbers, they are viewed as scalings with rotation like you say. But when adding them, they are viewed as translations. If you only took one of these perspectives, then complex numbers would be very simple and not express anything interesting. Their complexity and power comes from alternating between the scaling with rotation perspective (multiplication) and the translation perspective (addition).

>The 2 dimensions of a complex number do not correspond to translations, but to scalings and rotations of the geometric plane.

So, polar coordinates basically?

That's exactly what I was trying to convey, thanks.

They are very much equivalent. You can express complex numbers with a 2x2 matrix of real numbers.

  R = 
     1 0 
     0 1
  I = 
     0 1
    -1 0
I and R form a basis that spans something that behaves like a complex plane.

You have the expected identities

R^2 = R

IR = I

I^2 = -R

I^3 = -I

I^4 = R

Transposition is complex conjugate. You can put them into exponentials, e^Ix = cos(x) + I sin(x); everything works as you would expect.

It's really inelegant though! To multiply two complex numbers, we need to embed one in a 2x2 matrix while keeping the other as a 2d vector, and then do the matrix vector multiplication. And furthermore, it's not like every 2x2 matrix corresponds to a complex number...

That is not correct.

When complex numbers are represented in the matrix form mentioned by the other poster above, the multiplication of the complex numbers corresponds to the multiplication of the matrices, not to multiplications of vectors by matrices.

The matrices of this form, having just 2 parameters instead of the 4 parameters of a 2x2 matrix, are obviously just a subset of the general 2x2 matrices.

However this subset is closed to matrix addition and multiplication, so it is a field isomorphic to the complex numbers.

This representation of the complex numbers is useful to understand that the multiplication rule for complex numbers is not some random arbitrary rule, but it is the same as the rule for matrix multiplication, which is also not a made-up rule, but it results from function composition, when you compose the functions of a vector expressed by the multiplication of a matrix with the vector.

Actually the construction of the complex numbers goes like this, if you consider the transformations of the 2-dimensional vectors that correspond to a scaling and a rotation of the vector (which are a subset of the linear transformations expressed by multiplication with a non-singular 2x2 matrix of general form), and you compute the matrices that perform such transformations, you arrive at 2x2 matrices with 2 parameters of the form shown by the poster above.

This set of transformations happens to have the algebraic structure of a commutative field. Because the matrices are defined by only 2 parameters, you can simplify the operations by keeping just the 2 parameters and using rules for operations expressed directly with them.

Thus you obtain the standard rules for operations with complex numbers.

To give a simple example - Mandelbrot set is a direct consequence of amazing properties of complex numbers and has nothing to do with two dimension.

Well, it looks great in two dimensions

Two-dimensional does not necessarily mean two spatial dimensions. It is dimensionality, or orthogonality, in the most abstract sense. For all practical purposes, complex numbers represent a two-dimensional system.

It is not a coincidence that they arose in 16th century Italy in the context of "completing the square" and related 2D methods/intuitions for solving equations.

> For all practical purposes, complex numbers represent a two-dimensional system

Yet operations on complex numbers are not the same as operations on vectors on simple two-dimensional plane. This is my point.

Complex numbers and (2D) vectors/matrices are different representations of multi-dimensional number systems. For each operation over one, you can find an analogous operation in the other.

You can even find attempts to mix the two representations, like i+j+k vector syntax. But matrices generalize better to higher dimensions and are easier to parse.

I agree with you here, but I don't agree on downplaying complex numbers to be just a base vector and orthogonal.

If we take a matrix representation of a complex number it is usually done as a 2x2 matrix of very specific structure. I completely agree that it is easier to work with. But looking at them this way misses very important place of them in the grand scheme of things.

Complex numbers are actually what real numbers really ARE under the hood, we just aren't taught to think this way. 'i' is what real numbers miss to be completed. And you don't need 'j's, 'k's and others.

If your point is that introducing 'i' above traditional real numbers syntax is ugly - I completely agree.

> Complex numbers are actually what real numbers really ARE under the hood, we just aren't taught to think this way. 'i' is what real numbers miss to be completed. And you don't need 'j's, 'k's and others.

This is an unnecessarily absolute statement. On what basis are you claiming that all number systems are fundamentally two-dimensional, and not one-dimensional, three-dimensional, or some other dimension?

I'm guessing that it is because you spent a lot of time working with mathematics in a 2D context, i.e. on paper or blackboard or screen.

> On what basis are you claiming that all number systems are fundamentally two-dimensional, and not one-dimensional, three-dimensional, or some other dimension?

I never said anything like this. I was talking about complex numbers only.

I suggest to stop here, we are talking about two different things. But, if anything, there is a comment in this thread by adrian_b which explains what I mean in more detail.

> It is not a coincidence that they arose in 16th century Italy in the context of "completing the square" and related 2D methods/intuitions for solving equations.

That's... not how it happened

According to the OP, it did. I'm sure that there were many independent inventions of higher dimensional number systems.

The complex numbers prefigured other "hypercomplex" number systems by several centuries. And the modern 2-dimensional view of them was only described close to the year 1800. Before that, they were purely algebraic.

The video (I briefly skimmed it) shows that they were invented to solve a problem in algebra. Nobody thought of them as being 2D back then.

I'm not a historian or anything, but your claim is textbook "whig history" -- and as far as I can understood you, I already proved you wrong in a previous comment.

I am not sure where this aggressive condescension is coming from. You appear to be conflating the inherent multi-dimensionality of complex numbers with their geometric/2D spatial visualization, which came later. I'm pretty sure we are in agreement here.

> I cannot agree with calling complex numbers just two dimensional.

I once read, imaginary number is isomorphic to 2d vectors.

Isomorphic as vector spaces, meaning that their additive structure is the same. But the complex numbers are usually not used as a vector space, but rather as an algebraic field, i.e. considering both their additive and multiplicative structure.

Thank you. But I am guessing I need to know a bit of abstract algebra to understand what you are saying which I don't have.

Historically, the complex numbers were discovered by accident while solving cubic equations using the general formula. The general formula only works in general if you can take the square root of a negative number.

One of the first attempts to provide a geometric meaning to the complex numbers was by John Wallis, and I haven't been able to make much sense of it. I suspect he didn't see it the way we do. Also, there's no indication that in spite of the work Euler did on the complex numbers, that he knew of their geometric meaning. The mystical sounding name "imaginary number" was coined by Descartes in the 1600s at least partly because he didn't have the modern view of them.

Of course, teaching by explaining historical developments is not a common thing in mathematics, and the above facts illustrate why. But you have to be aware that things do start off being mysterious before they're fully understood.

The mystery has been somewhat reawakened with the quaternions and octonions, and some other hypercomplex number systems. And mystery gets some people out of bed, so don't be too hard on it.

> It turns out that physical problems are much more accurately modeled in 2 dimensions than in 1 dimension, and even more so in higher dimensions.

Are you talking about matrices?

> Are you talking about matrices?

Yeah, matrices turn out to be a better general representation than adding more ambiguous symbols beyond `i`. Still not the "correct" one by any means, because e.g. they don't represent exotic (non-integer, etc.) dimensions well.

In mathematics in addition to real and complex numbers there are higher dimensional numbers namely hypercomplex numbers i.e quaternion and octanion numbers, and no other valid numbering systems beyond that. The latter hypercomplex numbers are better at accurately model reality compared to the former real and complex numbers inherent limitations. As an example, electromagnetic waves unlike sound waves has polarization component that can be easily modeled using quaternion numbers compared to just using imaginary numbers. Yet for centuries until now scholars have been reluctant to use quaternion numbers and insisted on conventional imaginary numbers technique for Maxwell's Equations electromagnetic solutions popularized by Heaviside rather than the quaternion techniques being used by Maxwell himself! History is repeating itself as our current scholars are very much reluctant to use hypercomplex numbers, not unlike our predecessors who were reluctant in using imaginary numbers although the advantages of the higher dimensional numbers are plain obvious.

> in addition to real and complex numbers there are higher dimensional numbers namely hypercomplex numbers i.e quaternion and octanion numbers, and no other valid numbering systems beyond that.

To be precise, no other multi-dimensional numbers beyond octonions (which lack associativity) that follow the behavior of complex numbers. There are indeed an infinite amount of multivector numbers with different algebraic rules.

> hypercomplex numbers i.e quaternion and octanion numbers, and no other valid numbering systems beyond that

The term "hypercomplex number" refers to more than those examples. Those examples are the normed unital algebras, which are a finite subset of the hypercomplex numbers.

A thousand times this!

Scientists and mathematicians should be embracing cold hard reality instead of wallowing in self-aggrandising mysticism. That's the point of science! Empiricism and rationality is are its distinguishing features. These elevate it above the near-worthless philosophies that it replaced.

The worst offender by far in the mainstream sciences is Quantum Mechanics, which is absolutely infested with woo and self-contradictory nonsense. Its overuse of complex numbers is just the beginning.

Mysterious action at a distance isn't mysterious and isn't action at a distance at all.

The electromagnetic field is made up of continuous waves filling space, there are no point-like photon particles. That was just a mathematical abstraction -- a convenient shorthand term for describing interactions with matter, and shouldn't be taken literally.

Electrons aren't little hard balls, which is why they don't rotate, and this is why spin isn't some huge mystery.


At some point you’re stuck with a bunch of equations that work and the only way to make progress is to try to interpret the equations.

> Electrons aren't little hard balls, which is why they don't rotate, and this is why spin isn't some huge mystery.

This doesn’t clarify spin at all. Spin is actually incredibly weird and anyone who writes it off probably doesn’t understand what’s interesting about spin quantization, spin/angular momentum noncommutativity, fermions obeying SU(2) instead of SO(3) rotation characteristics, etc.

Yet, the only instruments we can build pick up particles, and nothing but particles. Maxwell's electromagnetics, and his field waves, are a useful approximation. But as close as we have been able to measure, the Schroedinger equation is an exact description: of waves that resolve to probability of the detection of a particle.

There are things to complain about in our description of the universe with particles: magic numbers that can only be measured, not deduced. But incorrectness isn't one of them.

"Spin" is just a name.

> the only instruments we can build pick up particles

In general, this is not true. As in, there are experiments we can build that demonstrate that the wave nature of light is the true description.

Where people get confused is that practically all instruments are made of matter. Atoms, or at least a plasma or even individual particles, but matter nonetheless.

If "particle behaviours" is a property of matter, and you have only experiments made of matter, then this tells you nothing about the behaviour of light, which is not matter.

This is like basic experimental science 101. Beginner stuff.

Isolate the thing you are testing, and test it, and only it.

This is hard to do, but not entirely impossible.

Try harder to be a scientist and stop reiterating woo that you picked up from textbooks that copied it verbatim from people that died nearly 100 years ago. (The same people that thought that a human looking at an experiment was somehow an essential aspect, leading to a century of stupid people arguing about whether a brain is necessary for physics to occur!)

Nitpick: Anything in reality that can be modeled with complex numbers can be with real numbers. Complex numbers provide incredible convenience, but that's all. There's no phenomenon that requires complex numbers to explain - Schrodinger's Equation included.

There are real, complex, quaternion and octonion numbers. Saying that we can just get by only with real numbers is really being dishonest and in denial, it's just like saying we do not need OS and higher level languages because bare metal and assembly are all that we ever need.

As a rough analogy, certainly we can travel all over the world on foot by walking and swimming, eventually we will get there but in reality we will not. But the more correct analogy is that we certainly cannot go to space on foot, and higher dimensional complex and hypercomplex numbering systems is like our spacecrafts. Ironicaly the famous gimbal lock problem for spacecraft navigation can be easily solved using quaternion as opposed to disaster waiting to happen if we use only complex numbers [1]. Einstein certainly will have never come up with the infamous equation E= mc^2 and corresponding relativity theory if not because of quaternion numbers [2].

[1]Gimbal lock:


[2]Quaternions in University-Level Physics Considering Special Relativity:


It sounds like you're agreeing with my statement:

> There's no phenomenon that requires complex numbers to explain

I thought the discussion between "mathematics only models the world" vs. alternatives wasn't settled?

(most recently I'm referring to Roger Penrose's views on the "epistemic argument against realism"[0], although I don't fully know where I stand myself)

[0] https://www.wikiwand.com/en/Philosophy_of_mathematics#/Epist...

It is a bit odd that the video really talks up the idea that complex numbers sever algebra from geometry, then without pause goes straight into a geometric interpretation of complex numbers https://youtu.be/cUzklzVXJwo?t=1153 .

It goes to analytic geometry, not really the same.

> "rotate by tau radians"

tau/4 radians?


I've always found the geometric interpretation as easy as useful but felt I was not grasping the whole thing because it was too cute and easy.

Just want to commend you for using tau.

It's the first time time I've seen it outside of tau promotional material and I really hope it catches on!

So, hold on, where I have a point (x,y) with two "real" numbers, if I have (xi, yi) each of those numbers has a 2nd dimension, i.e. 'y part'? So it's sorta like ((x1,y1), (x2,y2)) where the second part is re-oriented about the y-axis?

Forgive me I still struggle with complex numbers, no matter how many videos I watch.

Well, I went through 5 years of electrical engineering which is almost entirely in the realm of complex numbers and I’m struggling to convey my intuition here to you.

I think it’s helpful to understand that complex numbers are two dimensional but they aren’t like an [x,y] where the two dimensions are completely 100% unrelated.

Here’s another way to think about it in a way that is familiar to you that I haven’t seen anyone else post yet.

Imagine natural numbers like 1,2,3… and so on. Now imagine those are just a subset or special case of fractional numbers like 1/2,4/3,… Those fractional (rational) numbers are expressed in terms of two natural numbers.

Now in the past they couldn’t solve all equations because they were frustrated that some numbers couldn’t be expressed as fractions. They called these irrationals. But irrational numbers can be expressed as an infinite sum of rational numbers. This includes numbers like pi and e.

Next you can make all those irrational and rational (real Numbers) to be negative. And then that’s still not the end of it because there’s more gaps! We need complex numbers :)

Then after complex you can go into tensor spaces like real^(3x3) where numbers are more like matrices. Perhaps the easiest to understand is a space of (r,g,b). Each pixel has an x,y coordinate but each point is vector valued (3 natural numbers on the range of 0-255). It should be easy to also visualize a complex number space of the same nature but with different properties.

Perhaps the best equation to understand complex numbers is Euler’s equation

e^(ix) = cos(x) + isin(x) Or if you put in pi you end up with the most beautiful equation ever.


> The idea that mathematics is "the language of the universe" that can be precisely "discovered" by brilliant minds is a ridiculous notion

This claim is at least as speculative as any claim that the universe works on some symbolic system.

Let me soften that claim. The probability that we humans are even capable of comprehending the fundamental nature of the universe, if such a thing exists, is approximately 0%.

Why do you think the rules of the universe are complicated? If you use any kind of kolmogorov prior that's not a reasonable thing to assume without evidence.

Yes, 'imaginary' numbers are invented syntax. I think imaginary numbers are just an additional tool tacked on to our existing tools in order to explore new areas. They are a patch written to cover some cases our previous tools failed to cover.

> The idea that mathematics is "the language of the universe" that can be precisely "discovered" by brilliant minds is a ridiculous notion that only serves the status of the mathematical elite. Mathematics is fundamentally about designing models and abstractions that help us reason about real phenomena with minimal cognitive resources.

Yes, creating abstractions and designing models are aspects of math, but I believe the modern definition of mathematics has expanded to include the study of abstract objects.

Abstractions can describe concepts (concepts that exist in many different places) but are not the same as the concepts. I believe these concepts are discoverable and independent of any notations/models/abstractions we create. So in that sense, I believe mathematics can be discovered.


exponential growth/decay (the spread of COVID in an unvaccinated population, bacterial growth, etc.) - invented or discovered?


fractals (pattern of rivers, trees, blood vessels, etc.) - invented or discovered?

I can describe the details of these with tools like 'geometric progression', and 'the Mandelbrot set'. Those are tools aiding my understanding of these concepts, but the concepts themselves certainly seem like they were discovered.

All of mathematics is invented syntax. For every bit of mathematics that is taught, there are a thousand bits somebody explored and, maybe, published, but it wasn't considered beautiful enough to teach. So, the maths that are taught are the bits enough mathematicians liked. Some that are taught were considered ugly by older mathematicians, but were embraced by their students. The really ugly stuff has either fallen away or just not been adopted into canon yet. Physicists have pushed mathematics in directions mathematicians hated to go, but those directions have borne surprising fruit, so mathematicians are beginning to accept them.

Mathematics is a tapestry being woven by living mathematicians guided by what they like.

I've never liked the framing of "imaginary" numbers as "not reflecting reality" or somehow being less "real" or something. It's just an ordered pair of numbers with a convenient extension of multiplication that preserves nice properties of the reals.

I think it's because of the name. If you called them double numbers or paired numbers nobody would say that.

Gauss complained about the name too. He suggested using "Direct, inverse and lateral numbers" instead of "positive, negative and imaginary numbers".

Imaginary numbers are not pairs. Complex numbers are. Continuing Gauss, I'd rename "complex numbers" as "planar numbers".

I kinda think of them as 'orthogonal' numbers.

That I think is the best alternative name I've heard yet. Since you describe two orthogonal dimensions.

Also vector numbers could be somewhat of a useful name, since they behave like a 2D vector (or even higher dimensional vectors for e.g. quaternions)

"Vector numbers" for something purely 2D seems dodgy to me. There are lots of unital algebras, even in 2 and 4 dimensions, and none of them should hog the name "vector numbers" as they all have equal entitlement to the name.

they'll always be twirly numbers to me

Dual numbers and double numbers are also planar.

Don't know how relevant this is, but I've been thinking about a better naming scheme for hypercomplex number systems. I came to it after seeing a paper about the "dual-complex numbers", which are not a straightforward complexification of the dual numbers as one might expect. Hopefully, the scheme should be pronounceable, and without the possibility of confusing it for something else. This town needs law! I'm thinking of asking for suggestions for what it should be exactly.

Huh? Clifford and Cayley-Dickson Algebras?

Those ones. What are you confused about?

wow, `naming is hard` is way older than I thought

In college, our professor said "We should be saying i is for invisible numbers, because you won't see them unless you know about them. But they're as real as death and taxes."

> I think it's because of the name.

indeed. the name is terrible; causes a lot of folks consternation, or tricks them into thinking strangely about the complex numbers.

anyone happen to know of any languages where they're not referred to as "imaginary", or anything that implies they're less "real"?

Interesting question. I had a scan of the articles in other languages, on Wikipedia.

Almost all the Latin alphabet languages use a variant of 'imaginary'. Icelandic appears to be an exception, where it means cross-number. https://en.wiktionary.org/wiki/þvertala#Etymology

(I don't know non-Latin alphabets I can't tell you anything about them.)

Agreed. It's a similar semantic problem to the use of the term "significant" in statistics. P<0.05 as being "significant", while P>0.05 as "non-significant" has a technical meaning that doesn't equate to the common use of the term "significant".

> If you called them double numbers or paired numbers nobody would say that.

I'm not sure that's true.

Just think of solving x^2+k=0. It's clear that for k<0 you get two solutions and k=0 you get one solution. But when k>0 the graph doesn't touch the x-axis... so why should I expect a "double number" or "paired number" to be the solution?

I'm teaching college algebra right now and introduce 'i' algebraically as a solution to x^2+1=0... but then we talk about graphing quadratics and there's no simple connection between the geometry/graph and the algebra.

Even if I had the time to talk about the geometry of multiplication and such, it's still a big leap to the graph of z^2+1 and its roots in the complex plane.

And it's this leap which, IMO, makes them seem "not real".

> I'm teaching college algebra right now and introduce 'i' algebraically as a solution to x^2+1=0... but then we talk about graphing quadratics and there's no simple connection between the geometry/graph and the algebra.

Take an equation like x^2 -2x + c

When C is some large negative number, the roots are symmetric about x = 1. As you increase C, roots are real until C = 1, basically the two roots meet in the middle.

When you increase it beyond 1, the roots become complex numbers, but they stay symmetric, they "lift off" from being real into being complex, but still symmetric (conjugates) where 1 is always the real part but the imaginary parts become sqrt(C - 1). You can conveniently imagine the complex plane on top of the real XY plane and the roots go orthonogal to the direction they were going when they were real, from the point where they met.

That's kind of how I visualize it for quadratics. For higher orders you cut a complex circle into n equal pieces. Haven't quite figured it out in my head yet.

My point is that we step off the real line to talk about the points where the function vanishes... but what about the neighborhood of the roots where we're plugging complex numbers into the function and not getting a real number as output?

There's a big conceptual leap here.

Oh yeah, Veritasium has some great visualizations of that too. I cant remember what the video was called but it was about fractals and the Newton Raphson method, and he had these input/output complex planes left and right to illustrate it.

Sure we can locally visualize a conformal mapping. But in the historical development of complex numbers, or pedagogically in a college algebra class… this would be putting the cart before the horse.

Again why would anyone posit that a pair of numbers suddenly appears when trying to solve x^2+k=0 as k goes from negative to positive?

I guess you mean a pair of pairs, as we expected to have two solutions that were normally real numbers. Yeah it's one of those times when the veil lifts and you find out you'd not seen the imaginary part until now. I guess enough has been said about completeness by other commentators, but there's no obvious answer as to why adding just one number solves your problem. Why won't it simply create new problems once you use those new complex numbers as coefficients in an expression? Surprisingly it's all we need.

I hate to “pull rank” but I have a PhD in mathematics so I’m personally well aware of all the nuances. Again my point is historical/pedagogical. Why should “pair numbers” or “double numbers” be the answer as the OP suggested? It’s not straightforward without getting into conformal mappings.

And why is two dimensions enough for third and higher degree equations? Were there a simpler geometric connection, you’d probably have a nicer proof of the Jordan curve theorem… but you don’t.

Not sure what you mean by pair of pairs… the OP said that calling imaginary numbers “double numbers” would clear up a lot of issues but that is not clear at all to me.

I'm don't doubt that you know more than me about this, just coming at it from my very basic knowledge.

By pair of pairs I just mean that you're looking for two solutions, and they turn out to be computed numbers, both of them.

The problem is not the term "imaginary", the problem is the term "number. Unlike any other numbers imaginary numbers cannot be ordered by size. So in that sense they are not numbers at all. In fact they represent rotations in two dimensions. They should be better called rotation operators instead of imaginary numbers.

I love Veritasium. Derek has done a number of videos that are awesome like this. The videos are great and educational, but he also manages to subtly weave in an almost spiritual subtext to them as well. For example, in this one, you learn about imaginary numbers in some cool ways, history, how awesome e^ix is, etc, but also this yen-yang balance between embracing not only what is real but taking a chance on what is imaginary.

Some of my other favorites:

- https://m.youtube.com/watch?v=rhgwIhB58PA

- https://m.youtube.com/watch?v=HeQX2HjkcNo

- https://m.youtube.com/watch?v=OxGsU8oIWjY

- https://m.youtube.com/watch?v=3LopI4YeC4I

- https://m.youtube.com/watch?v=pTn6Ewhb27k

I think he’s going to lose the $10,000 bet on the car though.

Some of popular educational/informational channels and podcasts make this patronizing assumption about the audience, watering everything down to make it "accessible". Veritasium doesn't do this, yet maintains huge popularity.

I didn't click through all (mobile, too slow) so don't know if it's in there, but one favourite i have is the one where he has an ice cube, a metal block and a plastic block and ask people on the street which block is colder.

It teaches a lot about what temperature vs feels cold vs heat is.

That's my favorite video of him. This one ties history, education, simple geometry, advanced abstractions in an entertaining script. Pretty high bar.

> I think he’s going to lose the $10,000 bet on the car though.

Are you referring to the wind powered propeller car bet or is there a new one?

Yes, the wind powered deal, but it appears it's actually been conceded by the physics guy. I guessed wrong. Glad I didn't bet. :D

Yeah, it's a fascinating question and I can see why it trips people up. Anyone that's done sailing/windsurfing will be familiar with the phenomena of apparent wind, and that's roughly what's going on here too.

Real numbers are for straight number lines, and imaginary numbers are for circular number lines. That is about all there is to it.

It's really a shame we introduce imaginary numbers as quirky work around for negative square roots. I would say that is more of a side effect. Multiplying positive numbers always results in positive numbers, but due to the circular nature of imaginary numbers you can get positive or negative results.

Complex roots are like saying this equation has no solution in a traditional Cartesian/Euclidean world, but does have solutions in a rotational one.

I think we if introduced imaginary numbers as rotations initially, the seemingly magical things like Euler's Identify and Formula would look much more trivial.

EDIT: I would also say the lie that we live in a 3+1D world (X, Y, Z + Time), helps confuse the mater. We probably live in a 6D + 1 world: X, Y, Z, Pitch, Yaw, Roll, and Time. Just as you can conceive of a flatland 2D universe without a Z axis, you could conceive of a 3D world without rotational orientation.

Oversimplifications like this are not helpful. There are a ton of ways that complex numbers are interesting, and they have important generalizations as e.g. an even subalgebra of G^2 (leading to quaternions as the even subalgebra of G^3, and so on). They’re not just “a circle”.

> EDIT: I would also say the lie that we live in a 3+1D world (X, Y, Z + Time), helps confuse the mater. We probably live in a 6D + 1 world: X, Y, Z, Pitch, Yaw, Roll, and Time.

If your model doesn’t treat time the same way as space up to the metric tensor, it’s probably not very well generalized. There should be an even number of parameters.

Rotation including the time dimension is just acceleration. The math is simpler if you’re in a ++++ metric - rotations are a bit weird in +++- because rotations into the time dimension are hyperbolic. I suggest reading Dichronauts, a book covering a ++-- tensor.

What does the ++++ and +++- notation mean? Looking at info about tensors I don't see much. I did order the book out of interest.

I really like this explanation. The negative square roots explanation never sat right with me in high school and I think this touches on why.

The way I think about it:

- To express direction in Cartesian space, you need signed numbers

- To express 2D rotation, you need imaginary numbers

- To express 3D rotation, you "need" quaternions (in the sense that they're the smallest way to represent an arbitrary rotation)

> quirky work around for negative square roots. I would say that is more of a side effect.

I agree it's not a perfect motivation, for teaching purposes. When you introduce an answer to the square root of a negative number, questions of soundness might arise. Obviously it's sound, but is that a distraction at the level where complex numbers are introduced?

Besides, many people like geometric interpretations: they might feel more 'real' or relevant than an algebraic motivation.

/Nonetheless/, I can't agree well defined square roots is just a side effect. The complex numbers are (up to isomorphism) the one and only algebraic closure of the reals. That's a very natural and and useful way look at them for, say, university level maths.

> introduced imaginary numbers as rotations initially

How would you do this without developing the exponential first?

Well any complex number a + bi can be described as a transformation matrix

    |a -b|
    |b  a|
So you could start by just asking what it means to transform the plane

Ok but in a college algebra class it’s very easy to ask “what is the inverse of the squaring operation?” Not so clear a path to the action of a matrix on the plane.

I have successfully discussed this in an advanced calculus class.

A Danish cartographer named Caspar Wessel came up with an early formal treatment of complex numbers, in his work "On the Analytical Representation of Direction" (wikipedia has a nice article about him). It was published in an obscure forum, and predates subsequent rediscovery of complex numbers by others. His formulation is IMHO beautiful, intuitive, and compelling. He did it in terms of directions on a map, replacing the "sign" of a conventional real number with a "direction" or "compass heading". So, one might say, "the nearest Starbucks is two blocks east and one block north". He was simply using what became known as the polar form of complex numbers. One can follow intuition and define reasonable notions of addition and multiplication by real values. But what of multiplying two "Directions"? Wessel derived what multiplication must mean, and went further in deriving a large number of identities involving his newly discovered "directional numbers".

If you pick a specific important pair of directional numbers, the multiplicative identity (call it "1") and a number 90 degrees away from it (call it "i"), it is convenient to represent any directional number as a the sum of scalar multiples of these two numbers. Then, one considers the simple formula "(i + 1)(i - 1) = i^2 - 1". A straightforward geometrical argument demonstrates that i^2 must be equal to -1. ("Show HN": gregfjohnson.com/complex)

I really enjoy looking into the history of mathematics and physics. I think it gives one a much better appreciation of why things are defined the way they are, and also the limitations of those definitions.

There is a really great book on the history of imaginary numbers. The history mostly focuses on how i was used to help solve algebra problems, so definitely one should be comfortable with high school algebra to get something from the text, but I don't think one needs much more math than that for the first half of the book. The second half gets more into how various use cases developed, in those chapters basic college level calculus would be a major plus. I read it more than 10 years ago though so no promises. :)

An Imaginary Tale: The Story of √-1 Paul J. Nahin


While I enjoyed the history part towards the end it got super hand wavy; especially as it went from describing "i" as a rotation to e^ix; Better Explained does a fantastic job of explaining these from the first principles.



Macdonald's Linear and Geometric Algebra is an excellent GA text which not only geometrically derives complex numbers (and rotation via their exponentiation), but also quaternions.

If we still had math duels, my life would have taken a very different trajectory. We need these.

Imaginary numbers were invented too early, that is why the usual explanations following the historical development are so mysterious. The notion of Sqrt(-1) does not help to explain what they actually are.

In fact imaginary "numbers" are not numbers. They cannot be ordered by size. Imaginary "numbers" are also not vectors in the geometrical sense. This is because you can multiply two geometrical vectors and get a scalar as result (scalar product). But if you multiply two imaginary "numbers" you get another imaginary "number", so this is different to a geometrical vector. In fact imaginary "numbers" are transformations or operators, because they multiply like operators/matrices and they act on geometrical vectors like rotation matrices. Group theory makes all that very clear: Imaginary "numbers" are just the elements of the group SO(2), the group of rotation operations in 2D (multiplied with a scaling factor).

However the invention of group theory started only with Galois in 1832 when trying to solve not the cubic, but the quintic. And the powerful notation of matrices as is ubiquitous nowadays was only invented in 1913 (by Cuthbert Edmund Cullis, see https://en.wikipedia.org/wiki/Matrix_(mathematics)#History)

Unfortunately mathematicians (and YouTubers) don't stop to teach the confusing outdated notions of imaginary/complex numbers rooted in the 18th and 19th century. And unfortunately this video belongs to the many videos just repeating the same old stuff over and over again.

> Imaginary numbers exist on a dimension perpendicular to the real number line.

Correct me if I'm wrong, but I don't think that imaginary numbers exist in a dimension with any sort of fixed spatial relation to real numbers. Isn't the fact that we chose to graph the imaginary axis perpendicular to the real axis on the complex plane simply arbitrary? Or maybe better described as "a functional way to visually plot the negative root component of a number."

My understanding is that visualizing the imaginary portion of complex numbers on a spatial plane with perpendicular axes also just happened to get a convenient 90 degree rotation by multiplying by i, and rotations like this just happen to show up all over nature. For example, the wave function they mention in the video, where they say it has an "imaginary component," seems like complete bullshit to me. You could write a program with real numbers in order to graph that wave...in fact I guarantee that's how a technical artist visualized it. In fact, it's the same with the Mandelbrot set, which supposedly "exists on the complex plane." No...it doesn't. It just happens that if you describe the arbitrary rotation operation that is performed in order to define whether or not a coordinate falls within the Mandelbrot set or not, the rotation from the way we've chosen to graph the complex plane means the equation can be written very concisely like this.

Every independent variable should be graphed perpendicular. X and Y when graphed are perpendicular. If they are not graphed perpendicular it indicates that a change in X position must change Y (or vice versa) whereas being perpendicular makes it clear that they are independent and you can change one without any thought that you're changing the value of the other.

Complex numbers have 2 independent variables. The real and the imaginary parts. You can change one part without being forced to change the other. So you graph them perpendicular.

It's not really a convention, it's a rule that each independent variable is perpendicular (or else they aren't independent!). Complex numbers have 2 independent variables.

Well yes, if we want to plot any sort of independent values, we have 3 spatial axes we are familiar with and will use them to graph accordingly. But I'm not going to argue that "housing prices therefore exist on an axes perpendicular to square footage." I just plotted the relationship that way.

I think focusing on the perpendicular doesn't make much sense since of course we graph different variables perpendicular.

The real point i think you're making is that complex numbers are just a notational shorthand for multidimensional vectors. And yeah. That's true.


This is why it's fun to know the history of math.

The complex plane was introduced by Caper Wessel in a paper that was published in 1799 so it would've existed before vector notation.

My guess is by the time vectors got popular the complex notation, and theorems that people had proved which used complex notation, had already stuck. But I'm only a hack math historian so I can definitely be wrong here.

I think it's important to keep in mind that math and science, much like the code base that I am trying my hardest to avoid, is evolved.

There are valid reasons to plot variables that aren't perpendicular. Maybe you want a plot of the radius of a cylinder and its volume, for example. Then a change in the x axis changes the value of the y axis and the axes wouldn't be drawn perpendicular.

Dimension in this sense means degree of freedom, not a spatial dimension. That is, a number on the "imaginary dimension" cannot be represented on the "real dimension". This is similar to how a length does not represent a width or height.

When visualizing the real and imaginary dimensions, it is convenient to represent them using x/y axes. However, some visualizations of complex functions use colour/hue as a way of representing the real or imaginary part of the result.

Dimensional analysis is used in physics, etc. when manipulating fundamental units (time, length, luminosity, etc.). That is, m/s^2 has dimensions length=1 and time=-2.

Right, my point is that length and time don't exist perpendicular to each other simply because there might be some utility in visualizing them that way. The complex plane seems to be generally taught as if imaginary numbers in some capacity exist on this perpendicular axis. Even Veritasium, who is supposed to be teaching deconstruction/first principles approaches to these concepts, is saying this.

The spatial unit vectors for a 3 spatial dimension [x,y,z] vector are i^=[1,0,0], j^=[0,1,0], and k^=[0,0,1]. These are all said to be orthogonal to each other, which is a generalization of perpendicularity [1] to non-spatial vector spaces.

As a lay person who is not familiar with the mathematics would recognize the term perpendicular, using that in the video is fine.

[1] https://en.wikipedia.org/wiki/Orthogonality

That is correct. In the Clifford formulation that is equivalent (and in general a superset) of complex numbers you can choose the dimension or behavior to be whatever you want it to be in the resulting algebra. You can even assign it a particular metric or nullify it. The 'square to -1' that implies quadrature is just a specialization in that case. Many algebraic things of complex numbers like the exponential map still work or have general counterparts.

I disagree, it's not simply arbitrary. Euler's formula eⁱˣ=cos(x)+i⋅sin(x) exists outside of any physical representation of the complex plane. And the fact that we already graphed trigonometic functions lent itself naturally to a corresponding graph of complex functions. As sibling comments have stated, any graphical visual representation of completely independent variables is most efficiently represented with perpendicular axes, so the mapping is inevitable.

There is also a great playlist by Welch Labs called "Imaginary Number are Real" that I really like: https://youtube.com/playlist?list=PLiaHhY2iBX9g6KIvZ_703G3KJ...

That was one of the most interesting 20 min history lessons I have ever watched.

I wish I was taught this history when I was taught the math.

My schooling provided next to no historical context when learning mathematics. My math teachers just taught the math, and my history teachers were probably largely unaware of math history.

Veritasium always makes great stuff

I'd say mathematical concepts are "discovered" rather than "invented". Even something like complex numbers.

Pure math concepts like complex numbers are not naturally existing, just waiting for us to find them, they're human-defined tools to describe things. Like new words, they're invented. They wouldn't be there without us, as they are, for the most part, artifacts of our cognition.

That's self-evidently untrue. The properties that make a circle would be true regardless of whether a human ever set eyes on a perfect circle. Us identifying those properties is an act of discovery via research. Codifying those mathematical truths into a written notation is the only component of the process that could really be called invention.

It's not self-evidently untrue, this is an incredibly complex philosophical question with no easy "right answer". There is also a spectrum of positions about this.

The Stanford Encyclopedia of Philosophy is a good resource for reading about this topic: https://plato.stanford.edu/entries/platonism-mathematics/#Ob...

Edit: another good link: https://plato.stanford.edu/entries/nominalism-mathematics/

No human has ever set eyes on a perfect circle, because it (most likely) doesn't exist in nature. As such, I would argue that a perfect circle is a concept (or a model, if you will) that we've invented to make it easier for us to deal with an imperfect real world. I would not call identifying properties of such a model an act of discovery: one can come up with any model, no matter how far from reality, and use some set of axioms to identify its properties, but none of it will make said model real or fundamental in any way. The best we can hope for is that the model will be useful for making predictions about the real world.

But we still chose the labels. We chose what the elevate to the level of a mathematical object. Heck, even the idea of "what is true" is not universal in mathematics. Intuitionist and classical logic have different ideas of what. it means.

Well it really depends on how far you push the definition. There are inherent properties that can be discovered, but the way they're calculated and described is purely arbitrary. You need to get the same result of an area of a circle in the end, but how you get there is invented. Far more feasible and evident for complex stuff than the basics of course.

As shown in the video, the depressed quadratic was basically solved by 3 people in 3 different ways, with today's description and definition being different from that too.

If that were the case, would you expect another civilization on some rock in different galaxy to arrive at entirely different concepts, or ones isomorphic to our own?

I’d expect other civilizations to have also invented rockets, microwave ovens, radio communications, and more. Does another civilization arriving at the same thing as us mean it isn’t an invention?

Their observations of nature and their ability to predict stuff should be consistent with ours.

They might not use the same mathematical tools or the same physical models, but they should make the same predictions. That is, we might not be able to understand their theory of gravity, but whatever their theory is, it has to be able to make predictions about orbits, black holes, etc.

I don't think we could assume much about the mathematical concepts they use beyond that.

That's a pretty hard question to answer, and I think impossible to answer definitively (at least unless we met a civilization that did arrive at different concepts). It's kinda like the allegory of the cave; it's hard to envision another way of looking at the world, but that doesn't necessarily mean it's impossible for there to be one.

It’s a good question to think about.

A constant refrain of mine is that our brains have convinced us that they’re universal understanders, but we really don’t know that to be true.

Imagine the difficulty of dealing with aliens who have a different mapping of the physical universe, different than our mathematics, which is both true but literally and physically incomprehensible to us.

Their mathematics will most likely have fundamental differences from ours that have led them to solve problems we haven't noticed. They will be appalled at how we have missed such basic features of the universe. And we will be appalled at some things they haven't noticed.

I wonder if one can place mathematical concepts on a spectrum from discovery to invention? To me, the pythagorean theorem feels much more like a discovery of a "hidden" eternal truth that was once beyond our grasp. But to me, complex numbers seem more like a notational choice, more akin to an invention. To put it another way, I would expect aliens to have the pythagorean theorem, but perhaps different notation for complex numbers, like representing them with matrices and computing over them with linear algebra. Such a representation would reduce the distinction of the "imaginary axis" which is really not that special.

I may be naive; I haven't worked or thought with complex numbers since completing my basic education.

Definitions are invented, theorems are discovered.

The Pythagorean theorem is a discovery about Euclidean geometry. But in order for that discovery to be meaningful, one must first invent Euclidean geometry, or at least something sufficiently similar to Euclidean geometry.

If you are not familiar, this is part of a very long-running discussion in the philosophy of mathematics: https://plato.stanford.edu/entries/philosophy-mathematics/#F... (For context on the intro to that section, Platonism is, roughly speaking, the idea that mathematical objects truly exist and mathematicians are discovering them)

I wasnt, thanks!

I'd say especially complex numbers. Few things seem more natural once you get used to them and see how they make everything fall into place.

I remember just playing with my good old TI-83 Plus Silver Edition, plugging a bunch of symbols together, like e, and i, etc... when I suddenly got the e^i*pi = -1.

How in the world do these completely unrelated (to me, at the time, at least) constants ended up with -1? It was mind blowing. This made me actually interested in math.

If you realize where e comes from, which is compounding and what we call an exponential process, being that "the current rate of change is proportional to the present value", then the exponential of a complex number is no different to the exponent of a real number. In the case of one-dimensional real numbers you assign the slope of the function to the current value. The complex case is exactly the same, but if you think instead in the two-dimensional Argand plane and complex algebra, the slope is the tangent to a circle proportional to the angle at that radius (which curve is the one that at every point the slope is equal to its complex value?).

Therefore with the rules of complex arithmetic the tangent provides the "curling" effect in the curvature of a periodic circle, while in the real case you get the common compounding shape of the exponential curve. The relation to sine and cosine is just the projection into direct(real)/quadrature(imaginary) components in either fixed or intrinsic coordinates. Same when you expand the exponential into its complex power series.

BTW, this is actually the essence of infinitesimal transformations in continuous groups and the exponential map, which generalizes this concept to other types of numbers or abstract objects (i.e. Lie group theory).

I've been wondering about the connection between the "curling effect" and the exponential curve of e for a while. Do you have any links where I can learn more in depth?

With complex numbers in polar form the tangent is easy. In the unitary circle, if you derive e^(i*t) you get i*e^(i*t), which maps the cos(t) real component to i*cos(t) imaginary, and the i*sin(t) imaginary component to -sin(t) real. This is effectively a 90 degree rotation, so if you integrate the tangent infinitesimally over its path parameterized by t you will recover the circle.

Here is some introductory material to what I referenced above and some generalizations into more dimensions (which, as Hamilton discovered when stumbling into quaternions trying to augment complex numbers, is not as straightforward as you would think):

- Why i? [http://www.stat.physik.uni-potsdam.de/~pikovsky/teaching/stu...]

- An Introduction to Geometric Algebra with an Application in Rigid Body Mechanics [https://www.researchgate.net/profile/Terje-Vold/publication/...]

- Functions of Multivector Variables [https://journals.plos.org/plosone/article/file?id=10.1371/jo...]

- Lie Group Theory - A Completely Naive Introduction [https://jakobschwichtenberg.com/naive-introduction-lie-theor...]

- Previous HN discussion: Intuitive Understanding of Euler’s Formula [https://news.ycombinator.com/item?id=18325865]

Thanks to the way math is taught in India I still have very little understanding of so many of these concepts though I'm very good at remembering formulae/steps, mechanically calculating them, and applying them to real-world problems.

It wasn't until I stumbled upon Better Explained[1] I began to appreciate the need to understand math at a deeper/common-sensical level. Now I make it an effort to teach math to my son using examples, analogies so that he gets concepts like positional numbering system, why we need it, its advantages etc.,


My favorite mental model for imaginary numbers is that they are placeholders or stepping stones. A speed of (30 + i*10) miles per hour is nonsense, but sometimes in the course of calculating a nice real 40 mph, you might take a detour through some steps including complex numbers. Having them handy sure will make the derivation easier.

Same as negative numbers. I can’t be negative six feet tall, but negative feet (as a magnitude) make the calculations way easier.

You can always reformulate these derivations into derivations that don’t use e.g. imaginary numbers, but there’s no need to. If I were apointed Senior Exectutive Math Concept Namer For Humanity, I’d call them something like “algebraic placeholders” or “algebraic closure stepping stones” or something in that vein instead of “imaginary numbers.”

Yeah, this is the intuition I've come to, particularly after reading Quantum Computing Since Democritus. If you'll let me be slightly hand wavy, you can get to Quantum Mechanics by generalizing probabilities in two steps: allow them to be negative, then allow them to be complex. Negative probabilities are quite unintuitive but seeing them as an accounting placeholder similar to how negative numbers are vs counting numbers helps it all click imo.

The other big thing is that if you want algebraic closure you need complex numbers or something isomorphic to them.

I think that i is not interesting because i^2 = -1 but because i * conjugate(i) = 1.

It's the same thing

  i * i = -1  <=>  -i * i = 1

I know but the semantics are different.

In what way does multiplying an equation with -1 change its semantics?

It obviates the unit circle and more importantly, for split complex numbers, you’ll have j*conjugate(j) = -1 and the -1 obviates the connection between split-complex numbers and reflection.

Veritasium is fantastic, I can't think of many current or past TV shows that are of a similar quality to his videos. A while back he did a video of his own startup story, which was very cool: https://www.youtube.com/watch?v=S1tFT4smd6E

I really appreciated the demonstration of solving the equations with shapes. It was like watching algebra's origin story.

Ouch. There is some logic and sanity in e^ix.

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