> 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" 
 - https://en.wikipedia.org/wiki/All_models_are_wrong
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 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.)
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.
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).
So, polar coordinates basically?
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.
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.
Well, it looks great in two dimensions
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.
Yet operations on complex numbers are not the same as operations on vectors on simple two-dimensional plane. This is my point.
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.
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.
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.
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.
That's... not how it happened
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 once read, imaginary number is isomorphic to 2d vectors.
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?
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.
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.
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.
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.
> 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.
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.
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!)
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 . Einstein certainly will have never come up with the infamous equation E= mc^2 and corresponding relativity theory if not because of quaternion numbers .
Quaternions in University-Level Physics Considering Special Relativity:
> There's no phenomenon that requires complex numbers to explain
(most recently I'm referring to Roger Penrose's views on the "epistemic argument against realism", although I don't fully know where I stand myself)
It's the first time time I've seen it outside of tau promotional material and I really hope it catches on!
Forgive me I still struggle with complex numbers, no matter how many videos I watch.
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.
This claim is at least as speculative as any claim that the universe works on some symbolic system.
> 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.
Mathematics is a tapestry being woven by living mathematicians guided by what they like.
I think it's because of the name. If you called them double numbers or paired numbers nobody would say that.
Imaginary numbers are not pairs. Complex numbers are. Continuing Gauss, I'd rename "complex numbers" as "planar numbers".
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)
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.
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"?
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.)
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".
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.
There's a big conceptual leap here.
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?
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.
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.
Some of my other favorites:
I think he’s going to lose the $10,000 bet on the car though.
It teaches a lot about what temperature vs feels cold vs heat is.
Are you referring to the wind powered propeller car bet or is there a new one?
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.
> 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.
- 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)
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.
How would you do this without developing the exponential first?
I have successfully discussed this in an advanced calculus class.
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)
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
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.
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.
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.
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.
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.
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.
As a lay person who is not familiar with the mathematics would recognize the term perpendicular, using that in the video is fine.
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.
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/
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.
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.
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.
I may be naive; I haven't worked or thought with complex numbers since completing my basic education.
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.
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.
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).
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]
It wasn't until I stumbled upon Better Explained 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.,
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.”
The other big thing is that if you want algebraic closure you need complex numbers or something isomorphic to them.
i * i = -1 <=> -i * i = 1