What are the complex numbers, really?
robjlow.blogspot.com
robjlow.blogspot.com
At some point, one's limited biological brain will be unable to fully perceive or visualize some mathematical object. That's where intuition comes in.
QM went into blind math side way past any analogies. Yet it is currently matching observations which is as good as it gets with true.
Especially in mathematics it is fashionable to write proofs in a style that completely erases the reasoning that lead to the proof. I think that's a pity. So much unwritten knowledge gets lost that way when the originators die. Differential forms, for example, have a beautiful geometric interpretation, but the way they're taught nowadays obscures that completely and makes it seem like they're a formal algebraic tool only. In fact, we're now several generations later, so even some of the instructors may be unaware of that, because their own instructors failed to transfer the original geometric intuition!
Moreover, it very much depends on who your target group is. My brain, for example, works entirely inductively (in the beginning). I won't be able to develop an intuition of something if I don't start with examples. Pictures are often good examples. During my undergrad studies, my linear algebra prof was as critical about pictures as you and other commenters here. I hated it. I was never able to get an intuition about the more abstract topics until I saw concrete examples including pictures in later lectures and projects. Moreover, not everyone is going to be a theoretical mathematician or quantum physicist. I suspect that by not showing pictures, you usually lose more students along the way than pictures would ruin students that need a fully abstract understanding (later). It would be interesting to see some data on this, but I guess its going to be difficult to collect.
It's literally easier to describe Mandelbrot set without the picture than with it. The set of complex numbers c where recursive equation "z_n = z_(n-1)^2 + c, z_0 = 0" does not diverge.
In fact, the picture of a Mandelbrot set is actively misleading! You think you can see the set, but it's a fractal! It's infinitely complex in a way that cannot be shown in a picture.
>Eversion of a sphere?
Pretty sure that one was figured out way before anyone was able to visualize it. Same with Banach-Tarski paradox. These things resist visual intuition in the first place, so it's much safer to rely on equations when dealing with problems like these.
Thus pictures are not only tutorial but a fundamental necessity for understanding mathematics.
In the case of complex analysis, pictures are critical for explaining the contour integrals for example. The pictures do not mislead, they elucidate.
That said, there's nothing wrong with the OP article. It is an introductory piece and folks seeing this stuff for the first time do well with a variety of approaches to the topic-- this one happens to not use pictures. That's OK.
i is basically "rotate by 90" in the same way -1 is "head left from 0 instead of right"
It happens to be true that this field called the complex numbers is a two dimensional real vector space. It took humans quite a long time to come up with 0 and negative numbers. There’s no way the jump to the complex numbers was going to be as easy and it isn’t as simple as, “numbers in 2d”. Why isn’t 3D a finite dimensional associative division algebra?
[1]: https://math.stackexchange.com/questions/444475/whats-the-di...
I'm not sure that's true. It's a very good way to visualize complex numbers at first, but the set of points in a 2D plane does not define complex numbers.
That's a very interesting question actually, what is the difference between R^2 != C ? Here's someone's attempt at showing this (top answer): https://math.stackexchange.com/questions/444475/whats-the-di...
To me, trying to visualize complex numbers is an easy way to forget that R^2 != C.... They look the same but don't behave the same way (how do you multiply to members of R^2 with each other ? you need to define it, whereas multiplication in C is already defined). C doesn't behave like 2D vectors either. Since multiplying 2 complex is not the same as multiplying 2 vectors (whatever that means) although their components could be the same.
structure structure structure.
You can use the geometric product, and C falls out as a result.
There is no reason to limit to the plane either[0]. Or four dimensions for that matter[1]. Complex numbers have ways in which they are "more fundamental" than quaternions in the same ways that reals are "more fundamental" than complex numbers.
Once you’re working in k-dimensions, where k > 3 (or maybe 4), you can’t rely on pictures anymore because there is no satisfying depiction of 4 or greater dimensions on a 2 dimensional medium. Complex numbers can be learned this way because they’re (simplified) the set R^2. Number systems in general are sort of okay to learn this way because they typically just extend other number systems, so you can successively build intuition on top of previous abstractions.
But you won’t really “learn” a lot of much more complicated topological or geometric (and therein analytic) concepts if you can’t build intuition without “seeing” it. As a very simple example, contrast these two approaches to defining a trivial concept in topology, which you’d probably come across before complex numbers in an analysis course:
1. A neighborhood of a point p in a Euclidean space is the set of all q such that d(p, q) < r, where r is some radius greater than 0.
2. A neighborhood is a circle (or sphere) with a radius of r surrounding a point p.
One of those feels more immediately intuitive, but that’s only because we’re dealing with the simple case of k = 2 or 3. You can’t “dodge” the complexity by visualizing it once you get into higher k, you have to just formalize it to develop intuition rigorously. Definition 1 is useful because it abstracts to conceptual domains we cannot practically visualize. Definition 2 would be useful if you could build a visual intuition of a 3-sphere, or 4-sphere, and so on. But building a visual intuition of k > 3 dimensions just shifts the problem away from what you’re trying to learn to something that is just as difficult: eventually you have to just get comfortable staring at a page of frustrating numbers, unfortunately.
A pictorial formalism is possible for most if not all mathematical concepts as much as a symbolic.
It just won't be naive and may not have been systematically developed and communicated / taught anywhere.
The issue here is mostly just an awareness that, naively, the pictures change fast than the formula. (r^1, r^2, r^3... look more a like than lines, circles and spheres).
And for the pictorial representations which do exist, implicit in understanding them is the symbolic information. Conversely, you can understand common naive representations without an implicit understanding of the formalism. For example, we all know what a circle is before we know about irrational numbers, but it’s silly to try and understand a 5-cell or a tesseract without a broader understanding of k-dimensional space. It’s precisely that naive intuition that doesn’t generalize well.
In other words, my point more simply was caution: use pictures, but don’t replace symbolic formalism with them, just use them augment your understanding once you’ve at least sort of got it.
Isn't that what mathematicians call a ball rather than a neighbourhood? A name which rather suggests that they are visualising it.
It would not be correct to use terminology like “ball” to describe a “neighborhood”, even if, in specific instances, they are the same representation in different contexts (i.e. geometric vs analytic).
In a metric space M=(X,d), a set V is a neighbourhood of a point p if there exists an open ball with centre p and radius r>0, such that B_r (p)=B(p;r)={x∈X | d(x,p)<r} is contained in V. (My emphasis.)
That said, once we have a metric space, the metric balls give a natural basis for the topology, which means that every theorem about neighborhoods has a corresponding one that uses balls instead. So in practice, when in a metric space we just use balls everywhere because they let us exploit their extra structure if it's needed. This is why you see the abuse of language calling neighborhoods balls, despite that being technically incorrect.
However, 'ball' is from geometry, while 'neighborhood' is from analysis. They overlap, but they're not to be confused, because 'neighborhood' is used to emphasize that one should not visualize it.
You’re right, but in fairness I’ve seen multiple analysis textbook authors use terms like “circle” and “ball” to describe a neighborhood in topology. This is usually in the fashion, “...(in other words, E is a ball.”
This is precisely the sort of thing I dislike, because aside from being technically incorrect like you say, it can (for that section) encourage a student to think, “Oh, why didn’t he just say that?” and skip the heavier definition -> theorem -> proof sequence preceding it, to the detriment of understanding later material.
Build up your picture-mirroring skill on 2-dimensional and 3-dimensional pictures, and you can eventually imagine pictures with more dimensions. Maybe you could even manage an animation in 4-dimensions. About the only thing I can reliably imagine in 6 or more dimensions is a hypersphere, and even that might be incorrect if one of the dimensions is not mutually perpendicular to all the others or if its basis vector doesn't square to itself.
TL;DR: complex numbers are the unique algebraic extension to the real numbers
Logic example (transitive closure): given "A->B", "B->C" in a sytem then your system can be extended to also contain "A->C" if you decide to interpret "->" as transitive.
For example: The field of rational numbers is not closed since it does not contain the roots of x^3 - 3 (since we can prove the cube root of 3 is irrational). The real numbers are not closed since they do not contain the roots of x^2 + 1. The complex numbers however, are closed.
An algebraic closure of a field is an embedding of that field into a closed field. For example, the rational numbers into the complex numbers, or the real numbers into the complex numbers. The second case is special: while rational numbers have many possible closures, the only closure of the real numbers is the complex numbers.
Algebra is set of objects and operations on them. Operation is function that gets number of objects as argument and maps it to some other thing
A set has closure under an operation(s) if using those operations on the set members always gives you a member in the same set.
Example:
Natural numbers (0,1,2, ...) are closed under addition and multiplication, because you can't add two numbers and get something that is not natural number. Natural numbers are not closed under subtraction or division because you can get a result that is not a natural number.
THANK YOU. That line was key and just clicked for me. I'm actually liking everyone's completely-different ways of explaining the same things. When one, or five, are confusing, the sixth may be key to filling in the holes.
Maybe a different example would help. Imagine we are working with Integers modulo 3 (denoted Z/3Z). Z/3Z is a field, meaning we have all the expected arithmetic (+, -, , /) and we can look at polynomials. Here is a polynomial that is irreducible i.e. does not have roots in Z/3Z:
x^2+1
We can construct an extension field in which this has a root; let's call the root i and write the extension Z/3Z(i). Here is where finite fields like Z/3Z differ from the real numbers: extensions like Z/3Z(i) still have irreducible polynomials. On the other hand, in the real numbers, x^2+1 is irreducible, but if we construct an extension where it is reducible, R(i) i.e. the complex numbers, we get a field where there are no irreducible polynomials (that is the basic idea of the fundamental theorem of algebra).
> Complex numbers were introduced because they cropped up naturally in the solution of cubic equations. What might we need to introduce to solve higher order polynomial equations? What if we let the coefficients of our polynomials be complex numbers themselves?
> The astonishing answer is: nothing. Once you've allowed yourself complex numbers it turns out that you don't need anything else for higher order polynomials, or even polynomials with complex coefficients. The fundamental theorem of algebra tells us that any polynomial of degree n with complex coefficients (and that includes real ones, since a real number is a complex number with an imaginary part of zero) has n complex roots.
> That's a beautiful mathematical result.
The simplest, (or maybe most compact) extension.
You can also achieve closure by going from R to R^2 and adding the Geometric Product. This is also closed, but it includes vectors in R^2 as well as the complex numbers.
This ambiguity is where Galois theory starts. Analogous "theories of ambiguity" arise all throughout mathematics (Galois connections). This is really fascinating stuff, with deep connections to quantum physics...
I agree. I think complex numbers are best explained in the context of motivations for number systems (and I made a comment explaining complex numbers this way about a month ago[1]). This way takes longer than just answering the question, “what’s a complex number?”, but it also builds a better intuition of why complex numbers aren’t silly (or at least, why they’re only as silly as anything that isn’t a natural number). If you can get a student to be okay with the idea that we define new number systems to resolve problems in prior ones, you can get them to be okay with a construction of complex numbers from the reals, just as they’ll accept a construction of the irrationals from the rationals, and integers from the naturals, etc.
I find that a really rigorous (albeit...spartan) first pass for understanding complex numbers can be developed by reading through Chapter 1 of Rudin’s Principles of Mathematical Analysis. He doesn’t go quite as far as developing Peano arithmetic from first principles, but he does build up the number systems successively, beginning with the natural numbers if I recall correctly. That was the first book I read where I felt like I really understood what complex numbers were, because up until that point I was only dealing with them algebraically using an explicit i in the form a + bi. Rudin’s development of the complex plane and representation of real and complex numbers as points (a, b) is much better for intuition, in my experience, and it lights the way to an even deeper intuition of what complex numbers are in the context of Euclidean spaces later on.
That said, Rudin is terse to the point that many would call him pretentious, so if there is another book that does the same thing with better exposition, that might be a better choice...
> But there is one further thing to note, which is the inherent ambiguity between i and -i. We can't tell these apart. John Conway says this well, when asked about the square root of -1 his reply is "which square root of -1 do you mean?" (paraphrasing.)
It’s been a while since I did this exercise, but if I recall correctly this ambiguity is part of the proof that we cannot order the complex field, right? We end up with an absurdity because i should be greater than -i, but i^3 = -i, which evidently resists coherent ordering. And similarly i x i = -1, but -i x i = 1.
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Have a look at Conway's "Book of numbers". He has a nice description of the complex numbers there. Conway can be terse aswell, but at least he is funny!
It’s similar to the inherent ambiguity between left and right - the only defining characteristics of them are they are opposite to each other.
Note that 1 and -1 have different mathematical properties (in most cases - in Z/2 they would be equal) that allow us to distinguish. 1 * x = x but -1 * x = -x.
This is also true for x == i.
If by reality we mean the real world (not mathematics or some abstract model) then the article does not actually answer the question what the complex numbers are really -- they describe what complex numbers mean mathematically (geometrically, algebraically, number theory).
Here are some interpretations of imaginary and real parts in terms of the real (physical) world:
o Time vs. space. This means that they have to be measured using different constituents with different properties. For example, space is reversible and periodic (we can return to the same point where we were before many times while it is not possible to return to previous point in time).
o Periodic vs. monotonic behavior. For example, rotation frequencies or oscillators (electric or physical) vs. linear motion.
If beauty doesn't motivate you sufficiently, then take solace in the fact that the properties of some differential equations which arise in physics and engineering are determined by polynomials you can build out of them, and the systems modelled by these differential equations are stable if the real part of all the roots are negative. Google for control theory to see lots more.
and
This turns out to be a useful way of thinking about alternating current, and you can analyse AC circuits using complex arithmetic in much the same way as you use real algebra to analyse DC circuits.
and ... did you read it all?
Please don't insinuate that someone hasn't read an article. "Did you even read the article? It mentions that" can be shortened to "The article mentions that."
"Userful way of thinking" and "analyse" something is not a real object strictly speaking - it is a process or phenomenon (no doubt complex numbers are useful here).
But can you say what kind of real object or property a complex number represents or measures? For example, an integer can represent the number of apples and their weight is represented by a real number. What property of a real object can be expressed, for example, as 50i ?
The other descriptions were merely circular references to other mathematical inventions.
The bookkeeping explanation is still not satisfying for me though.
By pushing the common complexity into a shared location, problems become easier to solve. Complex numbers really do factor out a lot of the complexity from many problems, at comparatively little cost.
When people say they don't "get" complex numbers, usually they mean they've seen them mentioned, but have never actually done anything with them.
It's like staying that you've read a book on lisp and don't get it. Using these things is an essential part of coming to understand them.
What people complain about is that they do not have a (real or hypothetical) device which can display its measurements as a complex number while they do have devices which measure properties in terms of integer, rational and real numbers.
Even if real numbers are "real", only a countable number of them are computable, even in theory. The rest are probably completely inaccessible, barring hypercomputation. (Interestingly, access to true real numbers gives you hypercomputation as a side-effect)
Another way to say this: imaginary versus real exponentials.
[1] http://geometry.mrao.cam.ac.uk/1993/01/imaginary-numbers-are...
1. They don’t realize what you’re saying, i.e. number systems and mathematics in general are built from definitions, not observations, and any definitions work as long as they’re compatible with what we’ve already proven,
2. If they realize #1, even implicitly, they still do not know why mathematical definitions are motivated. A student might say, “Okay so I understand what complex numbers are, but why do we have them? What’s the point of all of this?”
In my opinion, new mathematical concepts should be taught by placing them in the context of the least complicated thing the student already knows, then motivating them by demonstrating what problem they resolve. In the case of complex numbers, this can be explained to a student via analogue to the irrational numbers, which are the set of all numbers p/q where p and q are positive integers. We “invent” and define irrational numbers to resolve real world problems involving numbers that cannot be in the integral or rational systems. Similarly, we define complex numbers to resolve real world problems involving negative numbers on a plane, and so on...
Interesting questions are 1) why do we care about polynomials and 2) how do polynomial roots motivate complex numbers.
1) Every non-constant polynomial has a root
2) Every holomorphic function is analytic
The only connection I know is that (2) implies Liouville's theorem which implies (1), but it doesn't seem any less miraculous.
Using the exterior derivative df = (df/dx) dx + (df/dy) dy. it is possibly to define (identifying C with R^2) a function to be holomorphic when it can be written df = g (dx + idy) for some function g. From the properties of the exterior derivative you then get 0 = d(df) = d( g dz) = df' ^ dz, which implies that dg = h dz for some h, hence g is holomorphic as well. Defining the complex derivative of 'f' to be 'g' at least shows that any holomorphic function is infinitely complex differentiable.
I haven't quite managed to figure out why this definition of the complex derivative coincides with the limit of f(w+z)/z as z->0 with z complex, but it should follow from the algebraic properties of C.
Showing holomorphic functions to be analytic is somewhat trickier, but you can probably use that the above definition of holomorphicity and Stokes' theorem together imply Cauchy's integral theorem (every integral around a loop vanishes).
Totally nonstandard terminology, but pedagogically very sound approach that was totally at odds with the official syllabus... she just “filled us in” with ‘synonyms’ a while later so that we would not be totally baffled when sitting our exams (the IB).
It has served me well ever since.
https://betterexplained.com/articles/a-visual-intuitive-guide-to-imaginary-numbers/There is no need to jump to quaternions (which is 4D, not 3D, and the point has been missed here), like the author does. There is a much simpler case that people are more familiar with. The problem with the article is that it does not gauge its audience well. As another commenter mentioned, pictures help. With this discussion on two dimensions you can use a lot of great graphics out there and talk about the beauty of Euler's formula. These topics all generalize, but it is ludicrous to talk about the generalization first.
https://www.youtube.com/watch?v=T647CGsuOVU
I like to tell my kids that imaginary numbers aren't really imaginary .. that in many ways they're more real than the Reals.