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.
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.