complex numbers. These numbers involve i, the square root of −1, and they take the form a + bi, where a is the real part and b is the imaginary part.
This is the shortest explanation I've seen of what a complex number is! (Reminds me of when a piece on some advancement in quantum physics in a pop-science magazine tries to explain to the reader what an atom is.)
Edit: OK, let me try then to explain what homology is: it is a way of discovering (and counting) interesting features of the space in consideration (like holes in a topological space or fixed points of a symplectic transformation) by looking at a series of algebraic structures, such as Abelian groups or vector spaces, which one can often conjure up from objects of the space (points, paths, etc.)
It's short, but I don't think that it's very helpful for someone who hears about it for the first time. When you see
a + bi
your first reaction, as a 12 year old, is to look at the plus sign and wonder - what's the actual result here is? How do these things add up?
I think that explaining it through geometry and vectors first is far more natural to begin with, and gives a solid intuition of what happens with complex numbers when they go through simpler operations.
This is perhaps true for the complex numbers, but I am not sure if any field extension can be explained geometrically in a sensible way. (For example, think about extending the rationals by adding the square root of 2.)
That's why you first develop a very good intuition of what complex numbers really are, understand Euler's formula (and probably asked to come up with a proof for it), and only then move forward.
In mathematics, it's really easy and dangerous to think that you understand a concept only because you can memorize it's definitions and properties; any math education should fight this pseudo-understanding and make sure that you really grok it.
What you are saying is true of course, except nobody in their right mind should think that algebra lacks intuition and that learning it consists of rote memorization.
Adding the square root of 2 gives you a way of describing coordinates in a symmetry system where you start with a square grid and then allow yourself to make 1/8 turn rotations.
I kind of disagree with this. The geometry implications are incidental (extremely interesting and useful, but incidental nonetheless). If you only understand complex numbers from a visual standpoint, you're going to be very confused when we go to (hypercomplex) number systems with > 3 dimensions.
No, the geometry is fundamental, and the rest is just bookkeeping. If you go through life only ever doing the bookkeeping without understanding its meaning you’re going to be very confused (as indeed many students are today, and many mathematicians were historically).
The proper way to understand complex numbers is as ratios of planar vectors. (Other interpretations can be layered on top of that one.)
A complex number z = v/u is the object which we can multiply the Euclidean vector u by to get the vector v, via scaling and rotation. That is, zu = (v/u)u = v (u\u) = v. Notice we have no coordinates here.
If we like, we can split the complex number as a sum z = x + yI into a scalar part x = ½(v/u + u\v) and a bivector part yI = x = ½(v/u – u\v), where x and y are scalars and I is a bivector representing a quarter turn in the u–v plane.
This version generalizes beautifully to higher (or lower) dimensions, to pseudo-Euclidean space, and even to non-metrical settings.
> No, the geometry is fundamental, and the rest is just bookkeeping.
I'd push back on this, but I think it's probably a foundational disagreement (perhaps between pure and applied mathematicians). For example, a key behavior of number systems as we move up in dimensionality is the loss of "nice" properties†: when going from the reals to the complex numbers, we lose total ordering. I've seen proofs of this done visually[1], and some are pretty clever, but it's much easier to simply "do the algebra" and end up with a contradiction. Getting to 1 ≺ 0 ≺ 1 is literally a two-line proof, while geometrically you might have various intuitions that would prove to be wrong: for example, why can't we use a spiral[2] to achieve total ordering on a plane?
Not to mention that going to the quaternions, we lose commutativity of multiplication -- I don't think anyone even bothers trying to prove this visually because it would probably end up looking like spaghetti.
† This is, right off the bat, not very intuitive: why would more degrees of freedom minimize nice properties? This is in the same vein as the volume of an n-dimensional ball tending to zero as n tends to infinity. To see why this happens, you need to put away the shapes, and look at the math.
Commutativity of multiplication is not “lost”. It’s something you shouldn’t generally expect to have. It would be good if people would spend more time learning about non-commutative multiplications at a younger age, which should be expected in general. Mathematicians’ bias towards commutativity is often so culturally ingrained from a lifetime of commutative algebraic structures that it’s very hard for them to break away from. If they spent more time on the geometry earlier on it could help quite a bit.
That multiplication of a vector by a scalar + bivector oriented in a plane containing the vector on the left happens to have the same result as multiplication by the reverse on the right Zv = vZ† comes down to the identity a(bc) = (bc)a for any three coplanar vectors a, b, c, which happens because vectors commute with both scalars and coplanar bivectors. This doesn’t work more generally because the trivector part of the product doesn’t vanish when the vectors are not coplanar. This convenient feature results in many nice results for plane geometry / complex numbers which don’t hold in more general settings. It would be good if students would start with the general version (in high school or early in undergraduate science/engineering programs) so they can separate out the special features in their minds and not take them for granted.
P.S. you can “graphically” demonstrate the non-commutativity of 3-dimensional rotation to a layperson (including children) in a few minutes by turning around an arbitrary 3-dimensional object in your hands.
P.P.S. The hypervolume of an n-dimensional ball of unit radius only tends to zero as n tends to infinity if you use an n-dimensional hypercube with unit side length as your unit for hypervolume. If you use a standard simplex as your unit for hypervolume, then the volume of the ball tends to infinity. If you use the volume of the ball as your unit, then the volume of the ball tends to 1 by definition. Using a unit-length n-cube as your unit hypervolume is a convenient but arbitrary cultural convention.
One main reason this ratio is ”counterintuitive” is because we conventionally use a unit-side-length square or cube but a unit-radius disk or ball, and the extraneous factors of 2 caused by the discrepancy cause confusion in low dimensions; people weirdly think of disks as being bigger than squares. If we used a unit-diameter disk/ball instead, we would see how the ratio of hypervolume of ball/cube goes strictly down as dimension goes up:
Sure, a useful representation is important, and it demonstrates the universal power of the complex numbers. (Same with groups vs. linear maps vs. matrices.) But the proper way to understand the field of complex numbers is indeed to do it in all its abstract generality, i.e. simply as the algebraically closed extension of the field of reals.
This is backwards. The “proper way” is to start with the right intuition, and then work out the bookkeeping and the generalizations and interconnections afterward. My version can be explained to 12-year-olds, and your version doesn’t really make sense until grad school. (Anything starting from “real numbers” and “algebraic closure” and “field extensions” is pedagogically horrendous, and calling that version “simple” has to be some kind of sick joke.)
All of the “abstract generality” is contained within the original idea of ratios of planar vectors, and this idea can be explored to whatever depth someone likes.
> (Same with groups vs. linear maps vs. matrices.)
I don’t know where you are going with this, but groups and linear maps are different concepts.
Linear transformations (and other kinds of transformations) should be studied geometrically long before anyone learns about matrices (and anyway, the matrix version involves the arbitrary choice of coordinate system). The basic ideas are plenty accessible to children.
In a similar way, symmetries of 2- and 3-dimensional shapes, patterns, tilings, arithmetic, etc. should be explored in detail before someone ever sees the definition of a group.
Well, one would certainly have a hard time trying to explain to a 12-year-old how to solve a quadratic equation using rotations (let alone division of vectors).
It’s certainly no harder than any other way of describing complex solutions to quadratic equations. (But you probably want to start with quadratic equations that have scalar solutions.)
It’s quite straight-forward to show graphically what it means to square an origin-preserving similarity transformation of the plane: just apply it twice. So if your original complex number transformed an arbitrary chosen vector by scaling it some amount and rotating it some amount, just scale and rotate the result again. This is no mostly the same idea as repeated multiplication by scalars, as taught in primary school, but with planar rotations thrown in.
Adding complex numbers can be shown graphically by demonstrating what effect the sum has on a vector (since the geometric product distributes over addition), or where convenient what effect it has on a square grid.
We can easily graphically apply some quadratic equation to arbitrary complex numbers, and see what it does to them (again, where we draw the complex numbers using points o + zv for some arbitrarily chosen origin o and “unit vector” v).
Showing that when we take a quarter-turn in our plane I, and apply it twice, we get II = –1 is a relatively trivial observation. It relies on the observation that a half-turn rotation is equivalent to a reflection across the axis of rotation. So that gives us I as one of the roots to xx + 1 = 0.
The root of a quadratic equation is just the complex number which gets mapped to 0. So try applying your quadratic equation to different complex numbers and see where they go. There will always be two complex numbers mapping to zero (except in the case of a double root). Once we have some intuition for the problem we can try to develop some concrete algorithms or strategies for finding those roots efficiently. We can also try to make other kinds of plots to help us, e.g. using a tool like https://observablehq.com/d/3a81942cd20f81e4
I haven’t tried teaching quadratic equations to middle schoolers, but since the graphical version mostly entails drawing pictures, I would guess it to be rather easier than a purely algebraic version. Either way there are quite a lot of tricky concepts to explore and unpack, so the students are going to need months of sustained effort.
This post shows clearly that geometric intuition (other, perhaps, than drawing a parabola crossing - or not crossing - the X-axis) is not helpful at all understanding and solving a quadratic equation, while elementary algebra indeed can and does; the imaginary unit, then, appears naturally as the square root of the negative unit (which leads to the need to extend the reals); if anything, the approach that you advocate would make the kid's head spin (no pun intended) with the result being the total loss of the ability to think about things in a simple and natural way.
I said essentially «there are many concepts involved which will take months to unpack, irrespective of which approach is taken, but in good faith here are some sketchy descriptions of what pictures you might draw if you wanted to start with a graphical interpretation ...»
To which you responded essentially «your sketch summary consisting of a few picture-free paragraphs aimed at an expert audience doesn’t completely describe all of a complicated set of ideas to a naïve middle school student anywhere near as well as 2 semesters of introductory algebra class does at covering a more limited subset of the same ideas, therefore it must be useless.» This after you earlier proposed an approach to complex numbers that optimistically will first be accessible for typical 3rd year undergraduate math majors.
This conversation is absurd. But all the best to you.
I know Eric Weinstein has introduced a (fairly undocumented) "theory of everything" which pins the fundamental structure of the universe to be geometric in nature (11 dimensions). Does anyone know if this is at all related? Or perhaps the geometric approach in physics is fairly typical?
I do not believe it's related. Symplectic spaces are useful in classical mechanics. Not sure about "typical", but yes, geometry in general is the most important part of the conceptual mathematical framework of modern theoretical physics.
This is the shortest explanation I've seen of what a complex number is! (Reminds me of when a piece on some advancement in quantum physics in a pop-science magazine tries to explain to the reader what an atom is.)
Edit: OK, let me try then to explain what homology is: it is a way of discovering (and counting) interesting features of the space in consideration (like holes in a topological space or fixed points of a symplectic transformation) by looking at a series of algebraic structures, such as Abelian groups or vector spaces, which one can often conjure up from objects of the space (points, paths, etc.)