Motivating complex structure on $\mathbb{R}^2$

Question

I'm giving a talk to a group of bright but not all that mathematically sophisticated students on the subject of complex numbers. I'd like to introduce complex numbers via geometric considerations about $\mathbb{R}^2$; in particular, I imagine an exposition going something like this:

You've all taken physics, so you know that two-dimensional vectors are sometimes really useful. We'd love if we could do everything on the plane that we can do on a number line.

Adding and subtracting can procede component-wise, but if we try to multiply component-wise, multiplication doesn't behave as we'd like it to. (E.g., zero divisors.)

This motivates the creation of some other "multiplication" of vectors.

Now here's where I get stuck: I want to define $$(a_1, b_1) \cdot (a_2, b_2) = (a_1a_2 - b_1b_2, a_1b_2+a_2b_1) \qquad (*)$$

play around with this definition to help the students understand what it means geometrically, and then eventually reach the punchline $(0, 1)\cdot (0, 1) = (-1,0)$.

I'm having trouble showing why $(*)$ might be a "natural" choice before you know about the connection to complex numbers. I also don't want to say right away that there's an interpretation based on dialation/rotation, since I'd prefer that that fact arise as a discovery along the way.

Even some easily comprehensible uniqueness claim would do — I know this is the only division algebra on $\mathbb{R}^2$, but is there anything simpler I can appeal to? I want to remove as much as possible the sense that complex numbers are needlessly abstract and arbitrarily constructed, and I fear this might be a weak point in the presentation.

Answer 1

If your students are familiar with the geometrical interpretation of $2\times 2 $ matrices then they should know or be able to find out that matrices of the form

$$ \left( \begin{array}{cc} a & b \\ -b & a \end{array} \right)$$

are a combination of rotation and scaling. Multiplication of two such matrices gives one of the same form:

$$ \left( \begin{array}{cc} a_1 & b_1 \\ -b_1 & a_1 \end{array} \right) \left( \begin{array}{cc} a_2 & b_2 \\ -b_2 & a_2 \end{array} \right) = \left( \begin{array}{cc} a_1 a_2 -b_1 b_2 & a_1 b_2 + a_2 b_1 \\ - (a_1 b_2 + a_2 b_1) & a_1 a_2 -b_1 b_2 \end{array} \right)$$

So too does addition of two such matrices.

You can then motivate $(a_1, b_1) \cdot (a_2, b_2) = (a_1a_2 - b_1b_2, a_1b_2+a_2b_1)$ as being a concise way of writing this which avoids effectively repeating the top rows in the bottom rows. So for example $(1,0)$ is the concise multiplicative identity.

If you wish, you can avoid explaining what such matrices are.

Answer 2

Well, you can give the above algebraic definition and show it fulfills some nice things similarly as the reals, like associativity, commutativity, existence of unit, inverses and etc.

Since you mention those students have studied some physics, they must have fooled around with polar coordinates and stuff, so the polar expression of complex numbers won't be neither out of the blue nor hard to understand for them. After this, the geometric intuition behind the product of complex in polar form will pop up pretty natural to them, I guess.

Answer 3

If you're introducing the via geometric considerations, why not just do this:

http://www.math.umn.edu/~hardy/1031/handouts/March.3.pdf

Answer 4

On the 2d Euclidean plane, complex numbers arise in the representation of rotations.

The following explanation uses geometric algebra, which I expect is far outside the scope of what you will want to introduce, but I'm hopeful that some of the ideas will stimulate some thought.

The geometric algebra of the 2d plane gives us four fundamental objects: a unit scalar 1, two unit vectors $e_1, e_2$, and a unit bivector $e_1 e_2$. This last object, in particular, represents an oriented planar object.

We can multiply these objects according to the following rules:

$$\begin{align*} e_1 e_1 &= e_2 e_2 = 1 \\ e_1 e_2 &= - e_2 e_1\end{align*}$$

And finally, $a(bc) = (ab)c$ for any three vectors $a, b, c$. What you'll note is that $e_1 (e_1 e_2) = e_2$, rotating the vector by 90 degrees, and similarly with $e_2$. In addition, $(e_1 e_2) (e_1 e_2) = -1$. It is for this reason that $e_1 e_2$ is often suggestively denoted by $i$, as in complex numbers.

Now, with this system of multiplication--the "geometric product"--we can build up various linear operators in fairly compact ways. For example,

$$\underline N(a) = -e_1 a e_1$$

is a linear operator that reflects over the y-axis. Here it is component by component:

$$\begin{align*} \underline N(a^1 e_1 + a^2 e_2) &= -e_1 (a^1 e_1 + a^2 e_2) e_1 \\ &= a^1 (-e_1 e_1 e_1) + a^2 (-e_1 e_2 e_1) \\ &= - a^1 e_1 + a^2 e_2 \end{align*}$$

Once you have an expression for a reflection, you can build up a rotation. This is a key point: any rotation can be built from two reflections in a plane.

So let's do just that. Let's reflect over the y-axis, and then reflect over the line $y=-x$. This is done by the following:

$$\underline R(a) = mnanm$$

where $n = e_1$ as before, and now $m = (e_1 + e_2)/\sqrt{2}$.

Note that $(mn)(nm) = 1$ because all the vectors are unit and by associativity. We can identify $(nm) = (mn)^{-1}$ and compact this slightly. See that

$$mn = e_1 \frac{e_1 + e_2}{\sqrt{2}} = \frac{1 + i}{\sqrt{2}} = \exp(i \pi /4)$$

The result is that we represent our rotation by

$$\underline R(a) = \exp(i \pi /4) a \exp(-i \pi/4)$$

This double-sided form is more common when dealing with quaternions. In fact, you can derive the properties of quaternions in the same way using the corresponding algebra for 3d space.

To rotate vectors on the 2d plane, we naturally end up using these objects, which are linear combinations of $1$ and $i=e_1 e_2$. Their multiplication rules are already determined by the multiplication rules of the algebra. They are perhaps best called spinors.

Now, is this the best way to introduce students to complex analysis? Probably not. Geometric algebra, though by no means new, is not traditional in any sense of the word. Still, perhaps it is possible to gloss over all the fine details. I think the big punchline to take away from all this is that complex numbers are intimately connected to rotations on the plane. Motivating the multiplication law, however, may just require some more fundamental supposition. Either you presuppose the existence of something that squares to $-1$, or you build up a multiplication system the way GA does and let the results all follow.