Convincing a high schooler that $i$ is a number

Question

I would like to convince a high school student that $i$ is a number, broadly put. I'm not going to define what I mean by "number" unless he asks, but I just want to convince him that it's somehow meaningful and useful. Part of this I've already done by showing the geometric interpretations of complex numbers as points on a complex plane with geometrically sensible and interesting interpretations of addition and multiplication. That's probably the most important part.

But I'd like to take it a step further if that's not asking too much, and I know that complex numbers started getting studied by studying cubic equations, particularly because we know that the curve $y=x^3$ must intersect any line $y=mx+b$ somewhere. Simple suggestive graphs convince us of this. So can someone give me an example of such a cubic equation that is most easily solved using complex numbers?

I'm not exactly sure what I have in mind for that (Will I ask him to use the complex root with polynomial division to find the other roots? Or is there some other assignment that is perhaps easier for a high schooler to grasp?). Any recommendations would be appreciated.

Icing on the cake is if the equation had some kind of meaning in Physics or Geometry, like modeling the volume of a certain box.

[Edit: Perhaps I should elaborate with how I came to be focused on getting some kind of an example with a cubic. I've read in some history of Math sources that the study of complex numbers arose from the study of certain cubics and the search for a general solution. I figure, if the study of cubics is what compelled people in history to take complex numbers at least a little seriously (and then later on, take them even more seriously) then it could serve as a helpful example. However, when I looked at the cubic and the method of solution Bombelli used to find the solutions of $x^3=15x+4$, it was clearly too complicated to give most of the high school students I work with. So I was wondering if there was some similar type of magic that was simple enough for an introductory lesson in complex numbers. But I'm starting to guess than the answer is no.]

Answer 1

I just want to convince him that it's somehow meaningful and useful.

This is probably going to require coming up with an example that is applied to something in the real world, not just an example that is applied to some abstract topic in mathematics (roots of polynomials, or some of the more advanced examples referred to in Paul Garrett's answer). You only need one example, but it has to be a convincing one.

The example I would pick is addition of sinusoidal functions. It is very easy to come up with real-world examples of this. If you want something very concrete (these are high school students, after all), you could make something out of mechanical linkages. Here is an animation of a slider-crank linkage: https://upload.wikimedia.org/wikipedia/commons/0/01/Slider_Crank_animation.svg . If you tie three of these together in series, you have a very concrete example of why you would want to take an expression such as

$\cos(t)+3\cos(t+\pi/2)+\sqrt{2}\cos(t+\pi/4)$

and simplify it into the form

$A\cos(t+\delta).$

Of course it's not totally obvious why you'd want to link together three such linkages in this way, but it's not implausible. For a more compelling motivation, you could use diffraction of a wave through three slits. However, this may be not as concrete and easy for high schoolers to get, since it lies outside their experience. Probably just showing a video of three-slit diffraction in a ripple tank would be enough motivation for a problem of this form. (I've been looking around for such a video for a while, and would be interested if anyone knows of one.)

So assuming you have the motivation, I think you could go ahead and solve the problem simply by using trig identities, and make the point that it's horribly nasty and grotty. Then you show them that the whole problem can be solved in a single line by writing down the sum of the amplitudes expressed as complex numbers:

$(1)+(3i)+(1+i).$

Answer 2

One of the historically compelling arguments for complex numbers was that they can be used to find real valued solutions to polynomials.

There's a nice discussion on this site. For example, the cubic $x^3=cx+d$ has, in general, a solution $$x=\sqrt[3]{d/2+\sqrt{(d/2)^2-(c/3)^3}}+\sqrt[3]{(d/2)-\sqrt{(d/2)^2-(c/3)^3}}.$$ (This formula can be verified by plugging in and doing a whole bunch of algebra, while completely ignoring whether or not $e=(d/2)^2-(c/3)^3$ is negative or not.)

If you try this with $x^3=15x+4$, you end up with $(d/2)^2-(c/3)^3=-121$, but $$x=\sqrt[3]{2+\sqrt{-121}}+\sqrt[3]{2-\sqrt{-121}}=4,$$ and you can easily check that this is a root.

This is a calculation which goes through the imaginary numbers, but ends up giving you a real number which actually works. This is like how one detects things in physics that can't be directly observed: the formula acts like the complex numbers really do exist.

Answer 3

If I were faced with this situation, I'd have a discussion with the student about what a number is. We believe that 1,2,3,4,… are numbers. It is a stretch to say that 0 is a number…which is why it took people so long to think about using it as a number. Think about it…0 behaves VERY DIFFERENTLY from any of the other familiar numbers. Admitting it "numberhood" is very strange. After this, why do we say -2 is a number? That's also bizarre. One may say that subtraction is the root of both of these problems…the point is that certain abstract properties of numbers are still seen if one generalizes the notion of addition and subtraction to include these new entities. The extended notion is a generalization of familiar properties to a larger class of objects.

In short, to me the problem with i is not that it is so terribly exotic. The problem is that we are too familiar and comfortable with 0 and negative numbers to realize how bizarre they are in comparison to the positive integers. If you can clarify the precise reasons why the non-positive integers are strange, i ought to be a bit more acceptable.

Answer 4

You probably cannot get away with having a different definition of "what a number is" than your student has. Your student almost certainly has a very clear understanding of what a number is. By this point in his education, your student probably defines a "number" to be "a real number". (But he might define it as "a positive integer", "a non-negative integer", "an integer", "a non-negative rational number", or "a rational number".)

You need to work with your student's understanding of what a number is.

First, you need to admit that i is not a real number.

Second, you can explain that i is almost a number -- it obeys most of the rules that real numbers obey. (Be able to show the student a list of the axioms that i obeys, and the axioms that i does not obey.)

From an engineer's perspective, i is useful for taking derivatives and integrals of sinusoidal functions. Specifically, we can pretend that sinusoids are exponential functions (with complex powers), and use the (relatively easy) method of taking derivatives (or integrals) of an exponential instead of the (relatively hard) method of taking the derivative of a sinusoid.

Thus, the key concept you should show them is:

• The slope of e^x = e^x.
• The slope of e^ax = a·e^ax.
• The slope of the slope of e^ax = a·a·e^ax.
• The slope of sin ax = a cos ax.
• The slope of cos ax = -a·sin ax.
• The slope of the slope of sin ax = -a·a·sin ax.
• Engineers need to do lots of calculations using exponentials and sinusoids. For example, alternating current has sinusoids, repeating waves can be thought of as sums of sinusoids, et cetera.
• Engineers need to do lots of calculations using slopes of functions, and slopes of slopes of functions. (distance -> velocity -> acceleration, momentum -> force, et cetera)
• That formula for calculating the slope of an exponential is awfully nice. Wouldn't it be nice if we could use a similar formula for the sines and cosines?
• We can, if we can imagine something like a number that is the square root of minus one.
• Thus, i. We define i to be that missing square root of minus one.

Some of the cool features of complex numbers (such as Euler's Rule, e^πi = -1, and rotation tricks) can be derived from the idea that a sinusoid is an exponential with an imaginary power.

The rotation tricks imply that the i axis is orthogonal to the real number line. Thus, the a + bi notation was an early way of expressing two-dimensional vectors. This was further extended to the a + bi + cj + dk notation for expressing three- and four-dimensional vectors. Treating a = 0 results in the i-axis being the x-axis, the j-axis being the y-axis, and the k-axis being the z-axis.

Answer 5

Ignoring the specifics of the question to concentrate on the title, I would explain that negative numbers, rational numbers, real numbers and complex numbers all answer the same need with respect to the precious number set.

You have non-negative integers, and you know how to add things. Then sometimes you need to substract, which exactly amounts to solve a certain equation: $a-b$ is the number x which solves $b+x=a$. Except sometimes such a number does not exist (in the nonnegative integers). At some point, one just introduces new numbers as solutions to such equations, and beautifully it all makes sense: with little adaptation, all previously known rules (distributivity, etc.) still holds.

Once one is used to work with possibly negative integer, the need for rational comes as soon as one wants to divide any number by (almost) any other number. We just introduce new numbers to ensure all equations $xb=a$ have a solution (when $b\neq 0$).

Then one needs to enlarge the set of numbers when one realizes that $\sqrt{2}$ is not a rational, i.e. there is no number (in the rationals) which solves $x^2=2$.

Introducing $i$ is not worse: we just want to have a set of numbers where $x^2=-1$ also have a solution. Very beautifully, just adding solutions to this equations gives solutions to all polynomial equations! This is an amazing theorem, when you think of it that way.

This is not exactly how things occurred in the history of maths, but it makes a lot of sense.

Answer 6

The formula (in terms of radicals) for roots of cubics generically can involve square roots of negative numbers, even when the coefficients are real and the three roots are real. This is the "casus irreducibilus" or something similar.

Indeed, in contrast, one could simply declare $x^2+1=0$ to have no solutions at all, etc.

A little more complicated: to understand the periodicity of the "sine" function defined indirectly by $x=\int_0^{{\rm arcsin}\, x} {dt\over \sqrt{1-t^2}}$, it is simplest to understand the ambiguity of this integral, up to path integrals enclosing both the $\pm 1$. These are integer multiples of $2\pi$, by complex analysis. Thus, the sine function itself, the inverse function, is $2\pi$-periodic. (This is in analogy with treatment of the doubly-periodic elliptic integrals...)

A more sophisticated example is Riemann's Explicit Formula, simplified by von Mangoldt, and generalized by Guinand, Weil, and others, giving asymptotics for various expressions involving primes in terms of the (non-real!) zeros of zeta functions and L-functions. If those zeros were "not real", or were somehow "optional", then these formulas would degenerate enormously.