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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.]

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    $\begingroup$ I'm not sure precisely what the question is. What do you want from the person answering this question? $\endgroup$ – DavidButlerUofA Nov 9 '15 at 5:06
  • $\begingroup$ @DavidButlerUofA In the simplest terms, I want an example problem. More specifically I am imagining solving for the roots of a cubic, where the easiest method somehow employs use of complex numbers. $\endgroup$ – Addem Nov 9 '15 at 5:09
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    $\begingroup$ @BenjaminDickman True, but I'm avoiding quadratic examples because you could dismiss them as "not having actual solutions" whereas the equation of a cubic with a line seems more compelling: There has to be at least one solution to that! $\endgroup$ – Addem Nov 9 '15 at 5:35
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    $\begingroup$ Maybe show/discuss that $a+bi$ is a root iff $a-bi$ is? This establishes that every cubic has a real root: If it has a non-real root, then it has as a second non-real root its complex conjugate. The Fundamental Theorem of Algebra says a cubic has three solutions in total, so if the other solution is $c+di$ then $c-di$ is also a root; but there's only that one last root, hence $c+di = c-di$, i.e., $d = 0$ and the root is real. This tells you, e.g., $x^3 + x^2 + x + 1$ must have a real root before you even find it... (though it is easily seen in that case to be $x=-1$). $\endgroup$ – Benjamin Dickman Nov 9 '15 at 10:26
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    $\begingroup$ What is neat about all of this is that the introduction of new objects sharing properties of familiar counting numbers (associativity etc.) has already been encountered (reification of the idea of negative number from subtraction process). The point is that we introduce new objects that act like familiar numbers by looking at processes that can be explained by these "generalized numbers". Abstract algebra can be regarded as the study of "generalized numbers" in this sense. As far as getting a high school student to buy this, talking about the negative numbers first might help. $\endgroup$ – Jon Bannon Nov 9 '15 at 14:56
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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:

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

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  • $\begingroup$ I agree up to a point, although I think the identification of complex numbers with points and mappings in the plane does a good job of establishing their "reality" since we all agree that Geometry is real (in some sense). I have further backed that up with proving certain trig laws and other facts by way of complex numbers, which I feel strongly suggests that they're mapped onto reality in a good way. But you're absolutely right that a physical example would only make the case for complex numbers more irresistible. $\endgroup$ – Addem Nov 9 '15 at 17:32
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    $\begingroup$ I definitely think this is what I'm going to go with. Probably the hunt for a cubic is doomed and perhaps not even as good as this idea. This I could even construct in GeoGebra and give students something to toy with if they're interested. Many thanks. $\endgroup$ – Addem Nov 9 '15 at 17:36
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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.

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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.

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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.

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    $\begingroup$ $i$ isn't a real number, that is to say, $i\not\in \mathbb{R}$. But it's a number just as much as $-\sqrt{2}$ is: namely it's an element in a system which consistently extends the notions of $+$ and $\times$ beyond their meanings in smaller number systems. $\endgroup$ – Addem Nov 9 '15 at 6:41
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    $\begingroup$ +1 I was taught about imaginary numbers this way. "What if we create a number that's defined as the square root of minus one. What happens?". What's great about this method is that it also opens up the notion that you can try things like this in mathematics just to see what happens. What if I try to define something that usually doesn't exist. Is it useful? Mathematicians do this all the time, but most school kids don't consider it as an option. When I realised this, maths became so much cooler because you can play around with it and see what you get. $\endgroup$ – user2138 Nov 9 '15 at 9:06
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    $\begingroup$ @Addem I'm inclined to disagree with the claim that $i$ is a number just as much as $-\sqrt2$. Both are indeed numbers in important systems, but $-\sqrt2$ is a number in (at least) one important system, namely $\mathbb R$, where $i$ isn't. Similarly, there's a reasonable system of axioms for numbers, the ordered field axioms, that allows for the existence of $-\sqrt2$ but not of $i$. $\endgroup$ – Andreas Blass Nov 9 '15 at 11:40
  • $\begingroup$ @AndreasBlass Fair enough, they are not exactly the same (of course) and one could with some reason argue that the ordering relation is somehow a "line in the sand" about what we're willing to call "number". But I would argue against that line in the sand, as being not as defensible as the idea that "numbers have to count something!" in which case we'd only allow the positive integers to be numbers--or maybe by a bit of a stretch one could perhaps encompass the non-negative rationals. $\endgroup$ – Addem Nov 9 '15 at 17:15
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    $\begingroup$ @Addem If I have to draw a line in the sand, then I'd probably include the complex numbers among the numbers, but I wouldn't include quaternions. I'm not sure, though, that a line in the sand is a good idea; reasonable people can disagree about where it should go. Perhaps that's part of the answer to the original question: For some purposes, it's useful to include $i$ among the numbers. For other purposes, one might restrict, say to integers $\geq2$ (which I think was the ancient Greek notion of number) or liberalize to allow quaternions and whatnot. $\endgroup$ – Andreas Blass Nov 9 '15 at 20:26
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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.

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    $\begingroup$ One thing that I think is sometimes overlooked in this frequent account of $\mathbb{N} \subset \mathbb{Z} \subset \mathbb{Q} \subset \mathbb{R} \subset \mathbb{C}$ is that, while we are gaining certain properties, we are also losing others. For example, the first jump loses a property like, "every nonempty set contains a smallest element." Similarly, given the elements' standard order, the next jump up loses the discreteness (i.e., suddenly there are infinitely many elements between any pair of distinct ones). Etc. So there is some judgment in weighing the potential gains against losses... $\endgroup$ – Benjamin Dickman Nov 9 '15 at 22:47
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    $\begingroup$ [Of course, I mean the words gain and loss in an ill-defined, colloquial sense. A reasonable contention would be that a "loss" of discreteness should really be interpreted as a "gain" to fill in gaps. But I am thinking about the exchanges made as we move from the naturals up to the complex numbers (and beyond)...] $\endgroup$ – Benjamin Dickman Nov 9 '15 at 22:51
  • $\begingroup$ @Benjamin: I quite like your observation, here! $\endgroup$ – Jon Bannon Nov 10 '15 at 1:46
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    $\begingroup$ @JonBannon and others: This sort of observation can be found in the Nov. MTMS here and in the call for other expiring rules here. But I know that it extends back further (even beyond the cited TCM article from 2014)... $\endgroup$ – Benjamin Dickman Nov 18 '15 at 1:04
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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.

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  • $\begingroup$ Well that's a pretty great answer, at least in the sense that it gives a lot of things to read up on! :) Many thanks for the help. $\endgroup$ – Addem Nov 9 '15 at 17:34
  • $\begingroup$ Another related question that maybe the students shouldn't see right away (or maybe they should...!?!) is whether or not $p$-adic numbers are "real". For me, by this time, they are at least 2/3 as "real" or complex numbers, and sometimes 100% as "real", and, on some days, more "real". If physicists truly find such stuff useful, as was already suggested a few decades back, then one will find oneself claiming that future-modern physics can't be done without such "numbers"... $\endgroup$ – paul garrett Nov 10 '15 at 1:34

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