I’ve been helping a high school student with his complex number homework (algebra, de Moivre’s formula, etc.), and we came across the question of the “usefulness” of “imaginary” numbers – If there not real, what are they good for?
Now, the answer is quite obvious to any math/physics/engineering major, but I’m looking for a simple application that doesn’t involve to much. The only example I’ve found so far is the formula for cubic roots applied to $x^3-x=0$, which leads to the real solutions by using $i$.
Ideally I’d like an even simpler example I can use as motivation.
Any ideas?
The fact that $\exp(i(\theta_1+\theta_2))=\exp(i\theta_1)\exp(i\theta_2)$ immediately leads to many trigonometric formulas, including the most basic of $\cos(\theta_1+\theta_2)$ and $\sin(\theta_1+\theta_2)$. Other good examples are $\sin 3\theta,\,\sin 4\theta,$ etc.
The easiest way to find the coordinates of a right polygon with $n$ vertexes is to find $n$ $n$th roots of 1.
First we can ask the student what may happen if we multiply a real number $b$ by $-a$, where $a$ is a positive real number.
Taking b as a vector, we can see that $a$ determines the product’s length, and $-1$ determines the direction—turning $b$ by $\pi$.
We then consider extending the number axis to a plane: what if we expand the dimensions and turn the vector by any other angle?
We can then construct the axis of $i$, which symbolizes the rotation by $\pi/2$ anticlockwise, give a few examples, multiplying $b$ by $ai$, where $b$ is any vector in this plane and $a$ is real, and see what happens. (Of course, by definition, $i*i$ means rotating the vector by $\pi$, and thus $i^2=-1$. )
One last step is to prove that on this plane we can construct any rotation with the help of $i$: take the unit vector $\cos\theta+i\sin\theta$, using the principle that $i^2=-1$, we can then get the desired result.
I think this is a most natural way of introducing imaginary numbers. For more you can refer to the documentary I recommend. Hope it can help you~
There are several convincing ways that we can help the student to “swallow” the complex number system, but, in terms of the simplest forms of application, the only places that we can turn are relatively “complex” themselves. For instance, people use complex numbers all the time in oscillatory motion. If you would like a concrete mathematical example for your student, cubic polynomials are the best way to illustrate the concept’s use because this is honestly where mathematicians even began needing this system. I don’t think much simpler of an actual MATHEMATICAL EXAMPLE exists (note: I am not talking about explanation/existence, just example).
So many things could be told about complex numbers…
It allows you to get out of the real line : $x^2+1$ doesn’t admit a solution on the real line and you need to get out of it.
In this new 2D space you may rotate ($z’=z e^{i\phi}$) with ease, scale, translate and combine all that just by writing $z’=\alpha(z-z_0)e^{i\phi}+z_1$. You may apply more complicated transformations in the complex plane (conformal transformations).
These transformations have very useful applications in 2D fluid mechanics and allowed to study with ease the shape of plane wings (Joukowsky transform) and so on… They were very useful during Maxwell’s investigations in electromagnetism. I won’t speak of Cauchy’s theorem multiple use…
Complex numbers allowed too to see very nice new worlds just starting with a very simple transformation $z’=z^2+c$ the Mandelbrot set.
They showed their true importance in quantum theory built on probability amplitudes
(‘real’ QM is a rather abstract construction by comparison).
Last (because we have to stop somewhere) they allowed Hamilton to find their natural extension : the quaternions. If you suppose that another equivalent to $i$ exists, $j$ perpendicular to the $(1,i)$ plane and apply the basic operations of algebra (except commutativity) then you’ll have to add a third one $k$ if you don’t want your construction to collapse. With this new tool you may investigate as well the 3 dimensions of space as the 4 of space-time (quaternions are very closely related to the Pauli matrices describing the spin and the $\gamma$ matrices of the Dirac relativistic equation).
Here
You might find some lucid and illustrative discussions within its first chapters.
Basically, this book intends exactly to make complex numbers friendly.^^
Point is that “imaginary” represents sine waveform that is quite real. So if you need so add or to subtract two trigonometric signals, you will do it more easily with complex numbers (phasor) approach, rather then directly. See more:
http://www.cirvirlab.com/index.php/electric/98-phasor-ac-signal-addition-and-subtraction.html
Contrary to the name, “imaginary” numbers are not imaginary at all. Sadly this name causes them to be viewed suspiciously.
First, consider the equation $x^2-2=0\in\mathbb{Q}$[x]. A solution to this equation does not exist in $\mathbb{Q}$, so we look for bigger fields where this equation has a solution, or the field extension $\mathbb{Q}(\sqrt{2})$. Similary, $x^{2}+1=0\in\mathbb{R}$ does not have a solution, so we adjoin $\mathbb{R}(i)$ and get the complex numbers. So actually from a purley mathematical point of view there is nothing suspicious with $i$ just like there is nothing wrong with $\sqrt{2}$.
Now for an example. Conformal transformations are used in physics and engineering to transform problems with difficult geometry into much simpler ones. Schwarzâ€“Christoffel mapping in particular are used in experimental aerospace engineering to model fluid flow. The paper linked to contains mathematics that will be advanced for a high school student. However, it really illustrates how important and practical the imaginary unit is.