Simple applications of complex numbers
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~