An introductory post to complex numbers.

This is an introductory post that will introduce you to the world of complex numbers. This is designed to teach the reader who doesn't know anything about it yet and to enhance the understanding of those who already know it. In addition, the problems will be designed to challenge you to see if you really know your stuff. As such, I've tagged the post with both the CosinesGroup tag as well as the TorqueGroup tag.

First off, what is $i$? $i$ is an arbitrary unit that we mathematicians "made up" in order to explain the phenomenon of negative square roots. In particular, $i = \sqrt{-1}$. Now, you might ask, why bother? This unit isn't just something that mathematicians decided to use because it would make us feel better about having negative numbers under square roots. Indeed, the world of complex numbers opens up endless new and exciting possibilities about numbers. But that's for another lesson.

A complex number, commonly noted with the variable $z$, is one can be expressed in the form $a + bi$ for $a, b \in \mathbb{R}$ (a and b are real numbers). Note that real numbers are complex numbers -- namely, when $b = 0$. That is, $\mathbb{R} \subset \mathbb{C}$ (the real numbers are a subset of the complex numbers). The complex conjugate of an imaginary number $z = a+bi$, denoted by $\overline{z}$, is equal to $a - bi$.

In the complex plane, the $x$ axis corresponds to the real part of the number, while the $y$ axis corresponds to the imaginary part of the number. Thus, the number $a+bi$ corresponds to the point $(a, b)$ in the complex plane.

In addition, the absolute value of a complex number is given by its distance from the origin. Using the euclidean distance formula, this is equal to $|z| = \sqrt{a^2 + b^2}$. Note that the absolute value of a complex number is positive and real -- after all, we're dealing with distance here.

Furthermore, every complex number can also be expressed in two other forms. The first I will introduce is trigonometric form. This is in the form $r(\cos \theta + i \sin \theta)$, commonly expressed as $r cis \theta$. $r$ corresponds to the radius -- that is, the distance of the point $(a,b)$ from the origin of the complex plane. Thus, we know that $r = \sqrt{a^2 + b^2}$. Additionally, $\theta$ corresponds to the angle that is formed by the point $(a, b)$, the origin, and the positive x-axis. Thus, it can be calculated as $\tan \theta = \frac{b}{a}$.

One of the most useful applications of this theorem is the fact that the following holds:

$(r_1 cis \theta_1 )(r_2 cis \theta_2) = r_1 r_2 cis (\theta_1 + \theta_2)$. In addition, in the special case where we are multiplying the same complex number together, we get the result of DeMoivre's Theorem that $(r cis \theta)^n = r^n cis n \theta$. This is one of the most commonly used theorems in math problems dealing with complex numbers. Know it and understand it inside and out. That's it for now, let's get started on some problems:

1. A complex number $|z|$ satisfies $z + |z| = 2 + 8i$. What is $|z|$?

2. A complex number $z$ is equal to $9 + bi$, where $b$ is a positive real number. Given that the imaginary parts of $z^2$ and $z^3$ are equal, find $b$.

3. It is a well known fact that $\cos 2x = cos^2 x - sin^2 x, \sin 2x = 2 \sin x \cos x$. How can we use our knowledge of complex numbers to find similar forms for $\cos 3x$ and $\sin 3x$?

4. Show that $\tan^{-1} 1 + \tan^{-1} 2 + \tan^{-1} 3 = \pi$.

5. Let $S$ denote the region in the complex plane that is made up of all points $z$ such that $\large{\frac{z}{40}}$ and $\large{\frac{40}{\overline{z}}}$ both have real and imaginary parts between $0$ and $1$ inclusive. What is the area of the region $S$?

Problems #2 and #5 are credit to $AIME$.

Image credit: Wikipedia, Mandelbrot Set