# Imaginary unit

The imaginary unit, $i=\sqrt{-1}$, is the fundamental component of all complex numbers. In fact, it is a complex number itself. It has a magnitude of 1, and can be written as $1 \text{cis } \left(\frac{\pi}{2}\right)$. Any complex number can be expressed as $a+bi$ for some real numbers $a$ and $b$.

## Trigonometric function cis

Main article: cis

The trigonometric function $\text{cis } x$ is also defined as $e^{ix}$ or $\cos x + i\sin x$.

## Series

When $i$ is used in an exponential series, it repeats at every four terms:

1. $i^1=\sqrt{-1}$
2. $i^2=\sqrt{-1}\cdot\sqrt{-1}=-1$
3. $i^3=-1\cdot i=-i$
4. $i^4=-i\cdot i=-i^2=-(-1)=1$
5. $i^5=1\cdot i=i$

This has many useful properties.

## Use in factorization

$i$ is often very helpful in factorization. For example, consider the difference of squares: $(a+b)(a-b)=a^2-b^2$. With $i$, it is possible to factor the otherwise-unfactorisable $a^2+b^2$ into $(a+bi)(a-bi)$.

## Problems

### Introductory

• Find the sum of $i^1+i^2+\ldots+i^{2006}$ (Source)
• Find the product of $i^1 \times i^2 \times \cdots \times i^{2006}$. (Source)

### Intermediate

• The equation $z^6+z^3+1$ has complex roots with argument $\theta$ between $90^\circ$ and $180^\circ$ in the complex plane. Determine the degree measure of $\theta$. (Source)

• Let $A\in\mathcal M_2(R)$ and $P\in R[X]$ with no real roots. If $\det(P(A)) = 0$ , show that $P(A) = O_2$. <url>viewtopic.php?t=78260 (Source)</url>