AoPSWiki

Art of Problem Solving's olympiad training program WOOT starts on Septebmer 8. Train with the top high school students in the the world! Click here to enroll today!

Personal tools

Log in

Search

Toolbox

Imaginary unit

From AoPSWiki

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 $b$ .

Contents

1 Trigonometric function cis
2 Series
3 Use in factorization
4 Problems
5 See also

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:

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

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)

Olympiad

Let $A\in\mathcal M_2(R)$ and $P\in R[X]$ with no real roots. If $\det(P(A)) = 0$ , show that $P(A) = O_2$ . (Source)

See also

Retrieved from "http://www.artofproblemsolving.com/Wiki/index.php/Imaginary_unit"

Category: Constants

NEW! NEW! NEW!
Want to learn how to tackle those tough AMC/AIME/Olympiad algebra problems? Check out Art of Problem Solving's NEW Intermediate Algebra by Richard Rusczyk and Mathew Crawford. Over 1600 problems!

© Copyright 2008 AoPS Incorporated. All Rights Reserved. • Foundation • Privacy • Contact Us