Difference between revisions of "Imaginary unit"

Revision as of 11:53, 27 October 2007

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:

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

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)

@@ Line 22: / Line 22: @@
 ===Intermediate===
-[[1984 AIME Problems/Problem 8]]
+*The equation <math>z^6+z^3+1</math> has complex roots with argument <math>\theta</math> between <math>90^\circ</math> and <math>180^\circ</math> in the complex plane. Determine the degree measure of <math>\theta</math>. ([[1984 AIME Problems/Problem 8|Source]])
 ===Olympiad===

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