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Imaginary unit

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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"

Categories: Constants | Complex numbers

Want to learn how to tackle those tough MATHCOUNTS and AMC counting and probability problems? Check out Art of Problem Solving's Introduction to Counting & Probability by David Patrick.

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