Art of Problem Solving
AoPS Online
Math texts, online classes, and more
for students in grades 5-12.
Visit AoPS Online ‚
Books for Grades 5-12 Online Courses
Beast Academy
Engaging math books and online learning
for students ages 6-13.
Visit Beast Academy ‚
Books for Ages 6-13 Beast Academy Online
AoPS Academy
Small live classes for advanced math
and language arts learners in grades 2-12.
Visit AoPS Academy ‚
Find a Physical Campus Visit the Virtual Campus

Difference between revisions of "Imaginary unit"

(→‎Olympiad: problem)
(making the article look nice)
Line 25: Line 25:
  
 
===Olympiad===
 
===Olympiad===
−
*Let <math>A\in\mathcal M_2(R)</math> and <math>P\in R[X]</math> with no real roots. If <math>\det(P(A)) = 0</math> , show that <math>P(A) = O_2</math>. (<url>viewtopic.php?t=78260 Source</url>)
+
*Let <math>A\in\mathcal M_2(R)</math> and <math>P\in R[X]</math> with no real roots. If <math>\det(P(A)) = 0</math> , show that <math>P(A) = O_2</math>. <url>viewtopic.php?t=78260 (Source)</url>
  
 
== See also ==
 
== See also ==

Revision as of 16:42, 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:

  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

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 $P(A) = O_2$. <url>viewtopic.php?t=78260 (Source)</url>

See also

Retrieved from "https://artofproblemsolving.com/wiki/index.php?title=Imaginary_unit&oldid=18973"