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"
m
 
(22 intermediate revisions by 6 users not shown)
Line 1: Line 1:
−
The '''imaginary unit''', <math>i=\sqrt{-1}</math>, is the fundamental component of all [[complex numbers]]. In fact, it is a complex number itself.
+
The '''imaginary unit''', <math>i=\sqrt{-1}</math>, 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 <math>1 \text{cis } \left(\frac{\pi}{2}\right)</math>. Any [[complex number]] can be expressed as <math>a+bi</math> for some real numbers <math>a</math> and <math>b</math>.
  
−
The imaginary unit shows up frequently in contest problems. The most common type of problem involving it are sums, i.e. problems such as "Find the sum of <math>i^1+i^2+\ldots+i^{2006}</math>."
+
==Trigonometric function cis==
−
Let's begin by computing powers of <math>i</math>.
+
{{main|cis}}
 +
The trigonometric function <math>\text{cis } x</math> is also defined as <math>e^{ix}</math> or <math>\cos x + i\sin x</math>.
  
−
<math>i^1=\sqrt{-1}</math>
+
==Series==
 +
When <math>i</math> is used in an exponential series, it repeats at every four terms:
 +
#<math>i^1=\sqrt{-1}</math>
 +
#<math>i^2=\sqrt{-1}\cdot\sqrt{-1}=-1</math>
 +
#<math>i^3=-1\cdot i=-i</math>
 +
#<math>i^4=-i\cdot i=-i^2=-(-1)=1</math>
 +
#<math>i^5=1\cdot i=i</math>
 +
This has many useful properties.
  
−
<math>i^2=\sqrt{-1}\cdot\sqrt{-1}=-1</math>
+
==Use in factorization==
 +
<math>i</math> is often very helpful in factorization. For example, consider the difference of squares: <math>(a+b)(a-b)=a^2-b^2</math>. With <math>i</math>, it is possible to factor the otherwise-unfactorisable <math>a^2+b^2</math> into <math>(a+bi)(a-bi)</math>.
 +
==Problems==
 +
=== Introductory ===
 +
*Find the sum of <math>i^1+i^2+\ldots+i^{2006}</math> ([[Imaginary unit/Introductory|Source]])
 +
*Find the product of <math>i^1 \times i^2 \times \cdots \times i^{2006}</math>. ([[Imaginary unit/Introductory|Source]])
  
−
<math>i^3=-1\cdoti=-i</math>
+
===Intermediate===
 +
*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]])
  
−
<math>i^4=-i\cdoti=-i^2=-(-1)=1</math>
+
===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>
  
−
<math>i^5=1\cdoti=i</math>
+
== See also ==
−
 
+
* [[Algebra]]
−
We can now stop because we have come back to our original term. This means that the sequence i, -1, -i, 1 repeats. Note that this sums to 0. That means that all sequences <math>i^1+i^2+\ldots+i^{4k}</math> have a sum of zero (k is a natural number). Since <math>2006=4\cdot501+2</math>, the original series sums to the first two terms of the powers of i which equals -1+i.
+
* [[Complex numbers]]
 +
* [[Geometry]]
 +
* [[Omega]]
 +
[[Category:Constants]]
 +
[[Category:Complex numbers]]

Latest revision as of 14:57, 5 September 2008

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=27796"