Difference between revisions of "Imaginary unit"
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− | 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 \ | + | 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>. |
+ | ==Trigonometric function cis== | ||
+ | {{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>. | ||
+ | |||
+ | ==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. | ||
+ | |||
+ | ==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== | ==Problems== | ||
=== Introductory === | === 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]]) | ||
+ | ===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]]) | ||
− | == | + | ===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> | |
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== See also == | == See also == | ||
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* [[Complex numbers]] | * [[Complex numbers]] | ||
* [[Geometry]] | * [[Geometry]] | ||
− | + | * [[Omega]] | |
[[Category:Constants]] | [[Category:Constants]] | ||
+ | [[Category:Complex numbers]] |
Latest revision as of 15:57, 5 September 2008
The imaginary unit, , 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
. Any complex number can be expressed as
for some real numbers
and
.
Contents
Trigonometric function cis
- Main article: cis
The trigonometric function is also defined as
or
.
Series
When is used in an exponential series, it repeats at every four terms:
This has many useful properties.
Use in factorization
is often very helpful in factorization. For example, consider the difference of squares:
. With
, it is possible to factor the otherwise-unfactorisable
into
.
Problems
Introductory
Intermediate
- The equation
has complex roots with argument
between
and
in the complex plane. Determine the degree measure of
. (Source)
Olympiad
- Let
and
with no real roots. If
, show that
. <url>viewtopic.php?t=78260 (Source)</url>