Difference between revisions of "Imaginary unit"
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=== Introductory === | === Introductory === | ||
*Find the sum of <math>i^1+i^2+\ldots+i^{2006}</math> ([[Imaginary unit/Introductory|Source]]) | *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=== | ===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]]) | ||
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===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> | ||
== See also == | == See also == | ||
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* [[Omega]] | * [[Omega]] | ||
[[Category:Constants]] | [[Category:Constants]] | ||
+ | [[Category:Complex numbers]] |
Latest revision as of 14:57, 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>