Difference between revisions of "Imaginary unit"
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===Intermediate=== | ===Intermediate=== | ||
− | [[1984 AIME Problems/Problem 8]] | + | *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=== | ===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 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>