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The '''complex numbers''' arise when we try to solve [[equation]]s such as <math> x^2 = -1 </math>.  

==Derivation==
We know (from the [[Trivial Inequality]]) that the square of a [[real number]] cannot be [[negative]], so this equation has no solutions in the real numbers.  However, it is possible to define a number, <math> i </math>, such that <math> i = \sqrt{-1} </math>.  If we add this new number to the reals, we will have solutions to <math> x^2 = -1 </math>.  It turns out that in the system that results from this addition, we are not only able to find the solutions of <math> x^2 = -1 </math> but we can now find ''all'' solutions to ''every'' polynomial.  (See the [[Fundamental Theorem of Algebra]] for more details.)

==Formal Definition==
We are now ready for a more formal definition.  A complex number is a number of the form <math> a + bi </math> where <math> a,b\in \mathbb{R} </math> and <math> i = \sqrt{-1} </math> is the [[imaginary unit]]. The set of complex numbers is denoted by <math>\mathbb{C}</math>.  The set of complex numbers contains the set <math>\mathbb{R}</math> of the [[real number]]s, since <math>a = a + 0i</math>, but it is much larger. 

==Parts==
Every complex number <math> z </math> has a ''[[real part]]'' denoted <math>\Re(z)</math> or <math>\mathrm{Re}(z)</math> and an ''[[imaginary part]]'' denoted <math> \Im(z)</math> or <math> \mathrm{Im}(z)</math>.  Note that the imaginary part of a complex number is real: for example, <math>\Im(3 + 4i) = 4</math>.  So, if <math>z\in \mathbb C</math>, we can write <math>z=\mathrm{Re}(z)+i\mathrm{Im}(z)</math>.  (<math>z</math> and <math>w</math> are traditionally used in place of <math>x</math> and <math>y</math> as [[variable]]s when dealing with complex numbers, while <math>x</math> and <math>y</math> (and frequently also <math>a</math> and <math>b</math>) are used to represent real values such as the real and imaginary parts of complex numbers.  This [[mathematical convention]] is often broken when it is inconvenient, so be sure that you know what set variables are taken from when dealing with the complex numbers.)

As you can see, complex numbers enable us to remove the restriction of <math>x\ge 0</math> from the [[domain]] of the [[function]] <math>f(x)=\sqrt{x}</math> (although some additional considerations are necessary).
== Operations ==

* Addition
* Subtraction
* Multiplication
* Division
* [[Absolute value]]/Modulus/Magnitude (denoted by <math>|z|</math>). This is the distance from the origin to the complex number in the [[complex plane]].
* [[Complex conjugate | Conjugation]]
* The [[argument]] function <math>\arg</math>

== Simple Example ==

If <math>z=a+bi</math> and <math>w = c + di</math>,

* <math>\mathrm{Re}(z)=a</math>,<math>\mathrm{Im}(z)=b</math>
* <math>|z|=\sqrt{a^2+b^2}</math>
* <math>\mathrm{Re}(w)=c</math>,<math>\mathrm{Im}(w)=d</math>
* <math>|w|=\sqrt{c^2+d^2}</math>
* <math>z+w=(a+c)+(b+d)i</math>
* <math>z-w=(a-c)+(b-d)i</math>

== Topics ==

* [[Complex plane]]
* [[De Moivre's Theorem]]
* [[Exponential form]]
* [[Roots of unity]]
* [[Geometry with complex numbers]]

== Problems ==
===Introductory===
*[[2007 AMC 12A Problems/Problem 18]]

===Intermediate===
*[[1984 AIME Problems/Problem 8|1984 AIME Problem 8]]
*[[1985 AIME Problems/Problem 3|1985 AIME Problem 3]]
*[[1988 AIME Problems/Problem 11|1988 AIME Problem 11]]
*[[1989 AIME Problems/Problem 14|1989 AIME Problem 14]]
*[[1990 AIME Problems/Problem 10|1990 AIME Problem 10]]
*[[1992 AIME Problems/Problem 10|1992 AIME Problem 10]]
*[[1994 AIME Problems/Problem 8|1994 AIME Problem 8]]
*[[1994 AIME Problems/Problem 13|1994 AIME Problem 13]]
*[[1995 AIME Problems/Problem 5|1995 AIME Problem 5]]
*[[1996 AIME Problems/Problem 11|1996 AIME Problem 11]]
*[[1997 AIME Problems/Problem 11|1997 AIME Problem 11]]
*[[1997 AIME Problems/Problem 14|1997 AIME Problem 14]]
*[[1998 AIME Problems/Problem 13|1998 AIME Problem 13]]
*[[1999 AIME Problems/Problem 9|1999 AIME Problem 9]]
*[[2000 AIME II Problems/Problem 9|2000 AIME II Problem 9]]
*[[2002 AIME I Problems/Problem 12|2002 AIME I Problem 12]]
*[[2004 AIME I  Problems/Problem 13|2004 AIME I Problem 13]]
*[[2005 AIME II Problems/Problem 9|2005 AIME II Problem 9]]
*[[2009 AIME I  Problems/Problem 2|2009 AIME I Problem 2]]

===Olympiad===

== See also ==
* [[Fundamental Theorem of Algebra]]
* [[Trigonometry]]
* [[Real numbers]]
* [[Imaginary unit]]

[[Category:Definition]]
[[Category:Complex numbers]]