Difference between revisions of "Complex number"

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== Problems ==
 
== Problems ==
*AIME
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===Intermediate===
** [http://www.artofproblemsolving.com/Forum/viewtopic.php?p=392620#p392620 1984 #8]
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*[[1984 AIME Problems/Problem 8|1984 AIME Problem 8]]
** [http://www.artofproblemsolving.com/Forum/viewtopic.php?p=421338#p421338 1985 #3]
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*[[1985 AIME Problems/Problem 3|1985 AIME Problem 3]]
** [http://www.artofproblemsolving.com/Forum/viewtopic.php?p=458040#p458040 1988 #11]
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*[[1988 AIME Problems/Problem 11|1988 AIME Problem 11]]
** [http://www.artofproblemsolving.com/Forum/viewtopic.php?p=436603#p436603  1989 #14]
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*[[1989 AIME Problems/Problem 14|1989 AIME Problem 14]]
** [http://www.artofproblemsolving.com/Forum/viewtopic.php?p=459508#p459508 1990 #10]
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*[[1990 AIME Problems/Problem 10|1990 AIME Problem 10]]
** [http://www.artofproblemsolving.com/Forum/viewtopic.php?p=430620#p430620 1992 #10]
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*[[1992 AIME Problems/Problem 10|1992 AIME Problem 10]]
** [http://www.artofproblemsolving.com/Forum/viewtopic.php?p=53847#p53847 1994 #8]
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*[[1994 AIME Problems/Problem 8|1994 AIME Problem 8]]
** [http://www.artofproblemsolving.com/Forum/viewtopic.php?p=394743#p394743 1994 #13]
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*[[1994 AIME Problems/Problem 13|1994 AIME Problem 13]]
** [http://www.artofproblemsolving.com/Forum/viewtopic.php?p=394478#p394478 1995 #5]
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*[[1995 AIME Problems/Problem 5|1995 AIME Problem 5]]
** [http://www.artofproblemsolving.com/Forum/viewtopic.php?p=394249#p394249 1996 #11]
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*[[1996 AIME Problems/Problem 11|1996 AIME Problem 11]]
** [http://www.artofproblemsolving.com/Forum/viewtopic.php?p=393654#p393654 1997 #11]
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*[[1997 AIME Problems/Problem 11|1997 AIME Problem 11]]
** [http://www.artofproblemsolving.com/Forum/viewtopic.php?p=393661#p393661 1997 #14]
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*[[1997 AIME Problems/Problem 14|1997 AIME Problem 14]]
** [http://www.artofproblemsolving.com/Forum/viewtopic.php?p=392484#p392484 1998 #13]
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*[[1998 AIME Problems/Problem 13|1998 AIME Problem 13]]
** [http://www.artofproblemsolving.com/Forum/viewtopic.php?p=392227#p392227 1999 #9]
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*[[1999 AIME Problems/Problem 9|1999 AIME Problem 9]]
** [http://www.artofproblemsolving.com/Forum/viewtopic.php?p=385894#p385894 2000 Alternate #9]
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*[[2000 AIME II Problems/Problem 9|2000 AIME II Problem 9]]
** [http://www.artofproblemsolving.com/Forum/viewtopic.php?p=378395#p378395 2002 #12]
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*[[2002 AIME I Problems/Problem 12|2002 AIME I Problem 12]]
** [http://www.artofproblemsolving.com/Forum/viewtopic.php?p=378129#p378129 2004 #13]
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*[[2004 AIME I  Problems/Problem 13|2004 AIME I Problem 13]]
** [http://www.artofproblemsolving.com/Forum/viewtopic.php?p=368277#p368277 2005 Alternate #9]
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*[[2005 AIME II Problems/Problem 9|2005 AIME II Problem 9]]
 
 
  
 
== See also ==
 
== See also ==

Revision as of 19:17, 25 November 2007

The complex numbers arise when we try to solve equations such as $x^2 = -1$.

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, $i$, such that $i = \sqrt{-1}$. If we add this new number to the reals, we will have solutions to $x^2 = -1$. It turns out that in the system that results from this addition, we are not only able to find the solutions of $x^2 = -1$ 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 $a + bi$ where $a,b\in \mathbb{R}$ and $i = \sqrt{-1}$ is the imaginary unit. The set of complex numbers is denoted by $\mathbb{C}$. The set of complex numbers contains the set $\mathbb{R}$ of the real numbers, since $a = a + 0i$, but it is much larger.

Parts

Every complex number $z$ has a real part denoted $\Re(z)$ or $\mathrm{Re}(z)$ and an imaginary part denoted $\Im(z)$ or $\mathrm{Im}(z)$. Note that the imaginary part of a complex number is real: for example, $\Im(3 + 4i) = 4$. So, if $z\in \mathbb C$, we can write $z=\mathrm{Re}(z)+i\mathrm{Im}(z)$. ($z$ and $w$ are traditionally used in place of $x$ and $y$ as variables when dealing with complex numbers, while $x$ and $y$ (and frequently also $a$ and $b$) 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 $x\ge 0$ from the domain of the function $f(x)=\sqrt{x}$ (although some additional considerations are necessary).

Operations

Simple Example

If $z=a+bi$ and $w = c + di$,

  • $\mathrm{Re}(z)=a$,$\mathrm{Im}(z)=b$
  • $|z|=\sqrt{a^2+b^2}$
  • $\mathrm{Re}(w)=c$,$\mathrm{Im}(w)=d$
  • $|w|=\sqrt{c^2+d^2}$
  • $z+w=(a+c)+(b+d)i$
  • $z-w=(a-c)+(b-d)i$

Topics

Problems

Intermediate

See also