Difference between revisions of "Complex number"

Revision as of 11:03, 8 July 2006

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

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, but is much larger. Every complex number $z$ has a real part, denoted by $\Re(z)$ or $\mathrm{Re}(z)$ and an imaginary part denoted by $\Im(z)$ or $\mathrm{Im}(z_$ (Error compiling LaTeX. Unknown error_msg). 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 $x\ge 0$ for the domain of $f(x)=\sqrt{x}$ .

The letters $z$ and $\omega$ are usually used to denote complex numbers.

Operations

Addition
Subtraction
Multiplication
Division
Absolute value/Modulus/Magnitude (denoted by $|z|$ ). This is the distance from the origin to the complex number in the complex plane.

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

AIME
- 1984 #8
- 1985 #3
- 1988 #11
- 1989 #14
- 1990 #10
- 1992 #10
- 1994 #8
- 1994 #13
- 1995 #5
- 1996 #11
- 1997 #11
- 1997 #14
- 1998 #13
- 1999 #9
- 2000 Alternate #9
- 2002 #12
- 2004 #13
- 2005 Alternate #9

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 The '''complex numbers''' arise when we try to solve [[equation]]s such as <math> x^2 = -1 </math>.  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.)
-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>. 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, but is much larger. Every complex number has a '''real part''', denoted by <math>\Re</math>, or simply <math>\mathrm{Re}</math>, and an '''imaginary part''', denoted by <math>\Im</math>, or simply <math>\mathrm{Im}</math>. So, if <math>z\in \mathbb C</math>, we can write <math>z=\mathrm{Re}(z)+i\mathrm{Im}(z)</math>, where <math>i</math> is the [[imaginary unit]].
+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, but is much larger. Every complex number <math> z </math> has a '''real part''', denoted by <math>\Re(z)</math> or <math>\mathrm{Re}(z)</math> and an '''imaginary part''' denoted by <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>.  (''z'' and ''w'' are traditionally used in place of ''x'' and ''y'' as [[variable]]s 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 <math>x\ge 0</math> for the domain of <math>f(x)=\sqrt{x}</math>.
@@ Line 60: / Line 60: @@
 * [[Trigonometry]]
 * [[Real numbers]]
+[[Media:Example.ogg]]

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