Difference between revisions of "Complex number"

Revision as of 22:14, 7 July 2006

We come about the idea of complex numbers when we trying to solve equations such as $x^2 = -1$ . We know that it's absurd for the square of a real number to be negative so this equation has no solutions in real numbers. However, if we define a number, $i$ , such that $i = \sqrt{-1}$ . Then we will have solutions to $x^2 = -1$ . It turns out that not only are we able to find the solutions of $x^2 = -1$ but we can now find all solutions to any polynomial using $i$ . (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 $i = \sqrt{-1}$ . 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 has a real part, denoted by $\Re$ , or simply $\mathrm{Re}$ , and an imaginary part, denoted by $\Im$ , or simply $\mathrm{Im}$ . So, if $z\in \mathbb C$ , we can write $z=\mathrm{Re}(z)+i\mathrm{Im}(z)$ , where $i$ is the imaginary unit.

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

@@ Line 1: / Line 1: @@
-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 wider. 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 come about the idea of '''complex numbers''' when we trying to solve equations such as <math> x^2 = -1 </math>.  We know that it's absurd for the square of a real number to be negative so this equation has no solutions in real numbers.  However, if we define a number, <math> i </math>, such that <math> i = \sqrt{-1} </math>.  Then we will have solutions to <math> x^2 = -1 </math>.  It turns out that not only are we able to find the solutions of <math> x^2 = -1 </math> but we can now find ''all'' solutions to ''any'' polynomial using <math> i </math>. (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]].
 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 55: / Line 57: @@
 == See also ==
+* [[Fundamental Theorem of Algebra]]
 * [[Trigonometry]]
 * [[Real numbers]]

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