Art of Problem Solving

AoPS Online

Difference between revisions of "Complex number"

Revision as of 09:32, 23 June 2006

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 wider. Every complex numbers has a real part, denoted by $\Re$ or simply $\mathrm{Re}$ , and a 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.

Contents

1 Operations
2 Simple Example
3 Topics
4 Problems
- 4.1 See also

Operations

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

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

See also

Retrieved from "https://artofproblemsolving.com/wiki/index.php?title=Complex_number&oldid=4175"

© 2024 AoPS Incorporated