Art of Problem Solving
AoPS Online
Math texts, online classes, and more
for students in grades 5-12.
Visit AoPS Online ‚
Books for Grades 5-12 Online Courses
Beast Academy
Engaging math books and online learning
for students ages 6-13.
Visit Beast Academy ‚
Books for Ages 6-13 Beast Academy Online
AoPS Academy
Small live classes for advanced math
and language arts learners in grades 2-12.
Visit AoPS Academy ‚
Find a Physical Campus Visit the Virtual Campus

Complex number

Revision as of 12:54, 23 June 2006 by Inscrutableroot (talk | contribs) (proofreading)

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 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 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


See also

Retrieved from "https://artofproblemsolving.com/wiki/index.php?title=Complex_number&oldid=4210"