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Complex number

Revision as of 12:58, 22 June 2006 by Fedja (talk | contribs)

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, the real part, denoted by $\Re$ or simply $\mathrm{Re}$, and the imaginary part, denoted by $\Im$ or simply $\mathrm{Im}$, so, if $z\in \mathbb C$, we can write it as $z=\Re z+i\Im z$ where $i=\sqrt{-1}$.

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 $\omega=c+di$,

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

Topics

See also

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