Complex number: Difference between revisions

Revision as of 21:31, 24 October 2006

Revision as of 03:34, 25 October 2006

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:::<math> = \left({ac + bd \over c^2 + d^2}\right) + i\left( {bc - ad \over c^2 + d^2} \right). </math>

=== Square roots ===

Each complex number has two square roots.

Consider

:<math>z=a+bi \ </math>

where

:<math>r=\sqrt{a^2+b^2}</math>

and

:<math> \phi = \arctan \left ( \frac b a \right )</math>.

The quadrant of <math>\phi \ </math> is determined by the signs of ''a'' and ''b''.

The square roots are

:<math>\pm\sqrt{\frac{r+a} 2 } \ \pm\ i \ \sqrt{\frac{r-a} 2}</math>

where the signs match if

:<math>0< \phi<\pi \ </math>,

but are different, if not.

===Matrix representation of complex numbers===