Complex Multiplication and Roots
- Author:
- Dr. Doug Davis, 3D
- Topic:
- Complex Numbers, Multiplication, Root
Complex Multiplication
Multiplication of complex numbers can be obtained as multiplication of binomials.
If an is included on the second terms , and the result is
since .
A complex number can be represented as a magnitude and angle in the complex plane.
The relations and taking care of the quadrant.
Taking and calculating the product
In other words to multiply in the complex plane multiply the magnitudes and add the angles.
In the following applet you can move the complex points 1 and 2 and the product is shown.
Complex Roots of Real Numbers
Raising a number to a power multiplies the angle by the power. Therefor an nth root will have n complex values. Each one multiplied n times would move completely around the circle. That is adding to a root when raised to the power will result in an extra transversal of a full circle ending at the original spot.
The following applet shows the roots of a number between -2 and 2.