Theory

Theory 1

The complex numbers are sums of real and imaginary numbers. Every complex number can be written uniquely in ‘Cartesian’ form:

To add, subtract, scale, and multiply complex numbers, treat ‘’ like a constant.

Simplify the result using .

For example:

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Complex conjugate

Every complex number has a complex conjugate:

For example:

In general, .

Conjugates are useful mainly because they eliminate imaginary parts:

In general:

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Complex division

To divide complex numbers, use the conjugate to eliminate the imaginary part in the denominator.

For example, reciprocals:

More general fractions:

Multiplication preserves conjugation

For any :

Therefore, one can take products or conjugates in either order.