Complex conjugate
The complex conjugate: z=a+bi⟹z∗=a−bi, its properties, and using it to simplify fractions with complex denominators.
Complex conjugates
The conjugate of a complex number z is the complex number with the same real component and the opposite imaginary component:
Since the real components of z and z∗ are the same, and the imaginary components are opposite, on the complex plane z∗ is the reflection of z in the x-axis.
Properties of the complex conjugate
The following properties hold for complex conjugates:
Fractions of complex numbers
Fractions with complex denominator can be made real using a process analogous to rationalizing the denominator. For a fraction with a complex denominator c+di, we multiply both the numerator and the denominator by the conjugate c−di to get the fraction in a more workable form:
This allows us to split z into its real and imaginary components.
Example
Nice work completing Complex conjugate, here's a quick recap of what we covered:
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