Complex conjugates

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Level: A Level Maths

Complex conjugates are pairs of numbers which have the same real part, and where the imaginary part differs only in sign. For example and are conjugates.

Notation and terms

For any complex number :

  • is the complex conjugate
  • is the complex conjugate
  • and are called a complex conjugate pair

For a complex number , the complex conjugate is written as .

Argand diagram

In the Argand diagram below:

Complex-conjugate.png

All conjugate pairs are symmetrical about the x (real) axis.

Special properties

The sum of any complex number , and its complex conjugate is always real. This is because the two imaginary parts cancel out:

The product of any complex number , and its complex conjugate is always real:

The two terms in cancel out, and since is -1, the final result is:

See also