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The complex conjugate for a given complex number is the number with the same real part but a negative imaginary part. In polars, this is the equivalent of having the radius remain the same but the argument becoming negative. For an example, the conjugate of

z=-4+3i=5e^{i\arctan\left(-\tfrac34\right)}=\left(5,\arctan\left(-\tfrac34\right)\right)

is

\bar{z}=-4-3i=5e^{i\arctan\left(\tfrac34\right)}=\left(5,\arctan\left(\tfrac34\right)\right)

Complex conjugates are useful for defining complex division and for finding roots of polynomials, as well as defining the inner product of complex vectors.

Properties

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