4.2: Conjugates and Magnitudes

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For each complex number \(z = x + iy\), its complex conjugate is a complex number whose imaginary part has the sign flipped: \[z^* = x - i y.\] Conjugation obeys two important properties: \[\begin{align} (z_1 + z_2)^* &= z_1^* + z_2^* \\ (z_1 z_2)^* &= z_1^* z_2^*.\end{align}\]

Example \(\PageIndex{1}\)

Let us prove that \((z_1 z_2)^* = z_1^* z_2^*\). First, let \(z_1 = x_1 + i y_1\) and \(z_2 = x_2 + i y_2\). Then, \[\begin{align} (z_1 z_2)^* &= \left[(x_1+iy_1)(x_2+iy_2)\right]^* \\ &= \left[\left(x_1 x_2 - y_1 y_2\right) + i\left(x_1y_2+y_1x_2\right)\right]^* \\ &= \left(x_1 x_2 - y_1 y_2\right) - i\left(x_1y_2+y_1x_2\right) \\ &= \left(x_1 - i y_1\right)\left(x_2 - i y_2\right) \\ &= z_1^* z_2^* \end{align}\]

For a complex number \(z = x + i y\), the magnitude of the complex number is \[|z| = \sqrt{x^2 + y^2}.\] This is a non-negative real number. A complex number and its conjugate have the same magnitude: \(|z| = |z^*|\). Also, we can show that complex magnitudes have the property \[|z_1 z_2| = |z_1| \, |z_2|.\] This property is similar to the “absolute value” operation for real numbers, hence the similar notation.

As a corollary, taking a power of a complex number raises its magnitude by the same power: \[|z^n| = |z|^n \;\;\;\textrm{for}\;\;n \in \mathbb{Z}.\]