For any complex number \[ z = x + iy ,\]

\[ z \bar{z} = (x + iy)(x -iy) \]

\[ = x^{2} + ixy - ixy + y^{2} = x^{2} + y^{2} .\]

That is, any complex number times its complex conjugate equals the square of its real part plus the square of its imaginary part.

Because both the real and imaginary parts of any complex number are real:

\[ z \bar{z} = x^{2} + y^{2} \geq 0, \forall z \in \mathbb{C}.\]

We define the positive square root of \[ z \bar{z}: \]

\[ +\sqrt{z\bar{z}} : = |z| ,\]

to be the **Modulus **of the complex number \[ z .\]