Modulus of a Complex Number

Just as the absolute value of a real number measures its distance from the origin along the real number line, the modulus of a complex number measures its distance from the origin in the complex plane.

Definition. If $z=x+iy$, the modulus of $z$, denoted $|z|$, is defined by

Geometrically, this is simply the length of the hypotenuse of the right triangle with legs $|x|$ and $|y|$, formed by the radial line from the origin to $z$.

For instance: $|i|=\sqrt{0^2+1^2}=1$; $|-12|=\sqrt{(-12)^2+0^2}=12$; $|12-5i|=\sqrt{12^2+(-5)^2}=\sqrt{169}=13$.

Note. If $z=x+iy$, then $\overline z=x-iy$, so

$z\overline z=(x+iy)(x-iy)=x^2+y^2=|z|^2.$ …