Aus Online Mathematik Brückenkurs 2

The magnitude of the number \displaystyle \text{3}+\text{4}i is the number's distance to the origin in the complex number plane.

If we treat the line from the origin to \displaystyle \text{3}+\text{4}i as the hypotenuse in a right-angled triangle which has its sides parallel with the real and imaginary axes, then Pythagoras' theorem gives that the magnitude is

\displaystyle \left| \text{3}+\text{4}i \right|=\sqrt{3^{2}+4^{2}}=\sqrt{9+16}=\sqrt{25}=5

NOTE: In general, the magnitude of a complex number \displaystyle z=x+iy is equal to

\displaystyle \left| z \right|=\left| x+iy \right|=\sqrt{x^{2}+y^{2}}