The absolute value for a complex number is like that of a real number in that it tells us the distance of that number to the origin. For a complex number we have to square the real part and the imaginary part, add them then take the square root of the whole thing.
Sometimes the signs and the [tex]i[/tex] confuse people. The magnitude is always going to be at least as big as the real part and at least as big as the imaginary part; there aren't any negative signs or subtraction involved.
So the magnitude of [tex]-4 - \sqrt{2} i[/tex] which we write [tex]|-4 - \sqrt{2} i|[/tex] is the same of the magnitude of some other complex numbers:
[tex]|-4 - \sqrt{2} i| = |4 - \sqrt{2} i| = |4 + \sqrt{2} i|=|\sqrt{2} + 4i|[/tex]
In other words for the magnitude we don't have to stress about the signs or which is the real part and which is the imaginary part. We just have to square each, add and take the square root:
[tex]|-4 - \sqrt{2} i| = \sqrt{4^2 + (\sqrt{2})^2} = \sqrt{16 +2} = \sqrt{18}=\sqrt{9 \times 2} = 3 \sqrt{2}[/tex]
