Why is the modulus of a complex number $a^2+b^2$? Why is the modulus not $\sqrt{a^2-b^2}$? Carrying out standard multiplication this would be the result-why is this not the case?  I know viewing the complex plane you can easily define the sum as being the distance to the points, but what meaning does $\sqrt{a^2-b^2}$ have? 

The geometric meaning of the modulus is the length of the vector from the origin to the point in the complex plane.  So it's just defined like the Euclidean distance to a point $(x, y)$, $\sqrt{x^2 + y^2}$.  It just happens that "y component" is the imaginary part of the complex number $a + i b$, so you use $\sqrt{a^2 + b^2}$.
