Modulus (Complex Number)

The modulus of a Complex Number $z$ is denoted $\vert z\vert$.

$$\vert x+iy\vert \equiv \sqrt{x^2+y^2}$$ (1)

$$\vert re^{i\phi}\vert = \vert r\vert.$$ (2)

Let $c_1\equiv Ae^{i\phi_1}$ and $c_2\equiv Be^{i\phi_2}$ be two Complex Numbers. Then

$$\left\vert{c_1\over c_2}\right\vert = \left\vert{Ae^{i\phi_1}\over Be^{i\phi_2}}\right\vert = {A\over B} \vert e^{i(\phi_1-\phi_2)}\vert = {A\over B}$$ (3)

$${\vert c_1\vert\over \vert c_2\vert} = {\vert Ae^{i\phi_1}\vert\over \vert Be^{i\phi_2}\vert} = {A\over B}{\vert e^{i\phi_1}\vert\over\vert e^{i\phi_2}\vert} = {A\over B},$$ (4)

so

$$\left\vert{c_1\over c_2}\right\vert={\vert c_1\vert\over \vert c_2\vert}.$$ (5)

Also,

$$\vert c_1c_2\vert = \vert(Ae^{i\phi_1})(Be^{i\phi_2})\vert = AB\vert e^{i(\phi_1+\phi_2)}\vert = AB$$ (6)

$$\vert c_1\vert\,\vert c_2\vert = \vert Ae^{i\phi_1}\vert\,\vert Be^{i\phi_2}\vert = AB\vert e^{i\phi_1}\vert\,\vert e^{i\phi_2}\vert=AB,$$ (7)

so

$$\vert c_1c_2\vert=\vert c_1\vert\,\vert c_2\vert$$ (8)

and, by extension,

$$\vert z^n\vert = \vert z\vert^n.$$ (9)

The only functions satisfying identities of the form

$$\vert f(x+iy)\vert=\vert f(x)+f(iy)\vert$$ (10)

are $f(z)=Az$, $f(z)=A\sin(bz)$, and $f(z)=A\sinh(bz)$ (Robinson 1957).

See also Absolute Square

References

Abramowitz, M. and Stegun, C. A. (Eds.). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, p. 16, 1972.

Robinson, R. M. ``A Curious Mathematical Identity.'' Amer. Math. Monthly 64, 83-85, 1957.