Complex Modulus

Question

Thanks everyone that answered my last question! I had one more question for a different concept. If the modulus of a complex number $z = |z| = \sqrt{(a^2 + b^2)}$,where $a$ and $b$ are the real and imaginary constants of $z$, are you allowed to use the similar property for conjugates and say that the modulus of $(z + 1)$, or $|z + 1| = |z| + |1|$ or would it instead be $\sqrt{(a^2 + b^2 + 1^2)}$?

Answer 1

In general if $z=x+iy$, then $|z|=\sqrt{x^{2}+y^{2}}$ so that if $z+1=(x+1)+iy$ we have $|z|=\sqrt{(x+1)^{2}+y^{2}}$.

Answer 2

$|z+1|=\sqrt{(z+1)(z'+1)}=\sqrt{zz'+z+z'+1}=\sqrt{a^2+b^2+2a+1}$ if $z=a+bi$.

Alternatively, $|z+1|=\sqrt{(z+1)(z'+1)}=\sqrt{(a+1+bi)(a+1-bi)}=\sqrt{(a+1)^2+b^2}$.