Q.If $|z+1|=z+2(1+i)$, then find $z$.

For a complex number $z = a + ib$, two quantities describe it from a geometric angle, and both come up constantly.

The modulus of $z$, written $|z|$, is defined as $|z| = \sqrt{a^2 + b^2}$ -- always a non-negative real number, never a complex one. Geometrically, on the Argand plane, $|z|$ is exactly the distance from the origin to the point $(a, b)$. For example, if $z = 3 + 4i$, then $|z| = \sqrt{3^2 + 4^2} = \sqrt{25} = 5$.

The conjugate of $z$, written $\bar{z}$, is defined as $\bar{z} = a - ib$ -- simply flip the sign of the imaginary part. Geometrically, $\bar{z}$ is the reflection of $z$ across the real axis. For $z = 3 + 4i$, $\bar{z} = 3 - 4i$.

Why they're defined together: multiplying a complex number by its own conjugate always produces a real, non-negative number equal to the square of the modulus:

This single identity is the workhorse of the whole topic. It is exactly how division of complex numbers is carried out: to simplify $\dfrac{1}{a+ib}$ into standard $x+iy$ form, multiply top and bottom by the conjugate $a - ib$, since that turns the denominator into the real number $a^2+b^2$:

Key properties, each directly checkable from the definitions above:

• $\overline{\overline{z}} = z$ -- conjugating twice returns the original.
• $|\bar{z}| = |z|$ -- reflecting across the real axis doesn't change the distance from the origin.
• $|z_1 z_2| = |z_1||z_2|$ -- the modulus of a product is the product of the moduli. …