# The Geometry of Complex Number and Its Multiplicative Inverse

I am confused that how a complex number and its multiplicative inverse are related geometrically. For example, the complex conjugate is found by reflecting z across the real axis. Please explain if there is such relationship between a complex number and its multiplicative inverse. It's a problem given in a book authored by Dennis G. Zill.

• Invert in unit circle, then reflect in real axis. – Lord Shark the Unknown Jun 10 '18 at 18:16
• Please explain properly and clearly, I don't understand. – Waqar Ali Shah Jun 11 '18 at 7:02

You can have a simple geometric representation using the geometric representation of the complex conjugate and the fact that $$\frac{1}{z}=\frac{\bar z}{|z|^2}$$

A complex number $a + bi$ when thought of as a point $(a,b)$ existing in a 2-D plan will have a magnitude $r = \sqrt{a^2 + b^2}$ which is it's distance from the origin, and an incidental angle $\theta$ measuring the angle of the point $(a,b)$ to the origin to the $x$ axis.

As $a$ is the real/$x$ value and $b$ is the imaginary/$y$ value and $r = \sqrt{a^2 + b^2}$ is the radius of a circle around the origin containing $a$ and $b$, we know that $a = r*\cos \theta$ and $b = r*\sin \theta$.

And so if $c + di$ has magnitude $s$ and angle $\eta$ then

$(c + di)(a + bi) =$

$(s*\cos \eta + i s*\sin \eta)(r*\cos \theta + i r*\sin \theta)=$

$rs[(\cos \theta\cos\eta - \sin\theta\sin\eta)+ i(\sin\eta\cos \theta + \sin \theta \cos \eta)]$

=rs(cos (\theta + eta) + i \sin(\theta + eta))$. This is a rather fascinating result that$(a+bi)(c + di)$is the result of multiplying their sizes together and adding their angles together. Is if$z * \frac 1z = 1= 1 + 0i$(which has size$1$and incidental angle$0^\circ$) and$z$has size$r$and incidental angle$\theta$, then$\frac 1z$will have size$s= \frac 1r$because we need$r*s = 1$, and$\frac 1z$will have an incidental angle of$\eta = -\theta$because we need$\theta + \eta=0$. So that's that. Geometrically,$\frac 1z$is the complex number that is the inverse of size, and the negative of angle to$z$. It's actually a fairly simple geometric construction. Define point$A$in the complex plane as your complex number,$O$as zero and$E$as$1+0i$. ($E$from German "Einheit" for "unity"). Construct the following two rays:$OB$such that$\angle EOB$is congruent with$\angle EOA$but on the opposite side of line$OE$.$EC$such that$\angle OEC$is congruent with$\angle OAE$and on the same side of line$OE$as ray$OB$rendered above. Rays$OB$and$EC$intersect at point$Z$such that$\triangle OZE$is similar to$\triangle OEA$. This similarity plus the polar form for a complex inverse guarantees that$Z=A^{-1}\$.