Name: Geometric View of Complex Number
Uploaded: 2019-07-19
Duration: 10 min 19 s
Description: Complex numbers extend the concept of the one-dimensional number line to the two-dimensional complex plane by using the horizontal axis. Watch the video to l...

Question

If

\frac{1 - {\bar{z}}_{1} z_{2}}{z_{1} - z_{2}}

is a unimodular complex number, prove that at least one of the two complex numbers

z_{1}, z_{2}

, must be unimodular.

Accepted Answer

Given, $\frac{1 - {\bar{z}}_{1} z_{2}}{z_{1} - z_{2}}$ is unimodular complex number.

We know that, a complex number is unimodular if $| z | = 1$ .

$|\frac{1 - {\bar{z}}_{1} z_{2}}{z_{1} - z_{2}}| = 1 \Rightarrow {|\frac{1 - {\bar{z}}_{1} z_{2}}{z_{1} - z_{2}}|}^{2} = 1$

$\Rightarrow (\frac{1 - {\bar{z}}_{1} z_{2}}{z_{1} - z_{2}}) (\bar{\frac{1 - {\bar{z}}_{1} z_{2}}{z_{1} - z_{2}}}) = 1 [∵ | z |^{2} = z \bar{z}]$

$\Rightarrow (\frac{1 - {\bar{z}}_{1} z_{2}}{z_{1} - z_{2}}) (\frac{1 - \bar{{\bar{z}}_{1} z_{2}}}{\bar{z_{1} - z_{2}}}) = 1 [∵ \bar{z_{1} - z_{2}} = \bar{z_{1}} - \bar{z_{2}}]$

$\Rightarrow \frac{1 - {\bar{z}}_{1} z_{2}}{z_{1} - z_{2}} \cdot \frac{1 - z_{1} {\bar{z}}_{2}}{{\bar{z}}_{1} - {\bar{z}}_{2}} = 1 [∵ \bar{z_{1} z_{2}} = \bar{z_{1}} . \bar{z_{2}}]$

$\Rightarrow (1 - {\bar{z}}_{1} z_{2}) (1 - z_{1} {\bar{z}}_{2}) = (z_{1} - z_{2}) ({\bar{z}}_{1} - {\bar{z}}_{2})$

$\Rightarrow 1 - z_{1} {\bar{z}}_{2} - {\bar{z}}_{1} z_{2} + z_{1} {\bar{z}}_{1} z_{2} {\bar{z}}_{2} = {\bar{z}}_{1} z_{1} - z_{1} {\bar{z}}_{2} - {\bar{z}}_{1} z_{2} + z_{2} {\bar{z}}_{2}$

$\Rightarrow 1 + z_{1} {\bar{z}}_{1} z_{2} {\bar{z}}_{2} = {\bar{z}}_{1} z_{1} + z_{2} {\bar{z}}_{2}$

$\Rightarrow 1 + {|z_{1}|}^{2} {|z_{2}|}^{2} = {|z_{1}|}^{2} + {|z_{2}|}^{2} [∵ | z |^{2} = z \bar{z}]$

$\Rightarrow 1 + {|z_{1}|}^{2} {|z_{2}|}^{2} - {|z_{1}|}^{2} - {|z_{2}|}^{2} = 0$

$\Rightarrow {|z_{1}|}^{2} ({|z_{2}|}^{2} - 1) - 1 ({|z_{2}|}^{2} - 1) = 0$

$\Rightarrow ({|z_{2}|}^{2} - 1) ({|z_{1}|}^{2} - 1) = 0$

$\Rightarrow {|z_{1}|}^{2} = 1$ or ${|z_{2}|}^{2} = 1$

$\Rightarrow |z_{1}| = 1$ or $|z_{2}| = 1$ . (Modulus is always positive)

Hence, proved.

If $\frac{1 - {\bar{z}}_{1} z_{2}}{z_{1} - z_{2}}$ is a unimodular complex number, prove that at least one of the two complex numbers $z_{1}, z_{2}$ , must be unimodular.

Important Questions on Complex Numbers

Learn and Prepare for any exam you want !

If 1-z¯1z2z1-z2 is a unimodular complex number, prove that at least one of the two complex numbers z1,z2, must be unimodular.

Important Questions on Complex Numbers

Learn and Prepare for any exam you want !

If $\frac{1 - {\bar{z}}_{1} z_{2}}{z_{1} - z_{2}}$ is a unimodular complex number, prove that at least one of the two complex numbers $z_{1}, z_{2}$ , must be unimodular.