The Modulus and the Conjugate of a Complex Number

The modulus of a complex number gives the distance of the complex number from the origin in the Argand plane, whereas the conjugate of a complex number gives the reflection of the complex number about the real axis in the Argand plane.

Conjugate of a Complex Number

Fig 1 - Conjugate of a complex number

The representation of a complex number $z=x+iy$ and its conjugate ${\overline {z}}=x-iy$ in the Argand plane are, respectively , the points $P(x,y)$ and $Q(x,-y)$

Geometrically , the point $(x,-y)$ is the mirror image of the point $(x,y)$ on the real axis (Fig 1) .Here $XX$ is the real axis.

Example

Find the conjugate of ${\frac {(3-2i)(2+3i)}{(1+2i)(2-i)}}$

${\frac {(3-2i)(2+3i)}{(1+2i)(2-i)}}$

$={\frac {(6+9i-4i-6i^{2})}{(2-i+4i-2i^{2})}}$ $={\frac {(6+5i-6(-1))}{(2+3i-2(-1))}}$

$={\frac {(6+5i+6)}{(2+3i+2)}}$$={\frac {(12+5i)}{(4+3i)}}$ $={\frac {(12+5i)}{(4+3i)}}\times {\frac {(4-3i)}{(4-3i)}}$

$={\frac {(48-36i+20i-15i^{2})}{(16-9i^{2})}}$$={\frac {(48-16i-15(-1))}{(16-9(-1))}}$$={\frac {63-16i}{25}}$$={\frac {63}{25}}-{\frac {16i}{25}}$

Answer: The conjugate of ${\frac {(3-2i)(2+3i)}{(1+2i)(2-i)}}$$={\frac {63}{25}}+{\frac {16i}{25}}$

Properties of Conjugate of a Complex Number

• ${\overline {z_{1}z_{2}}}={\overline {z_{1}}}\ {\overline {z_{2}}}$
• ${\overline {z_{1}\pm z_{2}}}={\overline {z_{1}}}\pm {\overline {z_{2}}}$
• ${\overline {\left[{\frac {z_{1}}{z_{2}}}\right]}}={\frac {\overline {z_{1}}}{\overline {z_{2}}}}$provided $z_{2}\neq 0$

Modulus of a Complex Number

Fig 2 - Modulus of a complex number

In the Argand plane, the modulus of the complex number $z=x+iy$ is

$\left\vert z\right\vert ={\sqrt {x^{2}+y^{2}}}$ is the distance between the point $P(x,y)$ representing the complex number $z$ and $O(0,0)$ representing the origin (Fig 2).

Example

Find the modulus of the complex number ${\frac {3-2i}{2i}}$

${\frac {3-2i}{2i}}$

$={\frac {3}{2i}}-{\frac {2i}{2i}}$$={\frac {3}{2i}}-1$$={\frac {3}{2i}}\times {\frac {i}{i}}-1$

$={\frac {3i}{2i^{2}}}-1$$={\frac {3i}{2(-1)}}-1$$=-1+\left[-{\frac {3i}{2}}\right]$

$\left\vert z\right\vert ={\sqrt {x^{2}+y^{2}}}$

$={\sqrt {(-1)^{2}+\left[-{\frac {3}{2}}\right]^{2}}}$$={\sqrt {1+{\frac {9}{4}}}}$

$={\sqrt {\frac {4+9}{4}}}$ $={\sqrt {\frac {13}{4}}}$

$\left\vert z\right\vert ={\frac {\sqrt {13}}{2}}$

Multiplicative inverse of a Complex Number

Multiplicative inverse of the non-zero complex number $z=x+iy$ is given by

$z^{-1}={\frac {1}{x+iy}}$

Properties of Modulus of a Complex Number

• $\left\vert z_{1}z_{2}\right\vert =\left\vert z_{1}\right\vert \left\vert z_{2}\right\vert$
• $\left\vert {\frac {z_{1}}{z_{2}}}\right\vert ={\frac {\left\vert z_{1}\right\vert }{\left\vert z_{2}\right\vert }}$ provided $\left\vert z_{2}\right\vert \neq 0$