Division of Complex Numbers

Division of complex numbers is also a complex number.

In other words, the division of two complex numbers can be expressed in the standard form A + iB where A and B are real.

Division of a complex number z$_{1}$ = p + iq by z$_{2}$ = r + is ≠ 0 is defined as

$\frac{z_{1}}{z_{2}}$ = $\frac{pr + qs}{\sqrt{r^{2} + s^{2}}}$ + i$\frac{qr - ps}{\sqrt{r^{2} + s^{2}}}$

Proof:

Given z$_{1}$ = p + iq by z$_{2}$ = r + is ≠ 0

$\frac{z_{1}}{z_{2}}$ = z1 ∙ $\frac{1}{z_{2}}$ = z$_{1}$ ∙ z$_{2}$$^{-1}$ = (p + iq) . $\frac{r - is}{\sqrt{r^{2} + s^{2}}}$ =  $\frac{pr + qs}{\sqrt{r^{2} + s^{2}}}$ + i$\frac{qr - ps}{\sqrt{r^{2} + s^{2}}}$

Again,

$\frac{z_{1}}{z_{2}}$ = $\frac{p + iq}{r + is}$ = $\frac{p + iq}{r + is}$ × $\frac{r - is}{r - is}$ =  $\frac{(pr + qs) + i(qr - ps)}{\sqrt{r^{2} + s^{2}}}$ = A + iB where A = $\frac{pr + qs}{\sqrt{r^{2} + s^{2}}}$ and B = $\frac{qr - ps}{\sqrt{r^{2} + s^{2}}}$ are real.

Therefore, quotient of two complex numbers is a complex number.

For example, if z$_{1}$ = 2 + 3i and z$_{2}$ = 4 - 5i, then

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

= $\frac{(8 - 15) + (10 + 12)i}{16 + 25}$

= $\frac{-7 + 22i}{41}$

= $\frac{-7}{41}$ + $\frac{22}{41}$i

Solved example on division of two complex numbers:

Find the quotient when the complex number 5 + √2i divided by the complex number 1 - √2i.

Solution:

$\frac{5 + √2i}{1 - √2i}$

= $\frac{5 + √2i}{1 - √2i}$ × $\frac{1 + √2i}{1 + √2i}$

= $\frac{5 + 5√2i + √2i + 2i^{2}}{1^{2} – (√2i)^{2}}$

= $\frac{5 + 6√2i - 2}{1 - 2(-1)}$

= $\frac{3 + 6√2i}{3}$

= 1 + 2√2i

11 and 12 Grade Math 

From Division of Complex Numbers to HOME PAGE