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

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