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
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