3CComplexDivision


 * Division of Complex Numbers **

In order to divide, we must first know about complex conjugates.

To obtain the ** complex conjugate ** of a number
 * reverse the sign on the imaginary part of the number.

The notation for the complex conjugate of z is z with a bar above it.

math . \qquad \text{Conjugate of }z = \bar{z} math

Complex Conjugate Definition

The **complex conjugate** of

math . \qquad z = x + yi math

is : math . \qquad \bar{z}=x-yi math


 * Example 1 **

Find the conjugate of .. z=3 + 2i,


 * Solution:**

math . \qquad \bar{z}=3-2i math

When complex conjugates are multiplied together, the result is a real number. {because of the difference of two squares rule}


 * Example 2 **

Simplify: .. (3 + 2i)(3 - 2i)

math \\ (3 + 2i)(3 - 2i) = 9 - 6i + 6i - 4i^2 \\. \\ . \qquad \qquad \qquad \; = 9 + 4 \\. \\ . \qquad \qquad \qquad \; = 13 math

Division

To divide two complex numbers: multiply the fraction by the complex conjugate of the denominator over itself (equivalent to 1)

Simplify math . \qquad \dfrac {3 + 5i}{4 - 2i} math
 * Example 3 **


 * Solution:**

math \\ . \qquad \quad \dfrac {3 + 5i}{4 - 2i} \qquad \qquad \{ \overline{4-2i} = 4+2i \} \\. \\ . \qquad= \dfrac{3 + 5i}{4 - 2i} \times \dfrac{4 + 2i}{4 + 2i} \\. \\ . \qquad = \dfrac{(3 + 5i)(4 + 2i)}{(4 - 2i)(4 + 2i)} \\. math math \\ . \qquad = \dfrac{12 + 6i + 20i + 10i^2}{16 + 8i - 8i - 4i^2} \\. \\ . \qquad = \dfrac{2 + 26i}{20} math

{The denominator is real!! Now separate fraction into real and imaginary parts}

math \\ . \qquad = \dfrac{2}{20} + \dfrac{26i}{20} \\. \\ . \qquad = \dfrac{1}{10} + \dfrac{13i}{10} math

Note: Compare this process to Rationalising the Denominator of a Surd, which is the process for dividing two surds.

Rationalising the Denominator of a Surd

math \\ . \qquad \quad \dfrac {3+\sqrt{5}}{4-\sqrt{2}} \\. \\ . \qquad = \dfrac {3+\sqrt{5}}{4-\sqrt{2}} \times \dfrac {4+\sqrt{2}}{4+\sqrt{2}} \\. \\ . \qquad = \dfrac {(3+\sqrt{5})(4+\sqrt{2})}{(4-\sqrt{2})(4+\sqrt{2})} math

Having multiplied by the surd conjugate over itself, now expand brackets and simplify

math . \qquad = \dfrac {12 + 3\sqrt{2} + 4\sqrt{5} + \sqrt{10}}{16 + 4\sqrt{2} - 4\sqrt{2} - \sqrt{4}} \\. math math . \qquad = \dfrac {12 + 3\sqrt{2} + 4\sqrt{5} + \sqrt{10}}{14} math

{Denominator is rational}

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