1.5 Finding Real and Imaginary Parts of Complex Number of $(x + iy)^n$ and $\left(\dfrac{x_1 + iy_1}{x_2 + iy_2}\right)^n$, where $x_2 + iy_2 \neq 0$ and $n = \pm 1$ and $n = \pm 2$

1.5 Finding Real and Imaginary Parts of Complex Number of $(x + iy)^n$ and $\left(\dfrac{x_1 + iy_1}{x_2 + iy_2}\right)^n$, where $x_2 + iy_2 \neq 0$ and $n = \pm 1$ and $n = \pm 2$

$(x + iy)^n$ when $n = \pm 1$

Let $z = (x + iy)^n$ be a complex number.

(i) For $n = 1$

We have $z = (x + iy)^1 = x + iy$

$Re(z) = x, Im(z) = y$

(ii) For $n = -1$

We have $z = (x + iy)^{-1}$

$\therefore Re(z) = \dfrac{x}{x^2 + y^2}, Im(z) = \dfrac{-y}{x^2 + y^2}$

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Skilled Practice!
Find real and imaginary parts of $(x + iy)^n$ for $n = \pm 2$

Example 17

Find real and imaginary parts of $(4 + 3i)^{-1}$.

Solution Let $z = (4 + 3i)^{-1}$

$\therefore Re(z) = \dfrac{4}{25}, Im(z) = -\dfrac{3}{25}$

Example 18

Find real and imaginary parts of $z = (4 + 3i)^{-2}$.

Solution $z = (4 + 3i)^{-2}$

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? Do you know?
$Re(z) = \dfrac{z + \overline{z}}{2}$,
$Im(z) = \dfrac{z - \overline{z}}{2i}$

$\therefore Re(z) = \dfrac{7}{625}, Im(z) = -\dfrac{24}{625}$

Example 19

Find real and imaginary parts of $z = \left(\dfrac{4 + 3i}{3 + 2i}\right)^{-1}$.

Solution $z = \left(\dfrac{4 + 3i}{3 + 2i}\right)^{-1}$

$\therefore Re(z) = \dfrac{18}{25}, Im(z) = -\dfrac{1}{25}$

Example 20

Find real and imaginary parts of $z = \left(\dfrac{1 + i}{1 - i}\right)^{-2}$.

Solution $z = \left(\dfrac{1 + i}{1 - i}\right)^{-2}$

$\therefore Re(z) = -1, Im(z) = 0$