Real and Imaginary Parts and Exercise 1.3

Real and Imaginary Parts

Example: Find the real and imaginary parts of the complex numbers:

$(2-3i{)}^2\mathrm{b}\mathrm{.}(1+2i{)}^{-2}\mathrm{c}\mathrm{.}\frac{(\sqrt{2}+i\sqrt{3}{)}^2}{1+i}\mathrm{d}\mathrm{.}\frac{(1-i{)}^2}{(1+i{)}^2}$$(2 - 3i)^{2}\qquad \mathrm{b.}(1 + 2i)^{-2}\qquad \mathrm{c.}\frac{(\sqrt{2} + i\sqrt{3})^{2}}{1 + i}\qquad \mathrm{d.}\frac{(1 - i)^{2}}{(1 + i)^{2}}$

​+i3

​)2​d.(1+i)2(1−i)2​(2−3i)2b.(1+2i)−2c.1+i(2

​+i3

​)2​d.(1+i)2(1−i)2​

Solution:

Solution of System of Equations with Complex Coefficients

In this section, we shall find solution of different equations in complex variables either with real or complex coefficients. Consider the following equation:

$pz + qw = r\dots (i)$

Where $p,q$ and $r$ are complex numbers (every real number is a complex number). The equation (i) is called a linear equation in two complex variables $z$ and $w$ .

$p_{1}z + q_{1}w = r_{1}$ $p_{2}z + q_{2}w = r_{2}$

These two equations together form a system of linear equations in two variables $z$ and $w$ . The linear equations in two variables are also called simultaneous linear equations.

Example: Solve:

Solution: Given equations are:

$z + w = 3i\dots \dots \dots \dots (i)$ $2z + 3w = 2\dots \dots (ii)$

$2z+2w=6i$ $\pm 2z\pm 3w=\pm 2$$\scriptstyle \pm 2z\pm 3w = \pm 2$ $- w = 6i - 2$ $w = 2 - 6i$ (Substitute in (i) to find z) $z + 2 - 6i = 3i$ $z = 9i - 2$