Actions with complex numbers.

Complex numbers are numbers of the form $x + i y$ , where $x, y \in R$ , and $i$ is a number such that $i^{2} = - 1$ . The set of complex numbers is denoted by $C$ .

Operations with complex numbers.

Addition of complex numbers:

$(x_{1} + i y_{1}) + (x_{2} + i y_{2}) = (x_{1} + x_{2}) + i (y_{1} + y_{2}) .$

Multiplication of two complex numbers:

$(x_{1} + i y_{1}) (x_{2} + i y_{2}) = x_{1} x_{2} - y_{1} y_{2} + (x_{1} y_{2} + x_{2} y_{1}) i .$

Multiplication of a complex number by a real number:

$λ (x + i y) = λ x + i λ y .$

Division of complex numbers:

$\frac{x_{1} + i y_{1}}{x_{2} + i y_{2}} = \frac{(x_{1} + i y_{1}) (x_{2} - i y_{2})}{(x_{2} + i y_{2}) (x_{2} - i y_{2})} = \frac{x_{1} x_{2} + y_{1} y_{2} + i (y_{1} x_{2} - x_{1} y_{2})}{x_{2}^{2} + y_{2}^{2}} =$ $\frac{x_{1} x_{2} + y_{1} y_{2}}{x_{2}^{2} + y_{2}^{2}} + \frac{y_{1} x_{2} - x_{1} y_{2}}{x_{2}^{2} + y_{2}^{2}} i .$

The real numbers $x$ and $y$ of the complex number $z = x + i y$ are called the real and imaginary parts of the number $z$ , respectively, and are denoted by $R e z = x$ and $I m z = y$ .

Two complex numbers $z_{1} = x_{1} + i y_{1}$ and $z_{2} = x_{2} + i y_{2}$ are called equal if and only if $x_{1} = x_{2}$ and $y_{1} = y_{2}$ .

The expression $z = x + i y$ is called the algebraic form of the complex number $z$ .

The numbers $z_{1} = x + i y$ and $z_{2} = x - i y$ are called conjugate.

Examples:

Perform operations with complex numbers, presenting the result in algebraic form:

1.421. $(2 + 3 i) (3 - i) .$

Solution:

$(2 + 3 i) (3 - i) = 6 - 2 i + 9 i - 3 i^{2} = 6 + 7 i + 3 = 9 + 7 i .$

Answer: $9 + 7 i .$

1.424. $(2 i - i^{2})^{2} + (1 - 3 i)^{3} .$

Solution:

$(2 i - i^{2})^{2} + (1 - 3 i)^{3} = (2 i + 1)^{2} + 1 - 3 (3 i)^{2} + 3 (3 i) - (3 i)^{3} =$ $= 4 i^{2} + 4 i + 1 - 27 i^{2} + 9 i - 27 i^{3} = - 4 + 4 i + 1 + 27 - 9 i + 27 i = 24 + 22 i .$

Answer: $24 + 22 i .$

1.425. $\frac{2 - i}{1 + i} .$

Solution:

$\frac{2 - i}{1 + i} = \frac{(2 - i) (1 - i)}{(1 + i) (1 - i)} = \frac{2 - 2 i - i + i^{2}}{1 - i^{2}} = \frac{2 - 3 i - 1}{1 + 1} = \frac{1 - 3 i}{2} = \frac{1}{2} - \frac{3}{2} i .$

Answer: $\frac{1}{2} - \frac{3}{2} i .$

1.428. $\frac{(1 + i) (3 + i)}{3 - i} - \frac{(1 - i) (3 - i)}{3 + i} .$

Solution.

$\frac{(1 + i) (3 + i)}{3 - i} - \frac{(1 - i) (3 - i)}{3 + i} = \frac{(1 + i) (3 + i) (3 + i)}{(3 - i) (3 + i)} -$ $- \frac{(1 - i) (3 - i) (3 - i)}{(3 + i) (3 - i)} = \frac{9 + 15 i + 7 i^{2} + i^{3}}{9 - i^{2}} - \frac{9 - 15 i + 7 i^{2} - i^{3}}{9 - i^{2}} =$ $= \frac{9 + 15 i - 7 - i - 9 + 15 i + 7 - i}{10} = \frac{28}{10} i = \frac{14}{5} i .$

Answer: $\frac{14}{5} i .$

Find the real solutions of the following equation:

1. 430. $(1 + i) x + (- 2 + 5 i) y = - 4 + 17 i .$

Solution.

$(1 + i) x + (- 2 + 5 i) y = - 4 + 17 i \Rightarrow$

$x + x i - 2 y + 5 y i = - 4 + 17 i \Rightarrow$

$(x - 2 y) + (x + 5 y) i = - 4 + 17 i \Rightarrow$

${\begin{array}{lcl} x - 2 y = - 4 \\ x + 5 y = 17 \end{array} \Rightarrow {\begin{array}{lcl} x = 2 y - 4 \\ 2 y - 4 + 5 y = 17 \end{array} \Rightarrow {\begin{array}{lcl} x = 2 \\ y = 3 \end{array} .$