Methodical directions for solving problems and tasks

Problem 1. Show this number on the complex plane, and write it in algebraic, trigonometric and exponential forms.

Solution. Definition. The complex number z i s an ordered pair of real numbers (x, y) and write z=(x, y). The first number x is called real part of complex number z and denoted by x=Re z.

Second number y is called imaginary part of complex number z and denoted by y=Im z.

Each complex number corresponds to a point of the coordinate plane Oxy.

In the considered problem .

Definition. Two complex numbers that differ only in the sign of the imaginary part are called conjugate.

As is well known, except for a Cartesian coordinate system, the position of a point on the plane can be specified by polar coordinates .

Let are the polar coordinates of the point M, corresponding to the complex number z=(x, y).

Definition. Polar radius of point M, that is the number called the module of a complex number z=(x,y) and denoted by |z|:

Definition. The polar angle of the point M, that is the angle is called the argument of a complex number z=(x, y) and denoted by Arg z,

As is well known, the polar angle is not defined uniquely, and up to a term of the form .

The value of , satisfying the following condition is called the principal value of the argument and denoted by .

If satisfies the following condition , then

In the problem and

A complex number (0, 1) is called the imaginary unit and denoted by the symbol i.