Complex number call number of a type of z =x + i * y where x and y – real numbers, and i = imaginary unit (i.e. number which square is equal to-1). To define a concept of an argument of complex number, it is necessary to consider complex number on the complex plane in the polar system of coordinates.
Instruction
1. The plane on which represent complex numbers is called complex. On this plane the horizontal axis is occupied by real numbers (x), and a vertical axis – imaginary numbers (y). On such plane the number is set by two coordinates of z = {x, y }. In polar to system of coordinates coordinates of a point are the module and an argument. The module call distance |z| from a point prior to the beginning of coordinates. An argument call a corner ϕ between the vector connecting a point and the beginning of coordinates and a horizontal axis of a system of coordinates (see the drawing).
2. From the drawing it is visible that the module of complex number z = x + i * y is on Pythagorean theorem: |z| = √ (x^2 + y^2). Further the argument of number z is as an acute angle of a triangle – through values of the trigonometrical functions sin, cos, tg: sin ϕ = y / √ (x^2 + y^2), cos ϕ = x / √ (x^2 + y^2), tg ϕ = y/x.
3. For example, let number z = 5 * is given (1 + √3 * i). First of all allocate material and imaginary parts: z = 5 +5 * √3 * i. It turns out that material Part x = 5, and imaginary Part y = 5 * √3. Calculate the number module: |z| = √ (25 + 75) = √100 =10. Further find a sine of the angle ϕ: sin ϕ = 5/10 = 1/2. From here the argument of number z turns out it is equal 30 °.
4. Example 2. Let number z = 5 * be given to i. According to the drawing it is visible that a corner ϕ = 90 °. Check this value on the formula given above. Write down coordinates of this number on the complex plane: z = {0, 5}. Module of number |z| = 5. tg tangent of angle ϕ = 5/5 = 1. From this it follows that ϕ = 90 °.
5. Example 3. Let it is necessary to find an argument of the sum of two complex numbers of z1 = 2 + 3 * i, z2 = 1 + 6 * i. By rules of addition you put these two complex numbers: z = z1 + z2 = (2 + 1) + (3 + 6) * i = 3 + 9 * i. Further on the scheme given above you count an argument: tg ϕ = 9/3 = 3.