Instruction

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The plane, which is a complex

**number**called complex. On this plane the horizontal axis is the real**number**(x) and the vertical axis is the imaginary**number**(y). On this plane number is defined by two coordinates z = {x, y}. In polar coordinates the coordinates of the point are the module and argument. The module is called the distance |z| from the point to the origin. The argument is called the angle ϕ between the vector connecting the point and the origin and horizontal axis of the coordinate system (see figure).2

The figure shows that the modulus of the complex

cos ϕ = x / √ (x^2 + y^2),

tg ϕ = y / x.

**number**z = x + i * y is the Pythagorean theorem: |z| = √ (x^2 + y^2). Further, the argument**of the number**z is the acute angle of the triangle using the TRIG functions sin, cos, tg:sin ϕ = y / √ (x^2 + y^2),cos ϕ = x / √ (x^2 + y^2),

tg ϕ = y / x.

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For example, suppose that the number z = 5 * (1 + √3 * i). First, select a real and imaginary part: z = 5 +5 * √3 * i. It turns out that the real part of x = 5, and the imaginary part y = 5 * √3. Calculate the module

**number**: |z| = √(25 + 75) = √100 =10. Next, find the sine of the angle ϕ: sin ϕ = 5 / 10 = 1 / 2. This leads to the argument**of the number**z equal to 30°.4

Example 2. Let the number z = 5 * i. The drawing shows that the angle ϕ = 90°. Check this value in the formula given above. Record the coordinates of the given

**number**in the complex plane: z = {0, 5}. Module**number**|z| = 5. The tangent of the angle tg j = 5 / 5 = 1. It follows that ϕ = 90°.5

Example 3. Suppose you want to find the argument of the sum of two complex numbers z1 = 2 + 3 * i, z2 = 1 + 6 * i. According to the rules of addition put these two complex

**numbers**: z = z1 + z2 = (2 + 1) + (3 + 6) * i = 3 + 9 * i. Then as above counting argument: tg ϕ = 9 / 3 = 3.Note

If the number z = 0, the value of the argument to undefined.

Useful advice

The value of the argument of a complex number is defined with precision to 2 * π * k, where k is any integer. The value of the argument ϕ such that –π < ϕ ≤ π is called the principal value of argument.