The formula for calculating the modulus of a complex number. Trigonometric notation

Which represents a given complex number $ z \u003d a + bi $, is called the modulus of this complex number.

The modulus of a given complex number is calculated using the following formula:

Example 1

Calculate the modulus of the given complex numbers $ z_ (1) \u003d 13, \\, \\, z_ (2) \u003d 4i, \\, \\, \\, z_ (3) \u003d 4 + 3i $.

The modulus of the complex number $ z \u003d a + bi $ is calculated by the formula: $ r \u003d \\ sqrt (a ^ (2) + b ^ (2)) $.

For the original complex number $ z_ (1) \u003d 13 $ we get $ r_ (1) \u003d | z_ (1) | \u003d | 13 + 0i | \u003d \\ sqrt (13 ^ (2) + 0 ^ (2)) \u003d \\ sqrt (169) \u003d 13 $

For the original complex number $ \\, z_ (2) \u003d 4i $, we get $ r_ (2) \u003d | z_ (2) | \u003d | 0 + 4i | \u003d \\ sqrt (0 ^ (2) + 4 ^ (2)) \u003d \\ sqrt (16) \u003d 4 $

For the original complex number $ \\, z_ (3) \u003d 4 + 3i $ we get $ r_ (3) \u003d | z_ (3) | \u003d | 4 + 3i | \u003d \\ sqrt (4 ^ (2) + 3 ^ (2) ) \u003d \\ sqrt (16 + 9) \u003d \\ sqrt (25) \u003d 5 $

Definition 2

The angle $ \\ varphi $ formed by the positive direction of the real axis and the radius vector $ \\ overrightarrow (OM) $, which corresponds to the given complex number $ z \u003d a + bi $, is called the argument of this number and is denoted $ \\ arg z $.

Note 1

The modulus and argument of a given complex number are explicitly used when representing a complex number in trigonometric or exponential form:

• $ z \u003d r \\ cdot (\\ cos \\ varphi + i \\ sin \\ varphi) $ - trigonometric form;
• $ z \u003d r \\ cdot e ^ (i \\ varphi) $ - exponential form.

Example 2

Write down a complex number in trigonometric and exponential forms, given by the following data: 1) $ r \u003d 3; \\ varphi \u003d \\ pi $; 2) $ r \u003d 13; \\ varphi \u003d \\ frac (3 \\ pi) (4) $.

1) Substitute the data $ r \u003d 3; \\ varphi \u003d \\ pi $ into the corresponding formulas and get:

$ z \u003d 3 \\ cdot (\\ cos \\ pi + i \\ sin \\ pi) $ - trigonometric form

$ z \u003d 3 \\ cdot e ^ (i \\ pi) $ is an exponential form.

2) Substitute the data $ r \u003d 13; \\ varphi \u003d \\ frac (3 \\ pi) (4) $ into the corresponding formulas and get:

$ z \u003d 13 \\ cdot (\\ cos \\ frac (3 \\ pi) (4) + i \\ sin \\ frac (3 \\ pi) (4)) $ - trigonometric form

$ z \u003d 13 \\ cdot e ^ (i \\ frac (3 \\ pi) (4)) $ is an exponential form.

Example 3

Determine the modulus and argument of the given complex numbers:

1) $ z \u003d \\ sqrt (2) \\ cdot (\\ cos 2 \\ pi + i \\ sin 2 \\ pi) $; 2) $ z \u003d \\ frac (5) (3) \\ cdot (\\ cos \\ frac (2 \\ pi) (3) + i \\ sin \\ frac (2 \\ pi) (3)) $; 3) $ z \u003d \\ sqrt (13) \\ cdot e ^ (i \\ frac (3 \\ pi) (4)) $; 4) $ z \u003d 13 \\ cdot e ^ (i \\ pi) $.

We find the modulus and argument using the formulas for writing a given complex number in trigonometric and exponential forms, respectively

\ \

1) For the initial complex number $ z \u003d \\ sqrt (2) \\ cdot (\\ cos 2 \\ pi + i \\ sin 2 \\ pi) $ we get $ r \u003d \\ sqrt (2); \\ varphi \u003d 2 \\ pi $.

2) For the initial complex number $ z \u003d \\ frac (5) (3) \\ cdot (\\ cos \\ frac (2 \\ pi) (3) + i \\ sin \\ frac (2 \\ pi) (3)) $ we get $ r \u003d \\ frac (5) (3); \\ varphi \u003d \\ frac (2 \\ pi) (3) $.

3) For the initial complex number $ z \u003d \\ sqrt (13) \\ cdot e ^ (i \\ frac (3 \\ pi) (4)) $ we get $ r \u003d \\ sqrt (13); \\ varphi \u003d \\ frac (3 \\ 4) For the initial complex number $ z \u003d 13 \\ cdot e ^ (i \\ pi) $ we get $ r \u003d 13; \\ varphi \u003d \\ pi $.

The argument $ \\ varphi $ of a given complex number $ z \u003d a + bi $ can be calculated using the following formulas:

\\ [\\ varphi \u003d tg \\ frac (b) (a); \\ cos \\ varphi \u003d \\ frac (a) (\\ sqrt (a ^ (2) + b ^ (2))); \\ sin \\ varphi \u003d \\ frac (b) (\\ sqrt (a ^ (2) + b ^ (2))). \\]

In practice, to calculate the value of the argument of a given complex number $ z \u003d a + bi $, one usually uses the formula:

$ \\ varphi \u003d \\ arg z \u003d \\ left \\ (\\ begin (array) (c) (arctg \\ frac (b) (a), a \\ ge 0) \\\\ (arctg \\ frac (b) (a) + \\ or solve the system of equations

$ \\ left \\ (\\ begin (array) (c) (\\ cos \\ varphi \u003d \\ frac (a) (\\ sqrt (a ^ (2) + b ^ (2)))) \\\\ (\\ sin \\ varphi \u003d \\ frac (b) (\\ sqrt (a ^ (2) + b ^ (2)))) \\ end (array) \\ right. $. (**)

example 4

Calculate the argument of the given complex numbers: 1) $ z \u003d 3 $; 2) $ z \u003d 4i $; 3) $ z \u003d 1 + i $; 4) $ z \u003d -5 $; 5) $ z \u003d -2i $.

Since $ z \u003d 3 $, then $ a \u003d 3, b \u003d 0 $. Let's calculate the argument of the original complex number using the formula (*):

\\ [\\ varphi \u003d \\ arg z \u003d arctg \\ frac (0) (3) \u003d arctg0 \u003d 0. \\]

Since $ z \u003d 4i $, then $ a \u003d 0, b \u003d 4 $. Let's calculate the argument of the original complex number using the formula (*):

\\ [\\ varphi \u003d \\ arg z \u003d arctg \\ frac (4) (0) \u003d arctg (\\ infty) \u003d \\ frac (\\ pi) (2). \\]

Since $ z \u003d 1 + i $, then $ a \u003d 1, b \u003d 1 $. Let's calculate the argument of the initial complex number by solving the system (**):

\\ [\\ left \\ (\\ begin (array) (c) (\\ cos \\ varphi \u003d \\ frac (1) (\\ sqrt (1 ^ (2) + 1 ^ (2))) \u003d \\ frac (1) (\\ \\ frac (1) (\\ sqrt (2)) \u003d \\ frac (\\ sqrt (2)) (2)) \\ end (array) \\ right. \\]

It is known from the trigonometry course that $ \\ cos \\ varphi \u003d \\ sin \\ varphi \u003d \\ frac (\\ sqrt (2)) (2) $ for the angle corresponding to the first coordinate quarter and equal to $ \\ varphi \u003d \\ frac (\\ pi) ( 4) $.

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Since $ z \u003d -5 $, then $ a \u003d -5, b \u003d 0 $. Let's calculate the argument of the original complex number using the formula (*):

\\ [\\ varphi \u003d \\ arg z \u003d arctg \\ frac (0) (- 5) + \\ pi \u003d arctg0 + \\ pi \u003d 0 + \\ pi \u003d \\ pi. \\]

Since $ z \u003d -2i $, then $ a \u003d 0, b \u003d -2 $. Let's calculate the argument of the original complex number using the formula (*):

\\ [\\ varphi \u003d \\ arg z \u003d arctg \\ frac (-2) (0) \u003d arctg (- \\ infty) \u003d \\ frac (3 \\ pi) (2). \\]

Note 2

The number $ z_ (3) $ is represented by the point $ (0; 1) $, therefore, the length of the corresponding radius vector is 1, i.e. $ r \u003d 1 $, and argument $ \\ varphi \u003d \\ frac (\\ pi) (2) $ as per note 3.

The number $ z_ (4) $ is represented by the point $ (0; -1) $, therefore, the length of the corresponding radius vector is 1, i.e. $ r \u003d 1 $, and argument $ \\ varphi \u003d \\ frac (3 \\ pi) (2) $ as per note 3.

The number $ z_ (5) $ is represented by the point $ (2; 2) $, therefore, the length of the corresponding radius vector is $ \\ sqrt (2 ^ (2) + 2 ^ (2)) \u003d \\ sqrt (4 + 4) \u003d \\ sqrt (8) \u003d 2 \\ sqrt (2) $, i.e. $ r \u003d 2 \\ sqrt (2) $, and the argument $ \\ varphi \u003d \\ frac (\\ pi) (4) $ is right triangle property.

Definition 8.3 (1).

Length | z | of the vector z \u003d (x, y) is called the modulus of the complex number z \u003d x + yi

Since the length of each side of the triangle does not exceed the sum of the lengths of its other two sides, and the absolute value of the difference in the lengths of the two sides of the triangle is not less than the length of the third side, then for any two complex numbers z 1 and z 2 the inequalities hold

Definition 8.3 (2).

Complex number argument. If φ is an angle formed by a nonzero vector z with a real axis, then any angle of the form (φ + 2πn, where n is an integer, and an angle of only this kind, will also be an angle formed by a vector z with a real axis.

The set of all angles that a nonzero vector z \u003d (x, y) forms with the real axis is called the argument of the complex number z \u003d x + yi and is denoted by arg z. Each element of this set is called the value of the argument of the number z (Fig. 8.3 (1)).

Figure: 8.3 (1).

Since the nonzero vector of the plane is uniquely determined by its length and the angle that it forms with the x axis, then two complex numbers other than zero are equal if and only if their absolute values \u200b\u200band arguments are equal.

If we impose on the values \u200b\u200bof the argument φ of the number z, for example, the condition 0≤φ<2π или условие -π<φ≤π, то значение аргумента будет определено однозначно. Такое значение называется главным значением аргумента.

Definition 8.3. (3)

Trigonometric notation of a complex number. The real and imaginary parts of the complex number z \u003d x + yi ≠ 0 are expressed in terms of its modulus r \u003d | z | and the argument φ as follows (from the definition of sine and cosine):

The right side of this equality is called the trigonometric notation of the complex number z. We will use it for z \u003d 0; in this case r \u003d 0, and φ can take any value - the argument of the number 0 is not defined. So, any complex number can be written in trigonometric form.

It is also clear that if the complex number z is written in the form

then the number r is its modulus, since

And φ one of the values \u200b\u200bof its argument

It can be convenient to use the trigonometric form of writing complex numbers when multiplying complex numbers, in particular, it allows you to find out the geometric meaning of the product of complex numbers.

Let's find formulas for multiplication and division of complex numbers in the trigonometric form of their notation. If

then by the rule of multiplication of complex numbers (using the formulas for the sine and cosine of the sum)

Thus, when multiplying complex numbers, their absolute values \u200b\u200bare multiplied, and the arguments are added:

Applying this formula sequentially to n complex numbers, we get

If all n numbers are equal, we get

Where for

performed

Hence, for a complex number, the absolute value of which is 1 (hence, it has the form

This equality is called moivre formulas

In other words, when dividing complex numbers, their modules are divided,

and the arguments are subtracted.

Examples 8.3 (1).

Draw on the complex plane С the sets of points satisfying the following conditions:

A complex number is a number of the form z \u003d x + i * y, where x and y are real numbers and i \u003d imaginary unit (i.e. a number whose square is -1). To define the view argument integrated numbers , you need to see the complex number on the complex plane in the polar coordinate system.

Instructions

1. The plane on which complex numbers , is called complex. On this plane, the horizontal axis is occupied by real numbers (x), and the vertical axis is imaginary numbers (y). On such a plane, the number is specified by two coordinates z \u003d (x, y). In a polar coordinate system, the coordinates of a point are the modulus and the argument. The distance | z | from point to origin. Is the angle called an argument? between the vector connecting the point and the preface coordinates and the horizontal axis of the coordinate system (see figure).

2. The figure shows that the module of the complex numbers z \u003d x + i * y is found by the Pythagorean theorem: | z | \u003d? (x ^ 2 + y ^ 2). Further argument numbers z is found as an acute angle of a triangle - through the values \u200b\u200bof the trigonometric functions sin, cos, tg: sin? \u003d y /? (x ^ 2 + y ^ 2), cos? \u003d x /? (x ^ 2 + y ^ 2), tg? \u003d y / x.

3. Say, let the number z \u003d 5 * (1 +? 3 * i) be given. First, select the real and imaginary parts: z \u003d 5 + 5 *? 3 * i. It turns out that the real part is x \u003d 5, and the imaginary part is y \u003d 5 *? 3. Compute module numbers : | z | \u003d? (25 + 75) \u003d? 100 \u003d 10. Next, find the sine of the angle ?: sin? \u003d 5/10 \u003d 1 / 2. From this we get the argument numbers z is 30 °.

4. Example 2. Let the number z \u003d 5 * i be given. The picture shows that the angle? \u003d 90 °. Check this value using the formula above. Write down the coordinates of this numbers on the complex plane: z \u003d (0, 5). Module numbers | z | \u003d 5. The tangent of the angle tg? \u003d 5/5 \u003d 1. Hence it follows that? \u003d 90 °.

5. Example 3. Let it be necessary to find the argument of the sum of 2 complex numbers z1 \u003d 2 + 3 * i, z2 \u003d 1 + 6 * i. According to the rules of addition, add these two complex numbers : z \u003d z1 + z2 \u003d (2 + 1) + (3 + 6) * i \u003d 3 + 9 * i. Further, according to the above scheme, calculate the argument: tg? \u003d 9/3 \u003d 3.

Note!
If the number z \u003d 0, then the value of the argument for it is undefined.

Helpful advice
The value of the argument of a complex number is determined with an accuracy of 2 *? * k, where k is any integer. The meaning of the argument? is that -?

Corresponding to this number:.
The modulus of a complex number z is usually denoted | z | or r.

Let and be real numbers such that a complex number (usual notation). Then

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