Trigonometric. Modulus and argument of a complex number. Trigonometric notation How to find an argument

Definition 8.3 (1).

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

Since the length of each side of the triangle does not exceed the sum of the lengths of its two other 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 the angle formed by a non-zero vector z with the real axis, then any angle of the form (φ + 2πn, where n is an integer, and only an angle of this kind, will also be an angle formed by the vector z with the real axis.

The set of all angles formed by the non-zero vector z = = (x, y) with the real axis is called the argument of the complex number z = 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)).

Rice. 8.3(1).

Since a non-zero vector of a plane is uniquely determined by its length and the angle it forms with the x axis, then two complex numbers different from zero are equal if and only if their absolute values ​​and arguments are equal.

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

Definition 8.3.(3)

Trigonometric form of writing a complex number. The real and imaginary parts of a complex number z = x + уi ≠ 0 are expressed through its modulus r= |z| and the argument φ as follows (from the definition of sine and cosine):

The right side of this equality is called the trigonometric form of writing the complex number z. We will also use it for z = 0; in this case, r = 0, and φ can take any value - the argument of the number 0 is undefined. So, every 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 is one of the values ​​of its argument

The trigonometric form of writing complex numbers can be convenient to use 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 multiplying and dividing complex numbers in trigonometric form. If

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

Thus, when multiplying complex numbers, their absolute values ​​are 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 whose absolute value is 1 (hence, it has the form

This equality is called Moivre's 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 C a set of points satisfying the following conditions:

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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See what “Modulus of a complex number” is in other dictionaries:

modulus of a complex number- kompleksinio skaičiaus modulis statusas T sritis fizika atitikmenys: engl. modulus of complex number vok. Betrag der komplexen Zahl, m rus. modulus of a complex number, m pranc. module du nombre complexe, m … Fizikos terminų žodynas

- (modulus) The magnitude of a number in terms of its distance from 0. The modulus, or absolute value of a real number x (denoted by |x|), is the difference between x and 0, regardless of sign. Therefore, if x0, then |x|=x and if x 0, then |x|=–x... Economic dictionary

For a complex number, see Absolute value. The modulus of transition from a system of logarithms with base a to a system with base b is the number 1/logab... Big Encyclopedic Dictionary

The absolute value or modulus of a real or complex number x is the distance from x to the origin. More precisely: The absolute value of a real number x is a non-negative number, denoted by |x| and defined as follows: ... ... Wikipedia

Module in mathematics, 1) M. (or absolute value) of a complex number z = x + iy is the number ═ (the root is taken with a plus sign). When representing a complex number z in trigonometric form z = r(cos j + i sin j), the real number r is equal to... ...

- (in mathematics) a measure for comparing homogeneous quantities and for expressing one of them using another; m. is expressed as a number. Dictionary of foreign words included in the Russian language. Pavlenkov F., 1907. MODULE (lat.). 1) a number that multiplies... ... Dictionary of foreign words of the Russian language

MODULE of a complex number, see Absolute value (see ABSOLUTE VALUE). The modulus of transition from a system of logarithms with base a to a system with base b is the number 1/logab... encyclopedic Dictionary

I Module (from Latin modulus measure) in architecture, a conventional unit adopted to coordinate the sizes of parts of a building or complex. In the architecture of different peoples, depending on the characteristics of construction technology and the composition of buildings behind M... ... Great Soviet Encyclopedia

I; m. [from lat. modulus measure] 1. of what. Specialist. A quantity characterizing which l. property of a solid. M. compression. M. elasticity. 2. Math. Real number, the absolute value of a negative or positive number. M. complex number. M... encyclopedic Dictionary

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A complex number is a number of the form z =x + i * y, where x and y are real numbers, and i = imaginary unit (i.e. a number whose square is -1). To define the concept argument comprehensive numbers, it is necessary to consider a complex number on the complex plane in the polar coordinate system.

Instructions

The plane on which complex complexes are represented 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 given by two coordinates z = (x, y). In the polar coordinate system, the coordinates of a point are the modulus and the argument. The modulus is the distance |z| from a point to the origin. The argument is the angle between the vector connecting the point and the origin and the horizontal axis of the coordinate system (see figure).

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

For example, let the number z = 5 * (1 + ?3 * i) be given. First of all, select the real and imaginary parts: z = 5 +5 * ?3 * i. It turns out that the real part is x = 5, and the imaginary part is y = 5 * ?3. Calculate modulus numbers: |z| = ?(25 + 75) = ?100 =10. Next, find the sine of the angle: sin = 5 / 10 = 1 / 2. This gives the argument numbers z is equal to 30°.

Example 2. Let the number z = 5 * i be given. The figure shows that the angle = 90°. Check this value using the formula given above. Write down the coordinates of this numbers on the complex plane: z = (0, 5). Module numbers|z| = 5. Tangent of the angle tg = 5 / 5 = 1. It follows that = 90°.

Example 3. Let it be necessary 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, you add these two complex numbers: z = z1 + z2 = (2 + 1) + (3 + 6) * i = 3 + 9 * i. Next, using the above diagram, calculate the argument: tg = 9 / 3 = 3.

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

Instructions

1. The plane on which complex complexes are represented 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 given by two coordinates z = (x, y). In the polar coordinate system, the coordinates of a point are the modulus and the argument. The modulus is the distance |z| from a point to the origin. Is an angle called an argument? between the vector connecting the point and the coordinate preface and the horizontal axis of the coordinate system (see figure).

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

3. Let's say, let the number z = 5 * (1 + ?3 * i) be given. First of all, select the real and imaginary parts: z = 5 +5 * ?3 * i. It turns out that the real part is x = 5, and the imaginary part is y = 5 * ?3. Calculate modulus numbers: |z| = ?(25 + 75) = ?100 =10. Next, find the sine of the angle?: sin ? = 5 / 10 = 1 / 2. From there we get the argument numbers z is equal to 30°.

4. Example 2. Let the number z = 5 * i be given. From the picture you can see that the angle? = 90°. Check this value using the formula given above. Write down the coordinates of this numbers on the complex plane: z = (0, 5). Module numbers|z| = 5. Tangent of the angle tg? = 5 / 5 = 1. From there it follows what? = 90°.

5. Example 3. Let's say we need to find the argument for the sum of 2 complex numbers z1 = 2 + 3 * i, z2 = 1 + 6 * i. According to the rules of addition, you add these two complex numbers: z = z1 + z2 = (2 + 1) + (3 + 6) * i = 3 + 9 * i. Then, according to the above diagram, calculate the argument: tg? = 9 / 3 = 3.

Note!
If the number z = 0, then the value of the argument for it is not defined.

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? such that –?