DEFINITION

The algebraic form of a complex number is to write the complex number \(\ z \) as \(\ z=x+i y \), where \(\ x \) and \(\ y \) are real numbers, \(\ i \ ) is an imaginary unit that satisfies the relation \(\ i^(2)=-1 \)

The number \(\ x \) is called the real part of the complex number \(\ z \) and is denoted \(\ x=\operatorname(Re) z \)

The number \(\ y \) is called the imaginary part of the complex number \(\ z \) and is denoted \(\ y=\operatorname(Im) z \)

For example:

The complex number \(\ z=3-2 i \) and its associated number \(\ \overline(z)=3+2 i \) are written in algebraic form.

The imaginary value \(\ z=5 i \) is written in algebraic form.

In addition, depending on the problem being solved, you can convert a complex number into a trigonometric or exponential number.

A task

Write the number \(\ z=\frac(7-i)(4)+13 \) in algebraic form, find its real and imaginary parts, as well as the conjugate number.

Solution.

Applying the term division of fractions and the rule of addition of fractions, we get:

\(\ z=\frac(7-i)(4)+13=\frac(7)(4)+13-\frac(i)(4)=\frac(59)(4)-\frac( 1)(4) i \)

Therefore, the real part of the complex number \(\ z=\frac(5 g)(4)-\frac(1)(4) i \) is the number \(\ x=\operatorname(Re) z=\frac(59) (4) \) , the imaginary part is a number \(\ y=\operatorname(Im) z=-\frac(1)(4) \)

Conjugate number: \(\ \overline(z)=\frac(59)(4)+\frac(1)(4) i \)

Answer

\(\ z=\frac(59)(4)-\frac(1)(4) i \), \(\ \operatorname(Re) z=\frac(59)(4) \), \(\ \operatorname(Im) z=-\frac(1)(4) \), \(\ \overline(z)=\frac(59)(4)+\frac(1)(4) i \)

Actions of complex numbers in algebraic form comparison

Two complex numbers \(\ z_(1)=x_(1)+i y_(1) \) are equal if \(\ x_(1)=x_(2) \), \(\ y_(1)= y_(2) \) i.e. Their real and imaginary parts are equal.

A task

Determine for which x and y two complex numbers \(\ z_(1)=13+y i \) and \(\ z_(2)=x+5 i \) are equal.

Solution

By definition, two complex numbers are equal if their real and imaginary parts are equal, i.e. \(\ x=13 \), \(\ y=5 \).

Answer \(\ x=13 \), \(\ y=5 \)

addition

The addition of complex numbers \(\ z_(1)=x_(1)+i y_(1) \) is done by direct summation of the real and imaginary parts:

\(\ z_(1)+z_(2)=x_(1)+i y_(1)+x_(2)+i y_(2)=\left(x_(1)+x_(2)\right) +i\left(y_(1)+y_(2)\right) \)

A task

Find the sum of complex numbers \(\ z_(1)=-7+5 i \), \(\ z_(2)=13-4 i \)

Solution.

The real part of the complex number \(\ z_(1)=-7+5 i \) is the number \(\ x_(1)=\operatorname(Re) z_(1)=-7 \) , the imaginary part is the number \( \ y_(1)=\mathrm(Im) \), \(\ z_(1)=5 \) . The real and imaginary parts of the complex number \(\ z_(2)=13-4 i \) are \(\ x_(2)=\operatorname(Re) z_(2)=13 \) and \(\ y_(2 )=\operatorname(Im) z_(2)=-4 \) .

Therefore, the sum of complex numbers is:

\(\ z_(1)+z_(2)=\left(x_(1)+x_(2)\right)+i\left(y_(1)+y_(2)\right)=(-7+ 13)+i(5-4)=6+i\)

Answer

\(\z_(1)+z_(2)=6+i \)

Read more about adding complex numbers in a separate article: Adding complex numbers.

Subtraction

The subtraction of complex numbers \(\ z_(1)=x_(1)+i y_(1) \) and \(\ z_(2)=x_(2)+i y_(2) \) is done by direct subtraction of the real and imaginary parts:

\(\ z_(1)-z_(2)=x_(1)+i y_(1)-\left(x_(2)+i y_(2)\right)=x_(1)-x_(2) +\left(i y_(1)-i y_(2)\right)=\left(x_(1)-x_(2)\right)+i\left(y_(1)-y_(2)\right )\)

A task

find the difference of complex numbers \(\ z_(1)=17-35 i \), \(\ z_(2)=15+5 i \)

Solution.

Find the real and imaginary parts of complex numbers \(\ z_(1)=17-35 i \), \(\ z_(2)=15+5 i \) :

\(\ x_(1)=\operatorname(Re) z_(1)=17, x_(2)=\operatorname(Re) z_(2)=15 \)

\(\ y_(1)=\operatorname(Im) z_(1)=-35, y_(2)=\operatorname(Im) z_(2)=5 \)

So the difference of complex numbers is:

\(\ z_(1)-z_(2)=\left(x_(1)-x_(2)\right)+i\left(y_(1)-y_(2)\right)=(17-15 )+i(-35-5)=2-40 i \)

Answer

\(\ z_(1)-z_(2)=2-40 i \) multiplication

The multiplication of complex numbers \(\ z_(1)=x_(1)+i y_(1) \) and \(\ z_(2)=x_(2)+i y_(2) \) is performed by directly generating numbers in algebraic form, taking into account the property of the imaginary unit \(\ i^(2)=-1 \) :

\(\ z_(1) \cdot z_(2)=\left(x_(1)+i y_(1)\right) \cdot\left(x_(2)+i y_(2)\right)=x_ (1) \cdot x_(2)+i^(2) \cdot y_(1) \cdot y_(2)+\left(x_(1) \cdot i y_(2)+x_(2) \cdot i y_(1)\right)= \)

\(\ =\left(x_(1) \cdot x_(2)-y_(1) \cdot y_(2)\right)+i\left(x_(1) \cdot y_(2)+x_(2 ) \cdot y_(1)\right) \)

A task

Find the product of complex numbers \(\ z_(1)=1-5 i \)

Solution.

Complex of complex numbers:

\(\ z_(1) \cdot z_(2)=\left(x_(1) \cdot x_(2)-y_(1) \cdot y_(2)\right)+i\left(x_(1) \cdot y_(2)+x_(2) \cdot y_(1)\right)=(1 \cdot 5-(-5) \cdot 2)+i(1 \cdot 2+(-5) \cdot 5 )=15-23 i \)

Answer

\(\ z_(1) \cdot z_(2)=15-23 i \) split

The complex number factor \(\ z_(1)=x_(1)+i y_(1) \) and \(\ z_(2)=x_(2)+i y_(2) \) is determined by multiplying the numerator and denominator to the conjugate number with a denominator:

\(\ \frac(z_(1))(z_(2))=\frac(x_(1)+i y_(1))(x_(2)+i y_(2))=\frac(\left (x_(1)+i y_(1)\right)\left(x_(2)-i y_(2)\right))(\left(x_(2)+i y_(2)\right)\left (x_(2)-i y_(2)\right))=\frac(x_(1) \cdot x_(2)+y_(1) \cdot y_(2))(x_(2)^(2) +y_(2)^(2))+i \frac(x_(2) \cdot y_(1)-x_(1) \cdot y_(2))(x_(2)^(2)+y_(2 )^(2)) \)

A task

To divide the number 1 by the complex number \(\ z=1+2 i \).

Solution.

Since the imaginary part of the real number 1 is zero, the factor is:

\(\ \frac(1)(1+2 i)=\frac(1 \cdot 1)(1^(2)+2^(2))-i \frac(1 \cdot 2)(1^( 2)+2^(2))=\frac(1)(5)-i \frac(2)(5) \)

Answer

\(\ \frac(1)(1+2 i)=\frac(1)(5)-i \frac(2)(5) \)

Complex numbers are an extension of the set of real numbers, usually denoted by . Any complex number can be represented as a formal sum, where and are real numbers, is an imaginary unit.

Writing a complex number in the form , , is called the algebraic form of a complex number.

Properties of complex numbers. Geometric interpretation of a complex number.

Actions on complex numbers given in algebraic form:

Consider the rules by which arithmetic operations are performed on complex numbers.

If two complex numbers α = a + bi and β = c + di are given, then

α + β = (a + bi) + (c + di) = (a + c) + (b + d)i,

α - β \u003d (a + bi) - (c + di) \u003d (a - c) + (b - d)i. (eleven)

This follows from the definition of the operations of addition and subtraction of two ordered pairs of real numbers (see formulas (1) and (3)). We have obtained the rules for addition and subtraction of complex numbers: to add two complex numbers, one must separately add their real parts and, accordingly, the imaginary parts; in order to subtract another from one complex number, it is necessary to subtract their real and imaginary parts, respectively.

The number - α \u003d - a - bi is called the opposite of the number α \u003d a + bi. The sum of these two numbers is zero: - α + α = (- a - bi) + (a + bi) = (-a + a) + (-b + b) i = 0.

To obtain the multiplication rule for complex numbers, we use formula (6), i.e., the fact that i2 = -1. Taking into account this ratio, we find (a + bi)(c + di) = ac + adi + bci + bdi2 = ac + (ad + bc)i – bd, i.e.

(a + bi)(c + di) = (ac - bd) + (ad + bc)i . (12)

This formula corresponds to formula (2), which defined the multiplication of ordered pairs of real numbers.

Note that the sum and product of two complex conjugate numbers are real numbers. Indeed, if α = a + bi, = a – bi, then α = (a + bi)(a - bi) = a2 – i2b2 = a2 + b2 , α + = (a + bi) + (a - bi) = ( a + a) + (b - b)i= 2a, i.e.

α + = 2a, α = a2 + b2. (13)

When dividing two complex numbers in algebraic form, one should expect that the quotient is also expressed by a number of the same type, i.e., α/β = u + vi, where u, v R. Let us derive a rule for dividing complex numbers. Let numbers α = a + bi, β = c + di be given, and β ≠ 0, i.e., c2 + d2 ≠ 0. The last inequality means that c and d do not vanish simultaneously (the case when c = 0, d = 0). Applying formula (12) and the second of equalities (13), we find:

Therefore, the quotient of two complex numbers is given by:

the corresponding formula (4).

Using the obtained formula for the number β = c + di, you can find the reciprocal of it β-1 = 1/β. Assuming in formula (14) a = 1, b = 0, we obtain

This formula determines the reciprocal of a given non-zero complex number; this number is also complex.

For example: (3 + 7i) + (4 + 2i) = 7 + 9i;

(6 + 5i) - (3 + 8i) = 3 - 3i;

(5 – 4i)(8 – 9i) = 4 – 77i;

Actions on complex numbers in algebraic form.

55. Argument of a complex number. Trigonometric form of writing a complex number (output).

Arg.comm.number. – between the positive direction of the real X axis by the vector representing the given number.

trine formula. Numbers: ,

Complex numbers

Imaginary and complex numbers. Abscissa and ordinate

complex number. Conjugate complex numbers.

Operations with complex numbers. Geometric

representation of complex numbers. complex plane.

Modulus and argument of a complex number. trigonometric

complex number form. Operations with complex

numbers in trigonometric form. Moivre formula.

Basic information about imaginary and complex numbers are given in the section "Imaginary and complex numbers". The need for these numbers of a new type appeared when solving quadratic equations for the caseD< 0 (здесь Dis the discriminant of the quadratic equation). For a long time, these numbers did not find physical use, which is why they were called "imaginary" numbers. However, now they are very widely used in various fields of physics.

and technology: electrical engineering, hydro- and aerodynamics, the theory of elasticity, etc.

Complex numbers are written as:a+bi. Here a and b – real numbers , a i – imaginary unit. e. i 2 = –1. Number a called abscissa, a b - ordinatecomplex numbera + b .Two complex numbersa+bi and a-bi called conjugate complex numbers.

Main agreements:

1. Real numberacan also be written in the formcomplex number:a + 0 i or a - 0 i. For example, entries 5 + 0i and 5 - 0 imean the same number 5 .

2. Complex number 0 + bicalled purely imaginary number. Recordingbimeans the same as 0 + bi.

3. Two complex numbersa+bi andc + diare considered equal ifa = c and b = d. Otherwise complex numbers are not equal.

Addition. The sum of complex numbersa+bi and c + diis called a complex number (a+c ) + (b+d ) i .In this way, when added complex numbers, their abscissas and ordinates are added separately.

This definition follows the rules for dealing with ordinary polynomials.

Subtraction. The difference between two complex numbersa+bi(reduced) and c + di(subtracted) is called a complex number (a-c ) + (b-d ) i .

In this way, when subtracting two complex numbers, their abscissas and ordinates are subtracted separately.

Multiplication. The product of complex numbersa+bi and c + di is called a complex number.

(ac-bd ) + (ad+bc ) i .This definition stems from two requirements:

1) numbers a+bi and c + dishould multiply like algebraic binomials,

2) number ihas the main property:i 2 = – 1.

EXAMPLE ( a + bi )(a-bi) = a 2 +b 2 . Consequently, work

two conjugate complex numbers is equal to the real

positive number.

Division. Divide a complex numbera+bi (divisible) to anotherc + di(divider) - means to find the third numbere + fi(chat), which, when multiplied by a divisorc + di, which results in the dividenda + b .

If the divisor is not zero, division is always possible.

EXAMPLE Find (8+i ) : (2 – 3 i) .

Solution. Let's rewrite this ratio as a fraction:

Multiplying its numerator and denominator by 2 + 3i

And after performing all the transformations, we get:

Geometric representation of complex numbers. Real numbers are represented by points on the number line:

Here is the point Ameans number -3, dotB is the number 2, and O- zero. In contrast, complex numbers are represented by points on the coordinate plane. For this, we choose rectangular (Cartesian) coordinates with the same scales on both axes. Then the complex numbera+bi will be represented by a dot P with abscissa a and ordinate b (see fig.). This coordinate system is called complex plane .

module complex number is called the length of the vectorOP, depicting a complex number on the coordinate ( integrated) plane. Complex number modulusa+bi denoted by | a+bi| or letter r

Consider a quadratic equation.

Let's define its roots.

There is no real number whose square is -1. But if the formula defines the operator i as an imaginary unit, then the solution of this equation can be written in the form . Wherein and - complex numbers, in which -1 is the real part, 2 or in the second case -2 is the imaginary part. The imaginary part is also a real (real) number. The imaginary part multiplied by the imaginary unit means already imaginary number.

In general, a complex number has the form

z = x + iy ,

where x, y are real numbers, is an imaginary unit. In a number of applied sciences, for example, in electrical engineering, electronics, signal theory, the imaginary unit is denoted by j. Real numbers x = Re(z) and y=Im(z) called real and imaginary parts numbers z. The expression is called algebraic form notation of a complex number.

Any real number is a special case of a complex number in the form . An imaginary number is also a special case of a complex number. .

Definition of the set of complex numbers C

This expression reads as follows: set FROM, consisting of elements such that x and y belong to the set of real numbers R and is the imaginary unit. Note that etc.

Two complex numbers and are equal if and only if their real and imaginary parts are equal, i.e. and .

Complex numbers and functions are widely used in science and technology, in particular, in mechanics, analysis and calculation of AC circuits, analog electronics, signal theory and processing, automatic control theory, and other applied sciences.

1. Arithmetic of complex numbers

The addition of two complex numbers consists in adding their real and imaginary parts, i.e.

Accordingly, the difference of two complex numbers

Complex number called complex conjugate number z=x +i.y.

The complex conjugate numbers z and z * differ in the signs of the imaginary part. It's obvious that

.

Any equality between complex expressions remains valid if in this equality everywhere i replaced by - i, i.e. go to the equality of conjugate numbers. Numbers i and – i are algebraically indistinguishable because .

The product (multiplication) of two complex numbers can be calculated as follows:

Division of two complex numbers:

Example:

1. Complex plane

A complex number can be graphically represented in a rectangular coordinate system. Let us set a rectangular coordinate system in the plane (x, y).

on axle Ox we will arrange the real parts x, it is called real (real) axis, on the axis Oy– imaginary parts y complex numbers. She bears the name imaginary axis. Moreover, each complex number corresponds to a certain point of the plane, and such a plane is called complex plane. point BUT the complex plane will correspond to the vector OA.

Number x called abscissa complex number, number y – ordinate.

A pair of complex conjugate numbers is displayed as dots located symmetrically about the real axis.

If on the plane set polar coordinate system, then every complex number z determined by polar coordinates. Wherein module numbers is the polar radius of the point, and the angle - its polar angle or complex number argument z.

Complex number modulus always non-negative. The argument of a complex number is not uniquely defined. The main value of the argument must satisfy the condition . Each point of the complex plane also corresponds to the total value of the argument . Arguments that differ by a multiple of 2π are considered equal. The number argument zero is not defined.

The main value of the argument is determined by the expressions:

It's obvious that

Wherein
, .

Complex number representation z as

called trigonometric form complex number.

Example.

1. The exponential form of complex numbers

Decomposition in Maclaurin series for real argument functions looks like:

For the exponential function of a complex argument z decomposition is similar

.

The Maclaurin series expansion for the exponential function of the imaginary argument can be represented as

The resulting identity is called Euler formula.

For a negative argument, it looks like

By combining these expressions, we can define the following expressions for sine and cosine

.

Using the Euler formula, from the trigonometric form of the representation of complex numbers

available demonstrative(exponential, polar) form of a complex number, i.e. its representation in the form

,

where - polar coordinates of a point with rectangular coordinates ( x,y).

The conjugate of a complex number is written in exponential form as follows.

For the exponential form, it is easy to define the following formulas for multiplication and division of complex numbers

That is, in exponential form, the product and division of complex numbers is easier than in algebraic form. When multiplying, the modules of the factors are multiplied, and the arguments are added. This rule applies to any number of factors. In particular, when multiplying a complex number z on the i vector z rotates counterclockwise by 90

In division, the numerator modulus is divided by the denominator modulus, and the denominator argument is subtracted from the numerator argument.

Using the exponential form of complex numbers, one can obtain expressions for well-known trigonometric identities. For example, from the identity

using the Euler formula, we can write

Equating the real and imaginary parts in this expression, we obtain expressions for the cosine and sine of the sum of the angles

1. Powers, roots and logarithms of complex numbers

Raising a complex number to a natural power n produced according to the formula

Example. Compute .

Imagine a number in trigonometric form

’

Applying the exponentiation formula, we get

Putting the value in the expression r= 1, we get the so-called De Moivre's formula, with which you can determine the expressions for the sines and cosines of multiple angles.

Root n th power of a complex number z It has n different values ​​determined by the expression

Example. Let's find .

To do this, we express the complex number () to the trigonometric form

.

According to the formula for calculating the root of a complex number, we get

Logarithm of a complex number z is a number w, for which . The natural logarithm of a complex number has an infinite number of values ​​and is calculated by the formula

Consists of real (cosine) and imaginary (sine) parts. Such stress can be represented as a vector of length U m, initial phase (angle), rotating with angular velocity ω .

Moreover, if complex functions are added, then their real and imaginary parts are added. If a complex function is multiplied by a constant or a real function, then its real and imaginary parts are multiplied by the same factor. Differentiation/integration of such a complex function is reduced to differentiation/integration of the real and imaginary parts.

For example, the differentiation of the complex stress expression

is to multiply it by iω is the real part of the function f(z), and is the imaginary part of the function. Examples: .

Meaning z is represented by a point in the complex z plane, and the corresponding value w- a point in the complex plane w. When displayed w = f(z) plane lines z pass into the lines of the plane w, figures of one plane into figures of another, but the shapes of lines or figures may change significantly.