Geometric representation of complex numbers. Trigonometric form of a complex number.
2015-06-04
Real and imaginary axis
Complex Number Argument
Main argument of a complex number
Trigonometric form of a complex number
Specifying a complex number $z = a+bi$ is equivalent to specifying two real numbers $a,b$ - the real and imaginary parts of this complex number. But an ordered pair of numbers $(a,b)$ is represented in the Cartesian rectangular coordinate system by a point with coordinates $(a, b)$. Thus, this point can also serve as an image for the complex number $z$: a one-to-one correspondence is established between complex numbers and points of the coordinate plane.
When using the coordinate plane to represent complex numbers, the $Ox$ axis is usually called the real axis (since the real part of the number is taken to be the abscissa of the point), and the $Oy$ axis is the imaginary axis (since the imaginary part of the number is taken to be the ordinate of the point).
Fig.1
In Fig. 1, images of the numbers $z_(1) = 2 + 3i, z_(2)=1 =1,z_(3) = 4i, z_(4) = -4 + i, z_(5) = -2, z_( 6) = - 3 – 2i, z_(7) = -5i, z_(8) = 2 – 3i$.
Two complex conjugate numbers are represented by points symmetrical about the $Ox$ axis (points $z_(1)$ and $z_(8)$ in Fig. 1).
Rice. 2
Often associated with a complex number $z$ is not only the point $M$ representing this number, but also the vector $\vec(OM)$ leading from $O$ to $M$; The representation of the number $z$ as a vector is convenient from the point of view of the geometric interpretation of the action of addition and subtraction of complex numbers. In Fig. 2, and it is shown that the vector representing the sum of complex numbers $z_(1), z_(2)$ is obtained as the diagonal of a parallelogram constructed on the vectors $\vec(OM_(1)), \vec(OM_(2)) $ representing terms. This rule for adding vectors is known as the parallelogram rule (for example, for adding forces or velocities in a physics course). Subtraction can be reduced to addition with the opposite vector (Fig. 2, b).
Rice. 3
As is known, the position of a point on a plane can also be specified by its polar coordinates $r, \phi$. Thus, the complex number - the affix of a point - will also be determined by specifying $r$ and $\phi$. From Fig. 3 it is clear that $r = OM = \sqrt(x^(2) + y^(2))$ is at the same time the modulus of the complex number $z$: the polar radius of the point representing the number $z$ is equal to the modulus of this numbers.
The polar angle of a point $M$ is called the argument of the number $z$ represented by this point.
All values of the argument are collectively denoted by the symbol $Arg \: z$.
So, any complex number can be associated with a pair of real numbers: the modulus and the argument of the given number, and the argument is determined ambiguously. On the contrary, given the module $|z| = r$ and the argument $\phi$ corresponds singular$z$ having the given module and argument. Special properties has the number zero: its modulus is zero, and no specific value is assigned to the argument.
To achieve unambiguity in the definition of the argument of a complex number, one can agree to call one of the values of the argument the main one. It is denoted by the symbol $arg \: z$. Typically, the main value of the argument is chosen to be a value that satisfies the inequalities
$0 \leq arg \: z (in other cases the inequalities $- \pi
The real and imaginary parts of a complex number (as the Cartesian coordinates of a point) are expressed through its modulus and argument (polar coordinates of the point) using the formulas:
$a = r \cos \phi, b = r \sin \phi$, (1)
and a complex number can be written in the following trigonometric form:
$z = r(\cos \phi \phi + i \sin \phi)$ (2)
(we will call writing a number in the form $z = a + bi$ a record in algebraic form).
The transition from writing a number in algebraic form to writing it in trigonometric form and vice versa is made according to formulas (4):
$r = \sqrt(a^(2) + b^(2)), \cos \phi = \frac(a)(r)= \frac(a)(\sqrt(a^(2) + b^ (2)), \sin \phi = \frac(b)(r) = \frac(b)(\sqrt(a^(2) + b^(2))), tg \phi = \frac( b)(a)$ (3)
and formulas (1). When defining an argument (its main value), you can use the value of one of trigonometric functions$\cos \phi$ or $\sin \phi$ and take into account the sign of the second.
Example. Write the following numbers in trigonometric form:
a)$6 + 6i$; b) $3i$; c) $-10$.
Solution, a) We have
$r = \sqrt(6^(2) + (-6)^(2)) = 6 \sqrt(2)$,
$\cos \phi = \frac(6)(6 \sqrt(2)) = \frac(1)(\sqrt(2)) = \frac(\sqrt(2))(2)$,
$\sin \phi = - \frac(6)(6 \sqrt(2)) = - \frac(1)(\sqrt(2)) = - \frac(\sqrt(2))(2)$,
whence $\phi = \frac(7 \pi)(4)$, and, therefore,
$6-6i = 6 \sqrt(2) \left (\cos \frac(7 \pi)(4) + i \sin \frac(7 \pi)(4) \right)$;
b) $r = 3, \cos \phi = 0, \sin \phi = 1, \phi = \pi /2$;
$3i = 3 \left (\cos \frac(\pi)(2) + i \sin \frac(\pi)(2) \right)$
c) $r = 10, \cos \phi = -1, \sin \phi = 0, \phi = \pi$;
$-10 = 10 (\cos \pi + i \sin \pi)$
Complex numbers
Basic Concepts
The initial data on the number dates back to the Stone Age - Paleomelitic. These are “one”, “few” and “many”. They were recorded in the form of notches, knots, etc. Development labor processes and the emergence of property forced man to invent numbers and their names. Natural numbers appeared first N, obtained by counting objects. Then, along with the need to count, people had a need to measure lengths, areas, volumes, time and other quantities, where they had to take into account parts of the measure used. This is how fractions came into being. The formal substantiation of the concepts of fractional and negative numbers was carried out in the 19th century. Set of integers Z– these are natural numbers, natural numbers with a minus sign and zero. Whole and fractional numbers formed a set of rational numbers Q, but it also turned out to be insufficient for the study of continuously changing variables. Genesis again showed the imperfection of mathematics: the impossibility of solving an equation of the form X 2 = 3, which is why irrational numbers appeared I. Union of the set of rational numbers Q and irrational numbers I– set of real (or real) numbers R. As a result, the number line was filled: each real number corresponded to a point on it. But on many R there is no way to solve an equation of the form X 2 = – A 2. Consequently, the need arose again to expand the concept of number. This is how complex numbers appeared in 1545. Their creator J. Cardano called them “purely negative.” The name “imaginary” was introduced in 1637 by the Frenchman R. Descartes, in 1777 Euler proposed using the first letter of the French number i to denote the imaginary unit. This symbol came into general use thanks to K. Gauss.
During the 17th and 18th centuries, the discussion of the arithmetic nature of imaginaries and their geometric interpretation continued. The Dane G. Wessel, the Frenchman J. Argan and the German K. Gauss independently proposed to represent a complex number as a point on the coordinate plane. Later it turned out that it is even more convenient to represent a number not by the point itself, but by a vector going to this point from the origin.
Only towards the end of the 18th - beginning of the 19th century did complex numbers take their rightful place in mathematical analysis. Their first use is in theory differential equations and in the theory of hydrodynamics.
Definition 1.Complex number is called an expression of the form , where x And y are real numbers, and i– imaginary unit, .
Two complex numbers and equal if and only if , .
If , then the number is called purely imaginary; if , then the number is a real number, this means that the set R WITH, Where WITH– a set of complex numbers.
Conjugate to a complex number is called a complex number.
Geometric representation of complex numbers.
Any complex number can be represented by a dot M(x, y) plane Oxy. A pair of real numbers also denotes the coordinates of the radius vector 
, i.e. between the set of vectors on the plane and the set of complex numbers, one can establish a one-to-one correspondence: .
Definition 2.Real part X.
Designation: x=Re z(from Latin Realis).
Definition 3.Imaginary part complex number is a real number y.
Designation: y= Im z(from Latin Imaginarius).
Re z is deposited on the axis ( Oh), Im z is deposited on the axis ( Oh), then the vector corresponding to the complex number is the radius vector of the point M(x, y), (or M(Re z, Im z)) (Fig. 1).
Definition 4. A plane whose points are associated with a set of complex numbers is called complex plane. The abscissa axis is called real axis, since it contains real numbers. The ordinate axis is called imaginary axis, it contains purely imaginary complex numbers. The set of complex numbers is denoted WITH.
Definition 5.Module complex number z = (x, y) is called the length of the vector: , i.e. 
.
Definition 6.Argument complex number is the angle between the positive direction of the axis ( Oh) and vector: 
.
Note 3. If the point z lies on the real or imaginary axis, then you can find it directly.
Go) numbers.
2. Algebraic form of representation of complex numbers
Complex number or complex, is a number consisting of two numbers (parts) – real and imaginary.
Real Any positive or negative number is called, for example, + 5, - 28, etc. Let's denote a real number by the letter “L”.
Imaginary is a number equal to the product of a real number and Square root from a negative unit, for example, 8, - 20, etc.
A negative unit is called imaginary and is denoted by the letter “yot”:
Let us denote the real number in the imaginary number by the letter “M”.
Then the imaginary number can be written like this: j M. In this case, the complex number A can be written like this:
A = L + j M (2).
This form of writing a complex number (complex), which is an algebraic sum of the real and imaginary parts, is called algebraic.
Example 1. Represent in algebraic form a complex whose real part is 6 and whose imaginary part is 15.
Solution. A = 6 +j 15.
In addition to the algebraic form, a complex number can be represented by three more:
1. graphic;
2. trigonometric;
3. indicative.
Such a variety of forms is dramatically simplifies calculations sinusoidal quantities and their graphic image.
Let's look at the graphical, trigonometric and exponent in turn.
new forms of representing complex numbers.
Graphical form of representing complex numbers
For graphical representation of complex numbers, direct
carbon coordinate system. In a regular (school) coordinate system, positive or negative values are plotted along the “x” (abscissa) and “y” (ordinate) axes. real numbers.
In the coordinate system adopted in the symbolic method, along the “x” axis
real numbers are plotted in the form of segments, and imaginary numbers are plotted along the “y” axis
Rice. 1. Coordinate system for graphical representation of complex numbers
Therefore, the x-axis is called the axis of real quantities or, for short, real axis.
The ordinate axis is called the axis of imaginary quantities or imaginary axis.
The plane itself (i.e., the plane of the drawing), on which complex numbers or quantities are depicted, is called comprehensive flat.
In this plane, the complex number A = L + j M is represented by the vector A
(Fig. 2), the projection of which onto the real axis is equal to its real part Re A = A" = L, and the projection onto the imaginary axis is equal to the imaginary part Im A = A" = M.
(Re - from the English real - real, real, real, Im - from the English imaginary - unreal, imaginary).
Rice. 2. Graphical representation of a complex number
In this case, the number A can be written as follows
A = A" + A" = Re A + j Im A (3).
Using a graphical representation of the number A in the complex plane, we introduce new definitions and obtain some important relationships:
1. the length of vector A is called module vector and is denoted by |A|.
According to the Pythagorean theorem
|A| = 
(4) .
2. angle α formed by vector A and real positive half-
the axis is called argument vector A and is determined through its tangent:
tg α = A" / A" = Im A / Re A (5).
Thus, for a graphical representation of a complex number
A = A" + A" in the form of a vector you need:
1. find the modulus of the vector |A| according to formula (4);
2. find the argument of the vector tan α using formula (5);
3. find the angle α from the relation α = arc tan α;
4. in the coordinate system j (x) draw an auxiliary
straight line and on it, on a certain scale, plot a segment equal to the absolute value of the vector |A|.
Example 2. Present the complex number A = 3 + j 4 in graphical form.
Complex numbers, their representation on a plane. Algebraic operations on complex numbers. Complex pairing. Modulus and argument of a complex number. Algebraic and trigonometric forms of complex numbers. Roots of complex numbers. Exponential function complex argument. Euler's formula. Exponential form of a complex number.
When studying one of the basic methods of integration: the integration of rational fractions, it is required to consider polynomials in the complex domain to carry out rigorous proofs. Therefore, let us first study some properties of complex numbers and operations on them.
Definition 7.1. A complex number z is an ordered pair of real numbers (a,b) : z = (a,b) (the term “ordered” means that in writing a complex number the order of the numbers a and b is important: (a,b)≠(b,a )). In this case, the first number a is called the real part of the complex number z and is denoted a = Re z, and the second number b is called the imaginary part of z: b = Im z.
Definition 7.2. Two complex numbers z 1 = (a 1 , b 1) and z 2 = (a 2 , b 2) are equal if and only if their real and imaginary parts are equal, that is, a 1 = a 2 , b 1 = b 2 .
Operations on complex numbers.
1. Amount complex numbers z 1 =(a 1 , b 1) And z 2 =(a 2 , b 2 z =(a,b) such that a = a 1 + a 2, b = b 1 + b 2. Properties of addition: a) z 1 + z 2 = z 2 + z 1; b) z 1 +(z 2 + z 3) = (z 1 + z 2) + z 3; c) there is a complex number 0 = (0,0): z + 0 =z for any complex number z.
2. The work complex numbers z 1 =(a 1 , b 1) And z 2 =(a 2 , b 2) is called a complex number z =(a,b) such that a = a 1 a 2 – b 1 b 2, b = a 1 b 2 + a 2 b 1. Properties of multiplication: a) z 1 z 2 = z 2 z 1; b) z 1 (z 2 z 3) = (z 1 z 2) z 3, V) ( z 1 + z 2) z 3 = z 1 z 3 + z 2 z 3 .
Comment. A subset of the set of complex numbers is the set of real numbers, defined as complex numbers of the form ( A, 0). It can be seen that the definition of operations on complex numbers preserves the known rules for the corresponding operations on real numbers. In addition, the real number 1 = (1,0) retains its property when multiplied by any complex number: 1∙ z = z.
Definition 7.3. Complex number (0, b) is called purely imaginary. In particular, the number (0,1) is called imaginary unit and is designated by the symbol i.
Properties of the imaginary unit:
1) i∙i=i² = -1; 2) purely imaginary number (0, b) can be represented as the product of a real number ( b, 0) and i: (b, 0) = b∙i.
Therefore, any complex number z = (a,b) can be represented as: (a,b) = (a,0) + (0,b) = a + ib.
Definition 7.4. A notation of the form z = a + ib is called algebraic form writing a complex number.
Comment. Algebraic notation of complex numbers allows you to perform operations on them according to normal rules algebra.
Definition 7.5. A complex number is called the complex conjugate of z = a + ib.
3. Subtraction complex numbers is defined as the inverse operation of addition: z =(a,b) is called the difference of complex numbers z 1 =(a 1 , b 1) And z 2 =(a 2 , b 2), If a = a 1 – a 2, b = b 1 – b 2.
4. Division complex numbers is defined as the inverse operation of multiplication: number z = a + ib called the quotient of division z 1 = a 1 + ib 1 And z 2 = a 2 + ib 2(z 2 ≠ 0), if z 1 = z∙z 2 . Consequently, the real and imaginary parts of the quotient can be found from solving the system of equations: a 2 a – b 2 b = a 1, b 2 a + a 2 b = b 1.
Geometric interpretation of complex numbers.
Complex number z =(a,b) can be represented as a point on a plane with coordinates ( a,b) or a vector with origin at the origin and end at point ( a,b).
In this case, the modulus of the resulting vector is called module complex number, and the angle formed by the vector with the positive direction of the abscissa axis is argument numbers. Considering that a = ρ cos φ, b = ρ sin φ, Where ρ = |z| - module z, and φ = arg z is its argument, you can get another form of writing a complex number:
Definition 7.6. Recording type
z = ρ(cos φ + i sin φ ) (7.1)
called trigonometric form writing a complex number.
In turn, the modulus and argument of a complex number can be expressed through A And b: 
. Consequently, the argument of a complex number is not uniquely determined, but up to a term that is a multiple of 2π.
It is easy to verify that the operation of adding complex numbers corresponds to the operation of adding vectors. Let's consider the geometric interpretation of multiplication. Let then
Therefore, the modulus of the product of two complex numbers is equal to the product their modules, and the argument is the sum of their arguments. Accordingly, when dividing, the modulus of the quotient is equal to the ratio of the moduli of the dividend and the divisor, and the argument is the difference of their arguments.
A special case of the multiplication operation is exponentiation:
- Moivre's formula.
Using the obtained relations, we list the main properties of complex conjugate numbers:
