While addition and subtraction of complex numbers is more convenient to do in algebraic form, multiplication and division are easier to do using trigonometric form of complex numbers.
Let's take two arbitrary complex numbers given in trigonometric form:
Multiplying these numbers, we get:
But according to trigonometry formulas
Thus, when multiplying complex numbers, their modules are multiplied, and the arguments
fold up. Since in this case the modules are converted separately, and the arguments - separately, performing multiplication in trigonometric form is easier than in algebraic form.
From equality (1) the following relations follow:
Since division is the inverse action of multiplication, we get that
In other words, the modulus of a quotient is equal to the ratio of the moduli of the dividend and the divisor, and the argument of the quotient is the difference between the arguments of the dividend and the divisor.
Let us now dwell on the geometric meaning of multiplication of complex numbers. Formulas (1) - (3) show that to find the product, you must first increase the modulus of the number of times without changing its argument, and then increase the argument of the resulting number by without changing its modulus. The first of these operations geometrically means homothety with respect to the point O with a coefficient , and the second means a rotation relative to the point O by an angle equal to Considering here one factor is constant and the other variable, we can formulate the result as follows: formula
A complex number is a number of the form , where and are real numbers, the so-called imaginary unit. The number is called real part() complex number, the number is called imaginary part () complex number.
Complex numbers are represented by complex plane:
As mentioned above, a letter usually denotes the set of real numbers. A bunch of same complex numbers usually denoted by a “bold” or thickened letter. Therefore, the letter should be placed on the drawing, indicating the fact that we have a complex plane.
Algebraic form of a complex number. Addition, subtraction, multiplication and division of complex numbers
Addition of complex numbers
In order to add two complex numbers, you need to add their real and imaginary parts:
z 1 + z 2 = (a 1 + a 2) + i*(b 1 + b 2).
For complex numbers, the rule of the first class is valid: z 1 + z 2 = z 2 + z 1 – the sum does not change from rearranging the terms.
Subtracting Complex Numbers
The action is similar to addition, the only peculiarity is that the subtrahend must be put in brackets, and then the parentheses must be opened in the standard way with a change of sign:
z 1 + z 2 = (a 1 – a 2) + i*(b 1 – b 2)
Multiplying complex numbers
Basic equality of complex numbers:
Product of complex numbers:
z 1 * z 2 = (a 1 + i*b 1)*(a 2 + i*b 2) = a 1 *a 2 + a 1 *i*b 2 + a 2 *i*b 1 + i 2 *b 1 *b 2 = a 1 *a 2 - b 1 *b 2 +i*(a 1 *b 2 +a 2 *b 1).
Like the sum, the product of complex numbers is commutable, that is, the equality is true: .
Division of complex numbers
The division of numbers is carried out by multiplying the denominator and numerator by the conjugate expression of the denominator.
2 Question. Complex plane. Modulus and arguments of complex numbers
Each complex number z = a + i*b can be associated with a point with coordinates (a;b), and vice versa, each point with coordinates (c;d) can be associated with a complex number w = c + i*d. Thus, a one-to-one correspondence is established between the points of the plane and the set of complex numbers. Therefore, complex numbers can be represented as points on a plane. The plane on which complex numbers are depicted is usually called complex plane.
However, more often complex numbers are depicted as a vector with a beginning at point O, namely, the complex number z = a + i*b is depicted as a radius vector of a point with coordinates (a;b). In this case, the image of complex numbers from the previous example will be like this: 
The image of the sum of two complex numbers is a vector equal to the sum of the vectors representing the numbers and . In other words, when complex numbers are added, the vectors representing them are also added.
Let the complex number z = a + i*b be represented by a radius vector. Then the length of this vector is called module number z and is denoted by |z| .
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The angle formed by the radius vector of a number with the axis is called argument numbers and is denoted by arg z. The argument of the number is not determined uniquely, but to within a multiple of . However, usually the argument is specified in the range from 0 or in the range from -to. In addition, number has an undefined argument.
Using this relationship, you can find the argument of a complex number:
Moreover, the first formula is valid if the image of the number is in the first or fourth quarter, and the second, if it is in the second or third. If , then the complex number is represented by a vector on the Oy axis and its argument is equal to /2 or 3*/2.
Let's get another useful formula. Let z = a + i*b. Then ,
While addition and subtraction of complex numbers is more convenient to do in algebraic form, multiplication and division are easier to do using trigonometric form of complex numbers.
Let's take two arbitrary complex numbers given in trigonometric form:
Multiplying these numbers, we get:
But according to trigonometry formulas
Thus, when multiplying complex numbers, their modules are multiplied, and the arguments
fold up. Since in this case the modules are converted separately, and the arguments - separately, performing multiplication in trigonometric form is easier than in algebraic form.
From equality (1) the following relations follow:
Since division is the inverse action of multiplication, we get that
In other words, the modulus of a quotient is equal to the ratio of the moduli of the dividend and the divisor, and the argument of the quotient is the difference between the arguments of the dividend and the divisor.
Let us now dwell on the geometric meaning of multiplication of complex numbers. Formulas (1) - (3) show that to find the product, you must first increase the modulus of the number of times without changing its argument, and then increase the argument of the resulting number by without changing its modulus. The first of these operations geometrically means homothety with respect to the point O with a coefficient , and the second means a rotation relative to the point O by an angle equal to Considering here one factor is constant and the other variable, we can formulate the result as follows: formula
We define the product of two complex numbers similarly to the product of real numbers, namely: the product is considered as a number made up of a multiplicand, just as a factor is made up of a unit.
The vector corresponding to a complex number with modulus and argument can be obtained from a unit vector, the length of which is equal to one and the direction of which coincides with the positive direction of the OX axis, by lengthening it by a factor and rotating it in the positive direction by an angle
The product of a certain vector by a vector is the vector that will be obtained if the above-mentioned lengthening and rotation are applied to the vector, with the help of which the vector is obtained from a unit vector, and the latter obviously corresponds to a real unit.
If the moduli and arguments are complex numbers corresponding to vectors, then the product of these vectors will obviously correspond to a complex number with modulus and argument . We thus arrive at the following definition of the product of complex numbers:
The product of two complex numbers is a complex number whose modulus is equal to the product of the moduli of the factors and whose argument is equal to the sum of the arguments of the factors.
Thus, in the case when complex numbers are written in trigonometric form, we will have
Let us now derive the rule for composing a product for the case when complex numbers are not given in trigonometric form:
Using the above notation for modules and arguments of factors, we can write
according to the definition of multiplication (6):
and finally we get
In case the factors are real numbers and the product is reduced to the product aag of these numbers. In the case of equality (7) gives
i.e. the square of the imaginary unit is equal to
Calculating sequentially the positive integer powers, we obtain
and in general, with any overall positive
The multiplication rule expressed by equality (7) can be formulated as follows: complex numbers must be multiplied like letter polynomials, counting
If a is a complex number, then the complex number is said to be conjugate to a, and is denoted by a. According to formulas (3) we have from equality (7) it follows
and consequently,
that is, the product of conjugate complex numbers is equal to the square of the modulus of each of them.
Let us also note obvious formulas
From formulas (4) and (7) it immediately follows that addition and multiplication of complex numbers obey the commutative law, that is, the sum does not depend on the order of the terms, and the product does not depend on the order of the factors. It is not difficult to verify the validity of the combinational and distributive laws, expressed by the following identities:
We leave it to the reader to do this.
Note, finally, that the product of several factors will have a modulus equal to the product of the moduli of the factors, and an argument equal to the sum of the arguments of the factors. Thus, the product of complex numbers will be equal to zero if and only if at least one of the factors is equal to zero.
The product of two complex numbers is similar to the product of two real numbers, namely: the product is considered as a number made up of a multiplicand, just as a factor is made up of a unit. The vector corresponding to a complex number with modulus r and argument j can be obtained from a unit vector whose length is equal to one and whose direction coincides with the positive direction of the OX axis, by lengthening it by r times and rotating it in the positive direction by an angle j. The product of a certain vector a 1 by a vector a 2 is the vector that is obtained if we apply lengthening and rotation to the vector a 1, with the help of which the vector a 2 is obtained from a unit vector, and the latter obviously corresponds to a real unit. If (r 1 , ? 1), (r 2 , ? 2) are the modules and arguments of complex numbers corresponding to the vectors a 1 and a 2, then the product of these vectors will obviously correspond to a complex number with the module r 1 r 2 and argument (j 1 + j 2). Thus, the product of two complex numbers is a complex number whose modulus is equal to the product of the moduli of the factors and whose argument is equal to the sum of the arguments of the factors.
In the case where complex numbers are written in trigonometric form, we have
r 1 (cos? 1 + i sin? 1) * r 2 (cos? 2 + i sin? 2) = r 1 r 2.
In the case (a 1 + b 1 i)(a 2 + b 2 i) = x + yi, using the notation of modules and arguments of factors, we can write:
a 1 = r 1 cos? 1 ; b 1 = r 1 sin? 1 ; a 2 = r 2 cos? 2 ; b 2 = r 2 sin? 2 ;
according to the definition of multiplication:
x = r 1 r 2 cos(? 1 + ? 2); y = r 1 r 2 sin(? 1 + ? 2),
x = r 1 r 2 (cos? 1 cos? 2 - sin? 1 sin? 2) = = r 1 cos? 1 r 2 cos? 2 - r 1 sin? 1 r 2 sin? 2 = a 1 a 2 - b 1 b 2
y = r 1 r 2 (sin? 1 cos? 2 + cos? 1 sin? 2) = = r 1 sin? 1 r 2 cos? 2 + r 1 cos? 1 r 2 sin? 2 = b 1 a 2 + a 1 b 2,
and finally we get:
(a 1 + b 1 i)(a 2 + b 2 i) = (a 1 a 2 - b 1 b 2) + (b 1 a 2 + a 1 b 2)i.
In the case b 1 = b 2 = 0, the factors are real numbers a 1 and a 2 and the product is reduced to the product a 1 a 2 of these numbers. When
a 1 = a 2 = 0 and b 1 = b 2 = 1,
equality (a 1 + b 1 i)(a 2 + b 2 i) = (a 1 a 2 - b 1 b 2) + (b 1 a 2 + a 1 b 2)I gives: i???i = i 2 = -1, i.e. the square of the imaginary unit is -1. Calculating sequentially the positive integer powers i, we obtain:
i 2 = -1; i 3 = -i; i 4 = 1; i 5 = i; i 6 = -1; ...
and, in general, for any positive k:
i 4k = 1; i 4k+1 = i; i 4k+2 = -1; i 4k+3 = -i
The multiplication rule expressed by the equality (a 1 + b 1 i)(a 2 + b 2 i) = (a 1 a 2 - b 1 b 2) + (b 1 a 2 + a 1 b 2)I can be formulated as follows: complex numbers must be multiplied like alphabetic polynomials, counting i 2 = -1.
From the above formulas it immediately follows that addition and multiplication of complex numbers obey the commutative law, i.e. the sum does not depend on the order of the terms, and the product does not depend on the order of the factors. It is not difficult to verify the validity of the combinational and distributive laws, expressed by the following identities:
(? 1 + ? 2) + ? 3 = ? 1 + (? 2 + ? 3); (? 1 ? 2)? 3 = ? 1 (? 2 ? 3); (? 1 + ? 2)? = ? 1 ? + ? 2 ? .
The product of several factors will have a modulus equal to the product of the moduli of the factors, and an argument equal to the sum of the arguments of the factors. Thus, the product of complex numbers will be equal to zero if and only if at least one of the factors is equal to zero.
Example: given complex numbers z 1 = 2 + 3i, z 2 = 5 - 7i. Find:
a) z 1 + z 2; b) z 1 - z 2; c) z 1 z 2 .
a) z 1 + z 2 = (2 + 3i) + (5 - 7i) = 2 + 3i + 5 - 7i = (2 + 5) + (3i - 7i) = 7 - 4i; b) z 1 - z 2 = (2 + 3i) - (5 - 7i) = 2 + 3i - 5 + 7i = (2 - 5) + (3i + 7i) = - 3 + 10i; c) z 1 z 2 = (2 + 3i)(5 - 7i) = 10 - 17i + 15i - 21i 2 = 10 - 14i + 15i + 21 = (10 + 21) + (- 14i + 15i) = 31 + i (here it is taken into account that i 2 = - 1).
Example: follow these steps:
a) (2 + 3i) 2 ; b) (3 - 5i) 2 ; c) (5 + 3i) 3 .
a) (2 + 3i) 2 = 4 + 2Х2Ч3i + 9i 2 = 4 + 12i - 9 = - 5 + 12i; b) (3 - 5i) 2 = 9 - 2Х3Ч5i + 25i 2 = 9 - 30i - 25 = - 16 - 30i; c) (5 + 3i) 3 = 125 + 3Х25Ч3i + 3Ч5Ч9i 2 + 27i 3 ; since i 2 = - 1, and i 3 = - i, we get (5 + 3i) 3 = 125 + 225i - 135 - - 27i = - 10 + 198i.
Example: perform actions
a) (5 + 3i)(5 - 3i); b) (2 + 5i)(2 - 5i); c) (1 + i)(1 - i).
a) (5 + 3i)(5 - 3i) = 5 2 - (3i) 2 = 25 - 9i 2 = 25 + 9 = 34; b) (2 + 5i)(2 - 5i) = 2 2 - (5i) 2 = 4 + 25 = 29; c) (1 + i)(1 - i) = 1 2 - i 2 = 1 + 1 = 2.