# Complex numbers

*Imaginary and complex numbers. Abscissa and ordinate of a complex*

number. Conjugate complex numbers. Pure imaginary number. Pure

imaginary number. Operations with complex numbers. Geometric

representation of complex numbers. Complex plane. Modulus and

argument of a complex number. Trigonometric form of a complex

number. Operations with complex numbers in the trigonometric form.

Moivre's formula.

number. Conjugate complex numbers. Pure imaginary number. Pure

imaginary number. Operations with complex numbers. Geometric

representation of complex numbers. Complex plane. Modulus and

argument of a complex number. Trigonometric form of a complex

number. Operations with complex numbers in the trigonometric form.

Moivre's formula.

The initial information about
**
imaginary
**
and

**numbers has been presented above, in the section “Imaginaryand complex numbers”. A necessity of these new kind numbers has appeared at solving of quadratic equations in the case of**

*complex**D*< 0

(
*
D
*
– a discriminant of a quadratic equation). During a long time these numbers had no physical applications, therefore they were called “imaginary” numbers. But now these numbers have various applications in different physical and technical fields, such as: electrical engineering, hydro- and aerodynamics, theory of elasticity and others.

**
Complex numbers
**
are written in the shape:

*a+ bi*. Here

*a*and

*b*–

*real numbers*, and

*i*– an

*imaginary unit, i.e.*

*i*

^{ 2 }

*=*–1.

*A real number*

*a*is called an

*abscissa*of complex number

*a+ bi*, and

*b –*an

*ordinate*of complex number

*a+ bi.*Two

complex numbers
*
a+ bi
*
and
*
a – bi
*
are called the
*
conjugate complex
*
numbers.

**
**

**
Main agreements:
**

1. A real number
*
a
*
can also
be written in the shape
of a complex number:
*
a+
*
0
*
i
*
or
*
a
–
*
0
*
i
*
. For example, the records 5 + 0
*
i
*
and 5 – 0
*
i
*
mean
the same real number 5 .

2. A complex number 0
*
+
*
*
bi
*
is called a
*
pure imaginary number.
*
The record
*
bi
*
means the same as 0
*
+
*
*
bi
*
.

3. Two complex numbers
*
a+ bi
*
and
*
c+ di
*
are considered as equal ones, if
*
a=c
*
and
*
b=d
*
.
Otherwise, the complex numbers aren’t equal.

**
Addition.
**
A sum of complex numbers

*a+ bi*and

*c+ di*is called a complex number (

*a+ c*) + (

*b+ d*)

*i.*So,

*at addition of complex numbers their*

*abscissas and ordinates are added separately.*This definition corresponds to the rules of operations at usual polynomials.

**
Subtraction.
**
A difference of two complex numbers

*a+ bi*( a minuend ) and

*c+ di*( a subtrahend ) is called a complex number (

*a – c*) + (

*b – d*)

*i.*So,

*at subtraction of two complex numbers their abscissas and ordinates are subtracted separately.*

**
Multiplication.
**
A product of complex numbers

*a+ bi*and

*c+ di*is called a complex number: (

*ac – bd*) + (

*ad + bc*)

*i.*

This definition follows from two requirements:

1) the numbers
*
a+ bi
*
and
*
c+ di
*
must be multiplied as algebraic
bin
omials,

2) a number
*
i
*
has a main property:
*
i
*
²
=
*
–
*
1.

E x a m p l e . (
*
a+ bi
*
)(
*
a – bi
*
)
*
= a
*
²
*
+ b
*
²
*
.
*
Hence it follows, that

*
a product
*
*
of two conjugate complex numbers is a real positive number!
*

*
*

**
Division.
**

*To divide a complex number*

*a+ bi*( a dividend ) by another

*c+ di*( a divisor )

*means to find the third number*

*e+ f i*( a quotient ), which being multiplied by the divisor

*c+ di*, results the dividend

*a+ bi.*If divisor isn’t equal to zero, then division is always valid.

E x a m p l e . Find ( 8 +
*
i
*
) : ( 2 – 3
*
i
*
) .

S o l u t i o n . Rewrite this quotient as a fraction:

Multiplying its numerator
*
*
and denominator by 2 + 3
*
i
*
and

executing all operations, we’ll receive:

**
Geometric representationof complex numbers.
**
Real numbers are represented by points in a numerical line:

Here a point
*
A
*
means a number –3, a point
*
B
*
– a number 2, and
*
O
*
– zero.
In contrast this complex numbers are represented by points in a “numerical”
( coordinate ) plane. For this we select a system of rectangular ( Cartesian )
coordinates with the same scale in both axes. Then, a complex number
*
a+ bi
*
will be represented by point
*
P
*
with abscissa
*
a
*
and ordinate
*
b
*
( see figure ).
This coordinate system is called a
**
complex plane
**
.

___

**
Modulus
**
of a complex number is a length of vector

*OP*, representing this complex number in a coordinate(

*complex*) plane. Modulus of complex number

*a+ bi*is signed as |

*a+ bi*| or by letter

*r*and equal to :

Conjugate complex numbers havethe same modulus.
*
*

___

**
Argument
**
of a complex number is the angle
between

*x*-axis and vector

*OP*, representing this complex number.

*
*

**
Trigonometric form of a complex number.
**
Abscissa

*a*and ordinate

*b*of the complex number

*a + bi*can be expressed by its modulus

*r*and argument

**:**

**
Operations with complex numbers, represented in the trigonometric form.
**

This is the famous
*
*
*
Moivre’s formula.
*

Here
*
k
*
is any integer. To receive
*
n
*
different values of the
*
n
*
-th degree root of
*
z
*
it’s necessary to give
*
n
*
consecutive values for
*
k
*
( e.g.,
*
k
*
= 0, 1, 2,…,
*
n
*
– 1) .