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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.


The initial information about imaginary and complex 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  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 breal 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 binomials,

  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 – 3i ) .

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

                       Multiplying its numerator and denominator by  2 + 3and

                      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) .

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