Web Notes on Complex Number part 1 for NDA Exam Preparation

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Complex Number part 1

Mathematics (NDA) Complex Numbers
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    Complex Number part 1

    Complex Numbers 

    Definition - A number of the form x + iy , where x and y are real numbers, is called a complex number. 

    Usually, we denote it by z. Thus, we write  

                                              z = x + iy 

    Here, z is called a complex variable. 

    The real numbers x and y are called the real and imaginary parts of z, respectively. 

    Customarily, we write  

    Re(z) = x    

    Im(z) = y  

    #Note  

    (1) If y= 0, then z = x is purely a real number. 

    (2) If x = 0, then z = iy  is purely an imaginary number. 

    (3) Here, the symbol ‘i’ is such that i²= -1 and is called the imaginary unit. 

    (4) The set of all complex numbers is denoted by the symbol C. 

    Examples

    (1)Find the value of i101

    Sol.    i101 = i4×(25+1) 

                       =( i4)25 ×i 

                       = (1)25 ×i 

                       = 1×i 

                       = i 

    (2)Find the value of √-25 ×√-49 ? 

    Sol.   √-25 ×√-49 = √i25 ×√i49 

                                = (5i)×(7i) 

                                =35 i2

                                = -35 

    #Basic Properties  
     
    (1)Equality of Complex Numbers 
    Two complex numbers are said to be equal if their real and imaginary parts are separately equal. 
    Thus , z = x + iy ,  z’ = x’ + iy’ are said to be equal , if
           x= x’ and y = y’ 
    Thus , x + iy = 0 , if x = 0 and y = 0 . 
     
    (2)Addition and Subtraction of two Complex Numbers 
    Let z = x + iy  and z’ = x’ + iy’ be two complex Numbers .Then their sum is denoted by z + z’ and is defined by  
      z + z’ = ( x + x’) + i(y + y’) 
    and their difference is denoted by z – z’ and is defined by  
         z – z’ = ( x- x’) + i(y – y’) 
     
    (3)Multiplication of two Complex numbers 
    Let z1 = x1 + iy1 and z2 = x2 + iy2 be two complex numbers. Then their product is denoted by z1z2 and is defined by  
       z1z2 = ( x1x2 – y1y2 ) + i(x1y2 + x2y1 ) 
     
    (4)Commutative law for addition 
    Let   z1 = x1 + iy1 and z2 = x2 + iy2 be any two complex numbers.Then  
                  z1 + z2 = z2 + z1 
     
    (5) Associative law for addition 
    Let z1 , z2 and z3 be three complex numbers , then  
          z1 + ( z2 + z3 ) = (z1 + z2 ) + z3 
     
    (6)Additive inverse 
    The number -x + i(-y) is called the additive inverse of the complex number x + iy . 
     
    (7)Division of two Complex numbers 
      (x1 +i y1)/(x2 + iy2) = (x1x2 + y1y2)/(x22 + y22)    + i(y1x2 – x1y2)/(x22+ y22) 
     
    (8) (a)If z = z’ , z is purely real . 
         (b)If z = -z’ , z is purely imaginary . 
     
    (9) (z’)’ = z  
     
    (10) (a)Re(z) = Re(z’) = (z +z’)/2 
           (b)Im(z)=(z -z’)/2 
     
     
    #Complex conjugate 
    The complex conjugate or simply conjugate of a complex number z = x + iy is the complex number  
                  z’ = x – iy . 
     
    #Examples  
    If 2y + (3x – y )i = (5 - 2i) , then find the value of x and y . 
    Sol.  2y +(3x – y )i = (5 - 2i) 
          2y = 5 and 3x – y = -2 
          y =5/2 and 3x – y = -2 
          y = 5/2 and x = 1/6 
          x = 1/6 and y = 5/2 
     
          2.Find the sum of (3 - 4i) and (-5 + 7i) . 
          Sol.  (3 - 4i) + ( -5 + 7i) = (3+(-5)) + i (-4 +7) 
                                                = ( -2 + 3i) 
     
    #Modulus of a Complex Number 
    If x and y are real numbers,then the non – negative real number √x² + y² is called the modulus of the complex number z= x + iy. 
    The modulus of the complex number z is denoted by |z|.  
    Thus, 
                 |z| = | x + iy |  
                     = √x²+ y²
     
    #Note 
    (1)If z is a complex number, then |z|=|z’| 
    (2)If z1 and z2 are two complex numbers, then  
        |z1 z2| = |z1| |z2| 
     
    Argument of a Complex number 
    Argument of a complex number z = x + iy is given by  
        θ = tan¯¹( x/y) 
    Or   tanθ = x / y  
    Also, it is denoted by?θ and is defined by  
      arg(z) = θ 
                = tan¯¹ ( x/y) 
    We write, 
             θ = arg(z) = amp(z) 
     
    Basic properties 
    (1)The argument of the product of two complex numbers is equal to the sum of their arguments.
    (2)The argument of the quotient of two complex numbers is equal to the difference in their arguments. 
    (3)If z1 = z2 , then arg(z1) = arg(z2) 
    (4)If arg(z) = π/2 or   -π /2 , z is purely imaginary. 
    (5)If arg(z) = 0 or  π , z is purely real. 
     

    #Cube root of unity

    x = (1)1?3 

    or x3 = 1 

    Or  ( x3 -1 ) = 0 

    Or    (x -1)(x2 + x + 1 ) = 0 

    Thus, x = 1 ,( -1 + i√3)/2 , (-1 – i√3)/2 

                  = 1 , ω, ω2 

    #Properties

    1. 1 + ω + ω2= 1 
    1. ω3= 1 
     
     
    #IIlustrative Examples 
     
    (1)Find the modulus of the complex number  ((1-i))/((1+i))?
     
    Sol. Given,  ((1-i))/((1+i))
                   = ((1-i)²)/((1-i²))
     
                   = (-2i)/2
                   = -i 
     
    (2)Find the argument of the following complex number ((1-i))/((1+i))
     
    Sol.    arg{((1-i))/((1+i))}= arg(-i) 
                        = tan¯¹ ( -1/0) 
                        = - π /2 
     
    (3)If ,z= 2 + 3i and z’= 5 + 4i , then find (z + z’). 
    Sol.  z + z’ = ( 2+3i) + ( 5+4i) 
                      = (2+5) + i(3+4) 
                       = 7+7i 
     
    (5)If ,z= 1 + 2i and z’ = 2+ 3i , then find zz’. 
    Sol. zz’= (1+2i)(2+3i) 
                 = (1×2 – 2×3) + i (2×2 + 1×3) 
                 = (2-6) + i (4+3) 
                  = -4 + 7i 
     
    (5)Solve x2+ 2 = 0 . 
    Sol. We have ,
                       x2+ 2 = 0 
               Or    x2 = -2 
               Or    x = +√-2 , -√-2 
               Or    x = +i√2 , - i√2 
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