[<< wikibooks] Arithmetic/Types of Numbers/Complex Number
== Complex Number ==
Complex Number is a number that can be expressed mathematically as a sum of a Real Number and an Imaginary Number

  
    
      
        Z
        =
        A
        +
        j
        B
      
    
    {\displaystyle Z=A+jB}
  

  
    
      
        Z
        =
        
          |
        
        Z
        
          |
        
        ∠
        θ
      
    
    {\displaystyle Z=|Z|\angle \theta }
  

  
    
      
        
          |
        
        Z
        
          |
        
        =
        
          
            
              A
              
                2
              
            
            +
            
              B
              
                2
              
            
          
        
      
    
    {\displaystyle |Z|={\sqrt {A^{2}+B^{2}}}}
  

  
    
      
        θ
        =
        T
        a
        
          n
          
            −
          
        
        1
        
          
            B
            A
          
        
      
    
    {\displaystyle \theta =Tan^{-}1{\frac {B}{A}}}
  


== Complex Conjugate Number ==

  
    
      
        Z
        =
        A
        −
        j
        B
      
    
    {\displaystyle Z=A-jB}
  

  
    
      
        Z
        =
        
          |
        
        Z
        
          |
        
        ∠
        −
        θ
      
    
    {\displaystyle Z=|Z|\angle -\theta }
  

  
    
      
        
          |
        
        Z
        
          |
        
        =
        
          
            
              A
              
                2
              
            
            +
            
              B
              
                2
              
            
          
        
      
    
    {\displaystyle |Z|={\sqrt {A^{2}+B^{2}}}}
  

  
    
      
        θ
        =
        −
        T
        a
        
          n
          
            −
          
        
        1
        
          
            B
            A
          
        
      
    
    {\displaystyle \theta =-Tan^{-}1{\frac {B}{A}}}
  


== Rules ==
If there are two Complex Numbers 

  
    
      
        
          Z
          
            1
          
        
        =
        A
        +
        j
        B
      
    
    {\displaystyle Z_{1}=A+jB}
  

  
    
      
        
          Z
          
            2
          
        
        =
        C
        +
        j
        D
      
    
    {\displaystyle Z_{2}=C+jD}
  (A + jB) + (C + jD) = (A + C) + j (B + D)
(A + jB) - (C + jD) = (A - C) + j (B - D)
(A + jB) x (C + jD) = (AC + BD) + j (AD + BC)

  
    
      
        
          
            
              (
              A
              +
              j
              B
              )
            
            
              (
              C
              +
              j
              D
              )
            
          
        
      
    
    {\displaystyle {\frac {(A+jB)}{(C+jD)}}}
   = 
  
    
      
        
          
            
              (
              A
              +
              j
              B
              )
              (
              C
              −
              j
              D
              )
            
            
              (
              C
              +
              j
              D
              )
              (
              C
              −
              j
              D
              )
            
          
        
      
    
    {\displaystyle {\frac {(A+jB)(C-jD)}{(C+jD)(C-jD)}}}
   = 
  
    
      
        
          
            
              (
              A
              C
              +
              B
              D
              )
              +
              j
              (
              B
              C
              −
              A
              D
              )
            
            
              
                C
                
                  2
                
              
              +
              
                D
                
                  2
                
              
            
          
        
      
    
    {\displaystyle {\frac {(AC+BD)+j(BC-AD)}{C^{2}+D^{2}}}}
  
  
    
      
        
          Z
          
            1
          
        
        ×
        
          Z
          
            2
          
        
        =
        
          |
        
        
          Z
          
            1
          
        
        
          |
        
        
          |
        
        
          Z
          
            2
          
        
        
          |
        
        ∠
        (
        
          θ
          
            1
          
        
        +
        
          θ
          
            2
          
        
        )
      
    
    {\displaystyle Z_{1}\times Z_{2}=|Z_{1}||Z_{2}|\angle (\theta _{1}+\theta _{2})}
  

  
    
      
        
          Z
          
            1
          
        
        
          /
        
        
          Z
          
            2
          
        
        =
        
          
            
              
                |
              
              
                Z
                
                  1
                
              
              
                |
              
            
            
              
                |
              
              
                Z
                
                  2
                
              
              
                |
              
            
          
        
        ∠
        (
        
          θ
          
            1
          
        
        −
        
          θ
          
            2
          
        
        )
      
    
    {\displaystyle Z_{1}/Z_{2}={\frac {|Z_{1}|}{|Z_{2}|}}\angle (\theta _{1}-\theta _{2})}