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PROPERTIES OF COMPLEX NUMBER

 T here are some property of complex number under     +,     −,    ×,    ÷      we prove these properties any value of complex number we take variable    a₁, b₁,c₁,d₁,e₁, f₁ ADDITION ( + ) complex number multiplication ( i ): closure property (+)    when         (a₁+b₁) , ( c₁ + d₁ )       ∈  ℂ =    (a₁+  b₁)+  ( c₁  + d₁ )  = (a₁ + c₁ ) +  (b₁+ d₁ )      ∈ ℂ    (say closed   +  in complex number )  ( ii ): associative property  ( + ) once        (a₁+b₁) ,  ( c₁ ,d₁ ) , (e₁  , f₁ )     ∈  ℂ     [ (  (a₁+b₁ ) +  ( c₁ +d₁ )     ] + (e₁  + f₁ )   =   ( (a₁+ c₁ ) + e₁        (  b₁ + d₁  )   +  f₁  = [ a₁ +  (    c₁ +  e₁  ) ,  b₁   + ( d₁   +  f₁) ]   as a result of addition is associative in R what is complex plane  =     (a₁ , b₁) +(   c₁ +  e₁ ) + ( d₁    +  f₁)  =        (a₁ , b₁) + [(   c₁ +  d₁ ) + ( e₁    +  f₁) ]  thence verify (iii) additive identity ( + )          ∀  ( a₁ ,b₁) ∈ ℂ   we have ( 0  , 0 ) ∈ ℂ  such that ( a₁ , b₁) + ( 0 , 0 )   = ( a₁+0 , b₁+ 0)   = ( a₁ , b₁)  this
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LAW OF DITRIBUTION (LEFT AND RIGHT) IN COMPLEX NUMBER

    we have three complex number Z1,Z2,Z3  =      (a₁+b₁) ,  ( c₁ ,d₁ ) , (e₁  , f₁ )     ∈  ℂ THEN law of distribution addition over multiplication                         (a₁, b₁) .[  ( c₁ ,d₁ ) + (e₁  , f₁ )]  =    (a₁, b₁).  ( c₁ ,d₁ ) +  (a₁, b₁) . (e₁  , f₁ )     L.H.S        =       (a₁, b₁) .[  ( c₁ ,d₁ ) + (e₁  , f₁ )]     LEFT IAW OF DISTRIBUTION                                                 =    (a₁, b₁) (    c₁ + e₁ , d₁+ f₁ )                                   multiplication property of two numbers in complex number                                        = [ a₁ (  c₁ + e₁) -  b₁(  d₁+ f₁) , a₁ ( d₁+ f₁ ) +   b₁(   c₁ + e₁ ) ] by multiplying                                        =  a₁  c₁ + a₁ e₁ -  b₁  d₁ -  b₁  f₁     ,     a₁ d₁+ a₁  f₁  +   b₁   c₁ + b₁   e₁......................1       R.H.S         =       (a₁, b₁) .  ( c₁ ,d₁ ) +  (a₁, b₁) . (e₁  , f₁ )   RIGHT LAW OF DISTRIBUTION                                                       multiplication property in c
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WHAT IS IOTA AND POWER OF IOTA

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 IOTA  describes   the real quantity of complex form to solve complex number equation and equation may be trigonometry, algebra equation, quadric equation and any equation of complex number   mathematically IOTA  denoted by the sign  I or  ι  complex number Z  =  a +bi   where a and b are real number  iota is the imaginary  POWER OIOTA   when the power of iota change it describes the real number and complex number , usually real value of  IOTA  is    ι =  √⁻1 POWER OF IOTA IN DIAGRAM //// CUBE ROOT OF UNITY POWER? solve iota power EXAMPLE  if  we   find the iota power 8 SOLUTION                                   ᵢ⁸ = (ᵢ₄)² =(1)²=1     any power of iota divided by 4 to             EXAMPLE: 2                                    solve     ι⁹ SOLUTION                                                ι⁹  =        ι × ι⁸                                            =     ι ×( ι⁴ )²                                            =    ι × (1)2                                             =    ι × 1  
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UNION AND PROPERTIES OF UNION

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DEFINITION OF UNION The set of all elements of SET A and SET B is called the union of sets where repeating element eliminates symbolically we denote the sign " U  union of two set A and B is denoted by the sign AUB says set of all element A and B Words union mean together and we read AUB A union B AUB = { set of all elements, such that, all elements of A and B } mathematically we write AUB = { y: y is part of A or part of B } if we have set A = { 1,2,3,4,5,6,7 } and B = { 5,6,7,8,9 } AUB ={ 1,2,3,4,5,6,7 } U { 5,6,7,8,9 } AUB = { 1,2,3,4,5,6,7,8,9 } A U B diagram is shown as we take an example to more explanation Example 1 :   if we have set S = { 15,16 ,17,20,25} T = { 12, 13,14 }   S U T = { 12,13,14,15,16,17,20,25} Example 2   R = { 0,1,7,9,11.17 } and S = { 5,7,,11,15 }     R U S = { 0.1,7,9,11,15,17 }           we cannot repeat element in union of set EXAMPLE 3     if we have P se
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what is complex number addition and multiplication

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The addition of real and imaginary parts of the number is called a complex number a complex number is the builder of explained math to different group algebra, Trigonometry, set theory electronic, magnetics field, current measuring, DC and AC, construction building, etc. basically, complex number in the form of a real number, imaginary number a + bi if the real part is zero then the resultant value is imaginary if the imaginary part is zero then the resultant value is a  real number complex number diagram is shown below    what is complex plane CUBE ROOT OF UNITY PIWER ??? THE COMPLEX NUMBER PLAN SHOWN BELOW 👩                                                                    z = a + bi where a is the real number and b is also a real number                                                                  I form imaginary unit number √ -1                                                                            can be a value of a = 0  or  and b = o then the complex number in the f
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REAL NUMBER AND SET OF REAL NUMBER

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SET OF   .. { NATURAL ,WHOLE ,INTEGERS,RATIONAL,IRRATIONAL},NUMBER  IS CALLED REAL NUMBER .   if real number  +   iota  then the number  is called  complex number  "  complex number  "   is   briefly explain  in  another post MULTIPLICATIVE PROPERTY OF REAL NUMBER???  ''''    EXPLANATION''''                                                                                                                                                                                                DIFFERENT  KIND OF REAL NUMBER   SET                                                                            SET OF REAL NUMBER rational and irrational   number  of the set  is called set  of a real number ( it can be  set  of  natural number whole number, integers,  rational number, irrational number  or  all number in one set  or chosen number in one set ) EXAMPLE              A   =   {    -1 , -2 ,-3 ,-5 }    B   =   {   1/2 , 1/3 , 1/6  }    C =  { 1 , 1/2 , 2
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REAL NUMBER PROPERTIES

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POWER OF IOTA  HOW SOLVE USE OF REAL NUMBER PROPERTIES            natural    number properties EX. 1                                                              5 + 8 = 8 + 5                (COMMUTATIVE PROPERTY} EX. 2                                                              (b +1) +3 /4 = b +( 1 +3 /4 )           (associative property) b is any number prove an associative property of real number           a =3 , b = 5 , c =7                                                                                      ( 3+ 5 ) +7 = 3 +( 5 +7 )                                                                                                15 = 15 EX. 3                                                         90 + 0 = 90 =0 + 90                ( 0 is additive identity)  EX. 4                                                    800 * 1 = 800 =1 * 800                     ( 1 is multip
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