Transcription of COMPLEX NUMBERS
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COMPLEX NUMBERS
Chapter 5 COMPLEX Constructing the COMPLEX numbersOne way of introducing the fieldCof COMPLEX NUMBERS is via the arithmeticof 2 2 COMPLEX number is a matrix of the form x yyx ,wherexandyare real NUMBERS of the form x00x are scalar matrices and are calledrealcomplex NUMBERS and are denoted by the symbol{x}.The real COMPLEX NUMBERS {x}and{y}are respectively called therealpartandimaginary partof the COMPLEX number x yyx .The COMPLEX number 0 11 0 is denoted by the have the identities x yyx = x00x + 0 yy0 = x00x + 0 11 0 y00y ={x}+i{y},i2= 0 11 0 0 11 0 = 1 00 1 ={ 1}.8990 CHAPTER 5. COMPLEX NUMBERSC omplex NUMBERS of the formi{y}, whereyis a non zero real number, arecalledimaginary two COMPLEX NUMBERS are equal, we can equate their real andimaginaryparts:{x1}+i{y1}={x2}+i{y2} x1=x2andy1=y2,ifx1, x2, y1, y2are real NUMBERS .
plex number has a square root: THEOREM 5.2.1 If w is a non–zero complex number, then the equation z2 = w has a so-lution z ∈ C. Proof. Let w = a+ib, a, b ∈ R. Case 1. Suppose b = 0. Then if a > 0, z = √ a is a solution, while if a < 0, i √ −a is a solution. Case 2. Suppose b 6= 0. Let z = x + iy, x, y ∈ R. Then the equation z2 = w ...
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