Transcription of COMPLEX NUMBERS - Number theory
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COMPLEX NUMBERS - Number theory
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 . Noting that{0}+i{0}={0}, gives theuseful special case is{x}+i{y}={0} x= 0 andy= 0,ifxandyare real sum and product of two real COMPLEX NUMBERS are also real complexnumbers:{x}+{y}={x+y},{x}{y}={xy} .
5.2. CALCULATING WITH COMPLEX NUMBERS 91 The set C of complex numbers forms a field under the operations of matrix addition and multiplication.
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