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 .
The sum and product of two real complex numbers are also real complex numbers: {x}+{y} = {x+y}, {x}{y} = {xy}. Also, as real complex numbers are scalar matrices, their arithmetic is very simple. They form a field under the operations of matrix addition and multiplication. The additive identity is {0}, the additive inverse of {x} is
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