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**Unformatted text preview: **Chapter 5 COMPLEX NUMBERS 5.1 Constructing the complex numbers One way of introducing the Feld C of complex numbers is via the arithmetic of 2 × 2 matrices. DEFINITION 5.1.1 A complex number is a matrix of the form b x − y y x B , where x and y are real numbers. Complex numbers of the form b x x B are scalar matrices and are called real complex numbers and are denoted by the symbol { x } . The real complex numbers { x } and { y } are respectively called the real part and imaginary part of the complex number b x − y y x B . The complex number b − 1 1 B is denoted by the symbol i . We have the identities b x − y y x B = b x x B + b − y y B = b x x B + b − 1 1 Bb y y B = { x } + i { y } , i 2 = b − 1 1 Bb − 1 1 B = b − 1 − 1 B = {− 1 } . 89 90 CHAPTER 5. COMPLEX NUMBERS Complex numbers of the form i { y } , where y is a non–zero real number, are called imaginary numbers . If two complex numbers are equal, we can equate their real and imaginary parts: { x 1 } + i { y 1 } = { x 2 } + i { y 2 } ⇒ x 1 = x 2 and y 1 = y 2 , if x 1 , x 2 , y 1 , y 2 are real numbers. Noting that { } + i { } = { } , gives the useful special case is { x } + i { y } = { } ⇒ x = 0 and y = 0 , if x and y are 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 Feld under the operations of matrix addition and multiplication. The additive identity is { } , the additive inverse of { x } is {− x } , the multiplicative identity is { 1 } and the multiplicative inverse of { x } is { x − 1 } . Consequently { x } − { y } = { x } + ( −{ y } ) = { x } + {− y } = { x − y } , { x } { y } = { x }{ y } − 1 = { x }{ y − 1 } = { xy − 1 } = b x y B . It is customary to blur the distinction between the real complex number { x } and the real number x and write { x } as x . Thus we write the complex number { x } + i { y } simply as x + iy . More generally, the sum of two complex numbers is a complex number: ( x 1 + iy 1 ) + ( x 2 + iy 2 ) = ( x 1 + x 2 ) + i ( y 1 + y 2 ); (5.1) and (using the fact that scalar matrices commute with all matrices under matrix multiplication and {− 1 } A = − A if A is a matrix), the product of two complex numbers is a complex number: ( x 1 + iy 1 )( x 2 + iy 2 ) = x 1 ( x 2 + iy 2 ) + ( iy 1 )( x 2 + iy 2 ) = x 1 x 2 + x 1 ( iy 2 ) + ( iy 1 ) x 2 + ( iy 1 )( iy 2 ) = x 1 x 2 + ix 1 y 2 + iy 1 x 2 + i 2 y 1 y 2 = ( x 1 x 2 + {− 1 } y 1 y 2 ) + i ( x 1 y 2 + y 1 x 2 ) = ( x 1 x 2 − y 1 y 2 ) + i ( x 1 y 2 + y 1 x 2 ) , (5.2) 5.2. CALCULATING WITH COMPLEX NUMBERS 91 The set C of complex numbers forms a Feld under the operations of matrix addition and multiplication. The additive identity is 0, the additive inverse of x + iy is the complex number ( − x ) + i ( − y ), the multiplicative identity is 1 and the multiplicative inverse of the non–zero complex number...

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