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ma265 lecture notes

ma265 lecture notes - Chapter 5 COMPLEX NUMBERS 5.1...

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Chapter 5 COMPLEX NUMBERS 5.1 Constructing the complex numbers One way of introducing the field 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 bracketleftbigg x y y x bracketrightbigg , where x and y are real numbers. Complex numbers of the form bracketleftbigg x 0 0 x bracketrightbigg 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 bracketleftbigg x y y x bracketrightbigg . The complex number bracketleftbigg 0 1 1 0 bracketrightbigg is denoted by the symbol i . We have the identities bracketleftbigg x y y x bracketrightbigg = bracketleftbigg x 0 0 x bracketrightbigg + bracketleftbigg 0 y y 0 bracketrightbigg = bracketleftbigg x 0 0 x bracketrightbigg + bracketleftbigg 0 1 1 0 bracketrightbiggbracketleftbigg y 0 0 y bracketrightbigg = { x } + i { y } , i 2 = bracketleftbigg 0 1 1 0 bracketrightbiggbracketleftbigg 0 1 1 0 bracketrightbigg = bracketleftbigg 1 0 0 1 bracketrightbigg = {− 1 } . 89
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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 { 0 } + i { 0 } = { 0 } , gives the useful special case is { x } + i { y } = { 0 } ⇒ 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 field under the operations of matrix addition and multiplication. The additive identity is { 0 } , 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 } = braceleftbigg x y bracerightbigg . 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)
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5.2. CALCULATING WITH COMPLEX NUMBERS 91 The set C of complex numbers forms a field 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 x + iy is the complex number u + iv , where u = x x 2 +
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