3_5_ex_sol

3_5_ex_sol - Section 3.5 Example Solutions Instructor Ms...

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Instructor: Ms. Hoa Nguyen ([email protected]) 1 Reminder of the Complex Numbers Conjugate: If z = a + bi , then the conjugate of z is ¯ z = a - bi . i 2 = - 1 because - 1 = i . z ¯ z = a 2 + b 2 = | z | 2 . Why? Because z ¯ z = ( a + bi )( a - bi ) = a 2 - abi + abi - b 2 i 2 = a 2 + b 2 . So the multiplication of a complex number z with its conjugate ¯ z is equal to the sum of the real part a squared and the imaginary part b squared. Division: If z = a + bi and w = c + di , then z w = z · ¯ w w ¯ w = z · ¯ w c 2 + d 2 because w ¯ w = c 2 + d 2 . Example 1 This is the division of complex numbers z w where z = 2 + i and w = 1 - 3 i . As suggested in the reminder, multiply the conjugate ¯ w = 1 + 3 i to the numerator and denominator of the fraction, i.e., z w = z · ¯ w w ¯ w = (2+ i )(1+3 i ) 1 2 +( - 3) 2 = - 1+7 i 10 because z · ¯ w = (2+ i )(1+3 i ) = 2+6 i + i +3 i 2 = - 1 + 7 i and w ¯ w = (1 - 3 i )(1 + 3 i ) = 1 + 3 i - 3 i
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3_5_ex_sol - Section 3.5 Example Solutions Instructor Ms...

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