ss_3_5

ss_3_5 - Section 3.5 Complex Numbers The complex number i...

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Section 3.5 Complex Numbers The complex number i is the solution of the equation x 2 = - 1, i.e. i 2 = - 1. We write i = - 1. The complex number system consists of all numbers of the form a + bi where a , b are real numbers. The usual real-number algebraic operations (addition, subtraction, multiplication, division) also hold for complex numbers. For example, (3 + 2 i )+(2 - i )=5+ i and (3 + 2 i )(2 - i )=6 - 3 i +4 i - 2 i 2 =8+ i If z = a + bi is a complex number, then a is called the real part of z and b is called the imaginary part of z . The number ¯ z = a - bi is called the complex conjugate of z = a + bi . Example. 3+2 i =3 - 2 i Example. 2 - 3 i =2+3 i If z = a + bi ,then z z =2 a z - ¯ z bi z ¯ z = a 2 + b 2 = | z | 2 Example. Simplify 2 - i 3+ i . Solution. 2 - i 3+ i = 2 - i 3+ i 3 - i 3 - i = 6 - 2 i - 3 i + i 2 9+1 = 5 - 5 i 10 = 1 - i 2 = 1 2 - 1 2 i Some basics of complex conjugates: ¯ ¯ z = z z + w z w z · w z · ¯ w Powers of i i 2 = - 1 i 3 = i 2 i = - i i 4 =( i 2 ) 2 =1 i 5 = i 4 i = i Solving Quadratic Equations Involving Complex Numbers: Example. Solve for x : x 2 +2 x +4=0.
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This note was uploaded on 12/11/2011 for the course MAC 1140 taught by Professor Kutter during the Fall '11 term at FSU.

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ss_3_5 - Section 3.5 Complex Numbers The complex number i...

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