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Unformatted text preview: MA 265 LECTURE NOTES: FRIDAY, APRIL 18 Review of Complex Numbers Definitions. Consider the quadratic equation az 2 + bz + c = 0 where a , b , and c are real numbers. Formally, we can write the two roots of this equation as z =- b 2 a ± √ b 2- 4 ca 2 a . Recall that we may classify these roots using the discriminant b 2- 4 ac : two real roots one real root two imaginary roots if b 2- 4 ac > 0; b 2- 4 ac = 0; b 2- 4 ac < 0. When the discriminant b 2- 4 ac ≤ 0, we can express these roots in the form z = x + iy in terms of real numbers x =- b 2 a and y = ± p | b 2- 4 ac | 2 a . Here i = √- 1 is that number such that i 2 =- 1, i 3 =- i , i 4 = 1, etc. We call z a complex number , where x is the real part and y is the imaginary part of z . Just as we denote the collection of all real numbers by R , we denote the collection of all complex numbers by C . Addition of Complex Numbers. We can add and subtract complex numbers by keeping track of the real and imaginary parts: z = x + iy w = u + iv = ⇒ ( z + w = ( x + u ) + i ( y + v ) z- w = ( x- u ) + i ( y- v ) We have the following properties: • Identity: The complex number 0 = 0 + i 0 satisfies z + 0 = 0 + z = z . • Inverses: The negative of z = x + iy is- z = (- x ) + i (- y ). This satisfies z + (- z ) = 0....
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This note was uploaded on 03/11/2010 for the course MA 261A 0026100 taught by Professor ... during the Spring '10 term at Purdue University Calumet.
- Spring '10