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# lecture_39 - MA 265 LECTURE NOTES FRIDAY APRIL 18 Review of...

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MA 265 LECTURE NOTES: FRIDAY, APRIL 18 Review of Complex Numbers Definitions. Consider the quadratic equation a z 2 + b z + 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 c a 2 a . Recall that we may classify these roots using the discriminant b 2 - 4 a c : two real roots one real root two imaginary roots if b 2 - 4 a c > 0; b 2 - 4 a c = 0; b 2 - 4 a c < 0. When the discriminant b 2 - 4 a c 0, we can express these roots in the form z = x + i y in terms of real numbers x = - b 2 a and y = ± p | b 2 - 4 a c | 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 + i y w = u + i v = ( 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 + i y is - z = ( - x ) + i ( - y ). This satisfies z + ( - z ) = 0. Associativity: For all complex numbers z , w , and t we have ( z + w ) + t = z + ( w + t ). Multiplication of Complex Numbers. Similarly, we can multiply complex numbers: z = x + i y w = u + i v = z · w = ( x + i y ) ( u + i v ) = ( x u - y v ) + i ( x v + y u ) .

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