Complex Numbers

Complex Numbers - Complex Numbers The complex numbers are...

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Complex Numbers The complex numbers are an extension of the real numbers containing all roots of quadratic equations. If we define i to be a solution of the equation x 2 = -1, then the set C of complex numbers is represented in standard form as { a+bi | a,b R}. We often use the variable z = a+bi to represent a complex number. The number a is called the real part of z (Re z) while b is called the imaginary part of z (Im z). Two complex numbers are equal if and only if their real parts are equal and their imaginary parts are equal. We represent complex numbers graphically by associating z = a+bi with the point (a,b) on the complex plane . Basic Operations The basic operations on complex numbers are defined as follows: = (a+c) + (b+d)i = (a-c) + (b-d)i
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= (ac-bd) + (bc+ad)i a+bi c+di = a+bi c+di · c-di c-di = ac+bd c 2 +d 2 + bc-ad c 2 +d 2 i In dividing a+bi by c+di, we rationalized the denominator using the fact that (c+di)(c-di) = c 2 -cdi +cdi -d 2 i 2 = c 2 + d 2 . The complex numbers c+di and c-di are called complex conjugates .
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