ss_8_3

ss_8_3 - Section 8.3 The complex Plane There are many...

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Section 8.3 The complex Plane There are many practical applications of mathematics that require solutions of polynomial equations. Solutions to all polynomial equations would not be possible, if the values of the solutions were restricted to real numbers. For example, the simple equation x 2 + 1 = 0 has no solution in the set of real numbers. Hence, there is a need to extend the set of real numbers to a larger set of numbers (called the complex numbers), and the larger set should include (at least) a solution to the equation x 2 + 1 = 0. Thus, one defines the imaginary unit i to be the complex number such that i 2 = - 1 . It turns out that, not only does i give a solution to the equation x 2 + 1 = 0, but i supplies the basis for a set of complex numbers that give satisfactory solutions to all polynomial equations. The complex numbers are defined to be the set of all z = x + yi where x and y are real numbers and i 2 = - 1. Since i 2 = - 1, we write i = - 1 .I f z = x + yi is a complex number, then x is called the real part of z and y is called the imaginary part of z . The mathematical operations of addition, subtraction, multiplication, and division are defined for complex numbers so that these operations obey the corresponding rules for real numbers. For example, (3 + 2 i )+(2 - i )=5+ i and (3 + 2 i
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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_8_3 - Section 8.3 The complex Plane There are many...

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