Week_11 - Week 11 Appendix A Complex Numbers In this course...

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Unformatted text preview: Week 11 Appendix A: Complex Numbers In this course, we have looked at extending vectors that we can visualize in R 2 and R 3 to more abstract vector spaces. We can also look at the extension of scalars. So far in this course, we have assumed our scalars have been real numbers. We can think of real scalars as an extension of integers, which are in turn an extension of natural numbers. For example, if we are working in the Natural numbers, the equation x + 4 = 0 has no solutions. However if we extend our definition of scalar to include all the integers, the equation x + 4 = 0 has the solution x =- 4. But the equation 3 x + 2 = 0 has no solutions unless we allow fractions by extending our definition of scalar even further. In solving x 2 +1 = 0, there are no solutions in the real number system. But if we extend our definition of scalar to allow for square roots of negative numbers, we obtain the solution x = √- 1 or x = i in the complex numbers. Note that if we define i = √- 1, then i 2 =- 1. This process of extending scalars to capture all solutions to a polynomial equation ends here. Every polynomial has complex roots. Recall: A polynomial p ( x ) has root x = c if p ( c ) = 0. Example Find the roots of x 2 + 2 x + 5 = 0. Solution: Definition: A complex number is an expression of the form z = a + bi , where a and b are real numbers. We call a the real part of z or Re ( z ) and b is called the Imaginary part of z or Im ( z ). We denote the set of all complex numbers C . Note: The Real numbers are a subset of the Complex numbers. We wouldn’t call them a subspace since we might wonder which set of numbers to use as our scalars in checking for closure under scalar multiplication. In fact, scalars are part of their own separate abstract space called a field . In a field we allow elements to be multiplied together. So while a vector space is closed under addition and scalar...
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This note was uploaded on 08/05/2008 for the course MATH 115 taught by Professor Dunbar during the Fall '07 term at Waterloo.

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Week_11 - Week 11 Appendix A Complex Numbers In this course...

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