Calculus 1000A/B Chapter Notes - Chapter Appendix: Imaginary Number, Partial Fraction Decomposition, Complex Analysis
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CALC 1000A/B Full Course Notes
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The set r of real numbers can be extended to a bigger set of the so-called complex numbers. This is done by introducing a single imaginary number i = 1. Real numbers can be viewed as a subset of complex numbers with zero imaginary part. Thus, denoting the space of complex numbers by c, we have the following chain of inclusions. We may extend the de nition of arithmetic operations on real numbers to the space of complex numbers as follows: One can verify that when b = b = 0, the above formulas provide the usual operations of addition, multiplication and division for reals. Note that i i = i2 = 1, which in particular means that the equation z2 + 1 = 0 over the set of complex numbers has two complex roots: i and i. This is in contrast with reals over which this equation has no solution.