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Complex Numbers - MIT Mathematics

LECTURE NOTES, SPRING 2014. BJORN POONEN. 7. Complex Numbers Complex Numbers are expressions of the form x + yi, where x and y are real Numbers , and i is a new symbol. Multiplication of Complex Numbers will eventually be defined so that i2 = 1. (Electrical engineers sometimes write j instead of i, because they want to reserve i for current, but everybody else thinks that's weird.) Just as the set of all real Numbers is denoted R, the set of all Complex Numbers is denoted C. Flashcard question: Is 9 a real number or a Complex number ? Possible answers: 1. real number 2. Complex number 3. both 4. neither Answer: Both, because 9 can be identified with 9 + 0i. Operations on Complex Numbers . real part Re(x + yi) := x imaginary part Im(x + yi) := y (Note: It is y, not yi, so Im(x + yi) is real ).

The arithmetic operations on complex numbers satisfy the same properties as for real numbers (zw= wzand so on). The mathematical jargon for this is that C, like R, is a eld. In particular, 1. for any complex number zand integer n, the nth power zn can be de ned in the usual way

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math.mit.edu

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