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18.03 LECTURE NOTES, SPRING 2014 - 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).

One can add, subtract, multiply, and divide complex numbers (except for division by 0). Addition, subtraction, and multiplication are as for polynomials, except that after multiplication one should simplify by using i2 = 1; for example, (2 + 3i)(1 5i) = 2 7i 15i2 = 17 7i:

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