MIT18_03S10_c05

MIT18_03S10_c05 - 18.03 Class 5, Feb 15, 2008 Complex...

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18.03 Class 5 , Feb 15, 2008 Complex Numbers, roots of unity [1] Complex algebra [2] Complex conjugation [3] Polar multiplication Complex numbers provide a tool for expressing aspects of the real world. Anything with an amplitude and a phase is secretly a complex number, and it is worthwhile bringing that secret identity out in the open. [1] Complex Algebra We think of the real numbers as filling out a line. The complex numbers fill out a plane. The point up one unit from 0 is written i (but written j by many engineers). Addition and multiplication by real numbers is as vectors. The new thing is i^2 = -1 . The usual rules of algebra apply. For example FOIL: (1 + i)(1 + 2i) = 1 + 2i + i - 2 = -1 + 3i. Every complex number can be written as a + bi with a and b real. a = Re(a+bi) the real part b = Im(a+bi) the imaginary part: NB this is a real number. Maybe complex numbers seem obscure because you are used to imagining numbers by giving them units: 5 cars, or -3 degrees. Complex numbers do not accept units. Also, there is no ordering on complex numbers, no "<." Question 1. Multiplication by i has the following effect
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This note was uploaded on 10/21/2011 for the course MATH 18.03 taught by Professor Vogan during the Spring '09 term at MIT.

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MIT18_03S10_c05 - 18.03 Class 5, Feb 15, 2008 Complex...

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