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Complex

# Complex - ECE 325 ALGEBRA OF COMPLEX NUMBERS Fall 2007 This...

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ECE 325 ALGEBRA OF COMPLEX NUMBERS Fall 2007 This handout will be brief. Its purpose is to “remind” everybody about some basic results from the algebra of complex numbers. The material, one hopes, will be more or less familiar to most people. Like good electrical engineers, we set j = - 1. A complex number c o is a number of the form c o = a o + jb o , where a o and b o are real numbers. We call a o the real part of c o (notation Re { c o } ) and b o the imaginary part of c o (notation Im { c o } ). You add and multiply two complex numbers c 1 = a 1 + jb 1 and c 2 = a 2 + jb 2 in the obvious way: c 1 + c 2 = ( a 1 + a 2 ) + j ( b 1 + b 2 ) and c 1 c 2 = ( a 1 + jb 1 )( a 2 + jb 2 ) = ( a 1 a 2 - b 1 b 2 ) + j ( a 1 b 2 + b 1 a 2 ) . The complex conjugate (notation c o ) of c o = a o + jb o is the complex number c o = a o - jb o . You can visualize complex numbers as points in the so-called complex plane. The horizontal axis is the real axis, and the vertical axis is the imaginary axis. In Figure 1, I’ve graphed an unspecified complex number c o along with the specified complex number ( 3 - j ). The real part of c o , namely a o , happens to be positive and c o ’s imaginary part b o is also positive. The length of the line from the origin to the point c o , by the Pythagorean Theorem, is simply a 2 o + b 2 o ; this quantity is called the magnitude of c o , and we denote it by | c o | . Note that | c o | 2 = c o c o ; that is, the magnitude-squared of c o is the product of c o and its complex conjugate c o . Observe that the magnitude of c o is the same as the magnitude of c 0 . The depicted angle φ o

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