Complex Numbers

# Complex Numbers - Imaginary Number Can we always find roots...

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EE110 review 1 Imaginary Number Can we always find roots for a polynomial? The equation x  2  + 1 = 0 has no solution for x in the set of real numbers.  If we define a number that satisfies the equation x  2  = -1 that is, x =  -1 then we can always find the n roots of a polynomial of degree n. We call the solution to the above equation the  imaginary  number , also known as i.   The imaginary number is often called j in electrical engineering. Imaginary numbers ensure that all polynomials have roots.

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EE110 review 2 Imaginary Arithmetic Arithmetic with imaginary works as expected: j + j = 2j 3j – 4j = -j 5 (3j) = 15 i To take the product of two imaginary numbers, remember  that j  2  = -1:  j = -1
EE110 review 3 Complex Numbers We define a  complex  number with the form z = x + jy where x, y are real numbers. The complex number z has a  real part , x, written Re{z}. The  imaginary part  of z, written Im{z}, is y.   Notice that, confusingly, the imaginary part is a real number. So we may write z as z = Re{z} + jIm{z}

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EE110 review 4 Set of Complex Numbers The set of complex numbers, therefore, is defined by Complex = {x + jy | x   Reals, y   Reals, and j =  -1} Every real number is in Complex, because x = x + j0; and every imaginary number jy is in Complex, because jy = 0 + jy.
EE110 review 5 Equating Complex Numbers Two complex numbers 2 2

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## This note was uploaded on 06/13/2009 for the course EE EE 110 taught by Professor Gupta during the Spring '09 term at UCLA.

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Complex Numbers - Imaginary Number Can we always find roots...

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