110_1_ComplexNumberRev

110_1_ComplexNumberRev - Imaginary Number Can we always...

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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 j = -1 i 3 = j j 2 = -i j 4 = 1 2j • 7j = -14 Dividing two imaginary numbers produces a real number: 6j / 2j = 3
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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.
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EE110 review 5 Equating Complex Numbers Two complex numbers z 1 = x 1 + j y 1 z 2 = x 2 + j y 2
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110_1_ComplexNumberRev - Imaginary Number Can we always...

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