Appendix_SB - Appendix SB Complex Numbers and Algebra SB.1...

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1 Appendix SB Complex Numbers and Algebra SB.1 Definitions and Notation Imaginary numbers arise when taking the square roots of negative numbers. Thus, 3 9 ± = , since multiplying +3 or –3 by itself gives 9. But what about 9 ? Whereas 9 is a real number ( 3 ± ), 9 is said to be an imaginary number. It is evaluated by defining a quantity j as being equal to 1 . Then j 2 = -1, and 9 = 9 2 j . Now we have two positive quantities under the square root, so the square root becomes 3 j ± . This is a valid answer, because ( j 3)( j 3) = j 2 9 = -9 = (- j 3)(- j 3). Defined in this way, imaginary numbers are a perfectly valid set of numbers, just like real numbers, integers, or rational numbers. j , the basis of all imaginary numbers, is of course itself an imaginary number, and all imaginary numbers are multiplied by j . Imaginary numbers can be manipulated according to certain logical and consistent rules. A complex number x is defined as the sum of a real number and an imaginary number: x = a + jb (SB.1.1) where a is referred to as the real part of x and b as the imaginary part. Complex numbers are commonly encountered in algebra and trigonometry. For example, the equation x 2 + x +1 does not have real roots, but it does have complex roots, i.e., roots that are complex numbers. The sine and cosine functions may be expressed in terms of complex quantities: 2 cos jx jx e e x + = , j e e x jx jx 2 sin = (SB.1.2) These relations can be readily verified using the infinite series representations of the exponential, sine, and cosine functions. The conjugate of a complex number x , denoted as x , is the number that has the same real part but a negated imaginary part. Thus, a + jb and a – jb are conjugates. Complex quantities play a central role in electric circuits. They are the basis for phasor notation. They are also encountered in Fourier series, Fourier and Laplace transforms, and in the extensive applications that derive from the theory of functions of complex variables.
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2 SB.2 Graphical Representation A complex number may be represented in rectangular form as a point in the complex plane , or Argand diagram . This is the familiar two- dimensional, cartesian coordinate plane except that the vertical axis is denoted as the imaginary axis (Fig. SB.2.1). A complex number a + jb is represented as a point whose horizontal coordinate is a and vertical coordinate is b . Fig. SB.2.1 illustrates several such examples where the number is real (such as 1, 4, -3), imaginary (such as j 2, - j 4), or complex (such as 4 + j 3, 4 – j 3, -3 + j 4, -5 – j 4). The numbers 4 + j 3 and 4 – j 3 are conjugates. Note that whereas in the real plane, a point is represented by a coordinate pair ( x, y ), the point is represented in the complex plane as a complex number, e.g., 4 + j 3.
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This note was uploaded on 02/17/2011 for the course EECE 210 taught by Professor Riadchedid during the Fall '07 term at American University of Beirut.

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Appendix_SB - Appendix SB Complex Numbers and Algebra SB.1...

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