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CHAPTER
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If you ve had a comprehensive algebra course such as the predecessor to this book, Algebra Know-It-All, then you ve been exposed to imaginary numbers and complex numbers In this chapter, we ll take a closer look at how these quantities behave
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A complex number consists of two components, the real part and the imaginary part Complex numbers can be defined as ordered pairs and mapped one-to-one onto the points of a coordinate plane They can also be represented as vectors
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The unit imaginary number The set of imaginary numbers arises when we ask, What is the square root of a negative real number This question poses a mystery to anyone who is familiar only with the real numbers Unless we come up with some new sort of quantity, we have to say, It s undefined In order to define the square root of a negative real number, mathematicians invented the unit imaginary number, called it i, and defined it on the basis of the equation
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i 2 = 1 Once they had set down this rule, mathematicians explored how this strange new number behaved, and a new branch of number theory evolved Engineers and physicists use j instead of i to denote the unit imaginary number That s what we ll use, because the lowercase italic i is found in other mathematical contexts, particularly in sequences and series The unit imaginary number j is equal to the positive square root of 1 That is, j = ( 1)1/2 When we use the symbol j to represent the unit imaginary number, we can also call it the j operator, a term commonly used by engineers
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The set of imaginary numbers We can multiply j by any real number, known as a real-number coefficient, and the result is an imaginary number The real coefficient is customarily written after j if it is positive or 0, and after j if it is negative Examples are
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j3 = j 3 = 3 j j5 = j ( 5) = 5 j j 2 /3 = j ( 2 /3) = 2 /3 j j0 = j 0 = 0 j = 0 The set of all possible real-number multiples of j composes the set of imaginary numbers For practical purposes, the elements of this set can be depicted along a number line corresponding one-to-one with the real-number line By convention, the imaginary-number line is oriented vertically, as shown in Fig 6-1 When either j or j is multiplied by 0, the result is equal to the real number 0 Therefore, the intersection of the sets of imaginary and real numbers contains one element, namely, 0
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Figure 6-1 Imaginary numbers
can be depicted as points on a vertical line As we go upward, we get more positiveimaginary numbers; as we go downward, we get more negativeimaginary numbers
j8 j6 j4 j2 j0 j 2 j 4 j 6 j 8 Negative-imaginary numbers Positive-imaginary numbers
Center of continuous line
Complex Numbers and Vectors
Complex numbers When we add a real number to an imaginary number, we get a complex number The general form for a complex number is
a + jb where a and b are real numbers If the real-number coefficient of j happens to be negative, then its absolute value is written following j, and a minus sign is used instead of a plus sign in the composite expression So instead of a + j( b) we should write a jb Individual complex numbers can be depicted as points on a Cartesian coordinate plane as shown in Fig 6-2 The intersection point between the real- and imaginary-number lines corresponds
Real part negative, imaginary part positive
j8 j6 j4 j2
Real part positive, imaginary part positive
2 j 2 j 4
Real part negative, imaginary part negative
j 6 j 8
Real part positive, imaginary part negative
Figure 6-2 Complex numbers can be depicted as
points on a plane, which is defined by the intersection of perpendicular real- and imaginary-number lines
Numbers with Two Parts
to 0 on the real-number line and j0 on the imaginary-number line This plane is called the Cartesian complex-number plane
An example If the imaginary part of a complex quantity is 0, we have a pure real quantity When the real part of a complex quantity is 0 and the imaginary part is something other than j0, we have a pure imaginary quantity Figure 6-3 shows nine complex numbers plotted as points on the Cartesian complex-number plane, as follows
0 + j0, whose ordered pair is (0,j0) and which is equal to the pure real 0 and the pure imaginary j0 5 + j0, whose ordered pair is (5,j0) and which is equal to the pure real 5 0 + j7, whose ordered pair is (0,j 7) and which is equal to the pure imaginary j 7 2 + j0, whose ordered pair is ( 2,j0) and which is equal to the pure real 2 0 j8, whose ordered pair is (0, j8) and which is equal to the pure imaginary j8 7 + j6, whose ordered pair is (7,j6) 8 + j5, whose ordered pair is ( 8,j5) 5 j5, whose ordered pair is ( 5, j5) 3 j7, whose ordered pair is (3, j7)