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Algebra of Complex Numbers
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In this chapter we study the algebra of complex numbers. This is a subject of great usefulness in all branches of engineering, being especially important in electrical and electronic engineering. We ll nd that the algebra is not di cult and is, in itself, a most interesting study.
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Let us begin with a bit of history. For a long time only ordinary positive and negative numbers were used, and the rules for working with these numbers were laid down and rmly established. Later, positive and negative exponents were invented, and tted into the established rules of mathematics. New rules for working with exponents had to be laid down, and these rules had to be in harmony with other rules previously established. Then the square root of a positive number was de ned. Thus, if the square root of y is equal to x, this is denoted by the symbol p y x which is de ned to mean that y x2 It was immediately noticed that the square root of a positive number must have TWO di erent values, equal in magnitude but opposite in sign. This had to be true because of a previous rule that had been laid down and established; this was the rule that the product of TWO POSITIVE NUMBERS or TWO NEGATIVE NUMBERS is always a POSITIVE number. Thus it had to be written that, for example, p 1 1
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CHAPTER 6 Algebra of Complex Numbers
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because 1 1 2 and 1 1 2 This naturally brought up the question, Does the square root of a NEGATIVE number have a meaning . What, for example, is the square root of minus 1 Thus, if we let x denote the value of the square root of minus 1 we must write that p 1 x which, by de nition, means that 1 x2 Notice that x cannot be equal to +1 and it cannot be equal to 1, because neither of these numbers when squared is equal to 1. It was therefore understandably stated, by the early investigators, that the square root of a negative number does not exist. As time passed, however, it became clear that a fully uni ed system of algebra was not possible unless the square roots of negative numbers were accepted as truly being a new kind of number. Such numbers are called imaginary numbers, the name simply re ecting the historical fact that they were originally thought not to exist. Thus we now have two sets of numbers, one being the set of ordinary positive and negative real numbers, the other being the set of positive and negative imaginary numbers. In the set of real numbers each individual number is composed of a certain number of digits arranged in a certain sequence, producing a unique number, di erent from any other. Thus we have the positive real numbers, 0, 1, 2, 3, . . . 25, 26, 27, . . . , and so on, endlessly. Then, for each positive real number there is a corresponding NEGATIVE real number of the same magnitude. In the real number system the number 1 is the basic unit of measure or value. Thus any other real number can be regarded as a multiple of 1; the number 23, for example, is equal to 23 times the value represented by the basic unit 1. We also note that
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Exactly the same procedure is used to denote the value of an imaginary number, except that now the basic IMAGINARY UNIT of value will be denoted by the letter j, which is de ned to be equal to the SQUARE ROOT OF MINUS ONE; that is, by de nition,
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