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Fig. 110
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Problem 92 Express each of the following complex numbers in exponential form. a b 3 j4 3 j4 c d 3 j4 3 j4
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Verify, by direct use of eq. (153), that your answers are correct. Problem 93 Write the answer to the following sum in exponential form. 14 j1128 8 j288 19 j1558
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Operations in the Exponential and Polar Forms. De Moivre s Theorem
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We have found that the algebraic ADDITION of complex numbers must be carried out in the a jb (rectangular) form. This is because, in addition, the REAL PARTS and the IMAGINARY PARTS of the numbers must be separately added together to get the nal resultant sum of the numbers (section 6.2).
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CHAPTER 6 Algebra of Complex Numbers
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It thus follows that complex numbers CANNOT BE DIRECTLY ADDED TOGETHER IN THE EXPONENTIAL FORM, because the real and imaginary parts are not shown separately in the exponential form (see problem 93). However, while the exponential form is not suited to the addition and subtraction operation, it is very de nitely suited to the MULTIPLICATION AND DIVISION operations. This is because in multiplication and division we can make use of the BASIC LAWS OF EXPONENTS, as the following will show. Consider any two complex numbers a jb A jp c jd B jq where A and B are the magnitudes of the numbers and p and q are the angular amplitudes of the numbers (eqs. (143) and (144), and Fig. 109). Now let us consider the PRODUCT of the above two complex numbers. First, in the rectangular form we have (section 6.2) a jb c jd ac bd j ad bc 160
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Now consider the same multiplication if the two numbers are expressed in exponential form. Remembering that in multiplication EXPONENTS ARE ADDED, we have the result 161 A jp B jq AB j p q Thus, if complex numbers are expressed in EXPONENTIAL FORM, the PRODUCT of the numbers is a complex number whose MAGNITUDE IS THE PRODUCT OF THE MAGNITUDES and whose ANGLE is the SUM OF THE ANGLES of the individual numbers. Comparison of eqs. (160) and (161) shows that multiplication in the exponential form is generally easier than multiplication in the rectangular form. Furthermore, the use of the exponential form can often simplify the mathematical work involved in theoretical investigations. Now let us consider the DIVISION or quotient of the same two complex numbers. First, in the rectangular form we have (section 6.3) a jb ac bd j ad bc c jd c2 d 2 162
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Now consider the same division if the two numbers are expressed in exponential form. Remembering that in division the exponent of the denominator is SUBTRACTED from the exponent of the numerator, we have the result   A jp A j p q 163  B B jq Thus, if two complex numbers are expressed in exponential form, the QUOTIENT of the two numbers is a complex number whose MAGNITUDE is equal to the QUOTIENT OF THE TWO MAGNITUDES and whose ANGLE is equal to the angle of the numerator MINUS the angle of the denominator. Comparison of eqs. (162) and (163) shows that division in the exponential form is generally easier than division in the rectangular form. Again, the use of the exponential form can often simplify the mathematical work involved in theoretical investigations.
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CHAPTER 6 Algebra of Complex Numbers
It follows that the same information contained in eqs. (161) and (163) can also be expressed in POLAR form; thus product: quotient: p B= AB= q q p A= A= p A = q p q B B= 164 165
It follows that eqs. (161) and (164) extend on to cover any number of factors; thus, for the case of three factors we take the product of the rst two times the third, and so on for any number of factors. Problem 94 Write, in rectangular form, the product of the three complex numbers 3 j1128 ; 4 j628 ; and 7 j1658
Problem 95 Write the product 4=198 3=398 in rectangular form. Problem 96 Making use of eq. (159) and the laws of exponents, raise the complex number 2 j3 to the sixth power. Answer in rectangular form. Problem 97 15 j628 36 j858 Problem 98 16=1028 9=3908 7=758 answer in rectangular form answer in rectangular form
Now, to continue, let us begin by writing down Euler s formula (eq. (153)) thus  j cos  j sin  Or, replacing  with n, it is also true that  jn cos n j sin n B A
Now raise both sides of ( A ) to the power n; noting that  j n  jn , we have  jn cos  j sin  n C
Since the right-hand sides of ( B ) and ( C ) are both equal to the same thing, they are equal to each other, and thus we have the important result that cos  j sin  n cos n j sin n 166
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