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qr code generator vb.net Fig. 110 in .NET framework
Fig. 110 Code 128C Recognizer In .NET Framework Using Barcode Control SDK for VS .NET Control to generate, create, read, scan barcode image in .NET framework applications. Code 128 Code Set C Encoder In .NET Framework Using Barcode generator for .NET Control to generate, create Code128 image in Visual Studio .NET applications. Problem 92 Express each of the following complex numbers in exponential form. a b 3 j4 3 j4 c d 3 j4 3 j4 Read Code128 In VS .NET Using Barcode recognizer for VS .NET Control to read, scan read, scan image in .NET applications. Barcode Maker In .NET Framework Using Barcode encoder for .NET framework Control to generate, create bar code image in VS .NET applications. 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 Bar Code Scanner In .NET Framework Using Barcode reader for .NET framework Control to read, scan read, scan image in .NET framework applications. Make Code 128 Code Set B In C#.NET Using Barcode generator for .NET framework Control to generate, create USS Code 128 image in VS .NET applications. Operations in the Exponential and Polar Forms. De Moivre s Theorem
Code128 Creator In .NET Framework Using Barcode generator for ASP.NET Control to generate, create Code 128A image in ASP.NET applications. Making Code 128 Code Set A In VB.NET Using Barcode printer for .NET framework Control to generate, create Code 128 image in Visual Studio .NET applications. 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). Linear Barcode Printer In Visual Studio .NET Using Barcode generator for Visual Studio .NET Control to generate, create Linear 1D Barcode image in .NET applications. Matrix 2D Barcode Encoder In VS .NET Using Barcode encoder for .NET Control to generate, create Matrix 2D Barcode image in .NET framework applications. @Spy
Code 128B Creator In .NET Framework Using Barcode generator for Visual Studio .NET Control to generate, create ANSI/AIM Code 128 image in .NET framework applications. ISSN Printer In VS .NET Using Barcode maker for VS .NET Control to generate, create ISSN  10 image in .NET applications. CHAPTER 6 Algebra of Complex Numbers
Drawing EAN13 In .NET Framework Using Barcode generator for ASP.NET Control to generate, create EAN13 Supplement 5 image in ASP.NET applications. Scanning DataMatrix In None Using Barcode decoder for Software Control to read, scan read, scan image in Software applications. 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 UCC.EAN  128 Creator In ObjectiveC Using Barcode creator for iPhone Control to generate, create GS1128 image in iPhone applications. Encode Bar Code In None Using Barcode generation for Software Control to generate, create barcode image in Software applications. 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 Recognizing Code 39 Full ASCII In Visual Studio .NET Using Barcode decoder for .NET framework Control to read, scan read, scan image in Visual Studio .NET applications. Barcode Generator In Visual Basic .NET Using Barcode drawer for .NET Control to generate, create barcode image in .NET applications. 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. Decoding UPC Code In Visual C#.NET Using Barcode decoder for Visual Studio .NET Control to read, scan read, scan image in .NET applications. USS128 Scanner In Visual C# Using Barcode reader for .NET Control to read, scan read, scan image in .NET framework applications. @Spy
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 righthand 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

