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vb.net generate qr code Complex Numbers and Vectors in .NET framework
CHAPTER Reading Code 3 Of 9 In VS .NET Using Barcode Control SDK for .NET framework Control to generate, create, read, scan barcode image in .NET applications. Code39 Creator In VS .NET Using Barcode generator for .NET framework Control to generate, create Code39 image in VS .NET applications. Complex Numbers and Vectors
Scanning Code 39 Extended In .NET Using Barcode reader for .NET framework Control to read, scan read, scan image in .NET framework applications. Barcode Encoder In Visual Studio .NET Using Barcode generator for .NET framework Control to generate, create barcode image in .NET applications. If you ve had a comprehensive algebra course such as the predecessor to this book, Algebra KnowItAll, 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 Bar Code Scanner In VS .NET Using Barcode scanner for .NET Control to read, scan read, scan image in .NET applications. Code 39 Generation In Visual C#.NET Using Barcode drawer for Visual Studio .NET Control to generate, create Code 39 Extended image in .NET applications. Numbers with Two Parts
Code 39 Extended Generation In VS .NET Using Barcode drawer for ASP.NET Control to generate, create Code 3 of 9 image in ASP.NET applications. ANSI/AIM Code 39 Creation In VB.NET Using Barcode printer for .NET framework Control to generate, create Code 39 Extended image in .NET framework applications. A complex number consists of two components, the real part and the imaginary part Complex numbers can be defined as ordered pairs and mapped onetoone onto the points of a coordinate plane They can also be represented as vectors Linear Barcode Maker In VS .NET Using Barcode creator for Visual Studio .NET Control to generate, create Linear image in Visual Studio .NET applications. USS Code 128 Creator In .NET Framework Using Barcode creator for .NET framework Control to generate, create Code 128 image in VS .NET applications. 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 Barcode Creation In VS .NET Using Barcode maker for Visual Studio .NET Control to generate, create barcode image in VS .NET applications. MSI Plessey Creator In Visual Studio .NET Using Barcode printer for .NET Control to generate, create MSI Plessey image in .NET framework applications. 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 ECC200 Scanner In Java Using Barcode scanner for Java Control to read, scan read, scan image in Java applications. Code 128 Code Set B Generator In None Using Barcode creator for Software Control to generate, create Code 128 image in Software applications. Numbers with Two Parts
Code 3 Of 9 Printer In ObjectiveC Using Barcode generator for iPhone Control to generate, create Code 39 Full ASCII image in iPhone applications. Code 128 Code Set B Drawer In VB.NET Using Barcode generation for .NET framework Control to generate, create Code 128 Code Set C image in VS .NET applications. The set of imaginary numbers We can multiply j by any real number, known as a realnumber 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 Draw UCC.EAN  128 In VB.NET Using Barcode drawer for .NET Control to generate, create GTIN  128 image in Visual Studio .NET applications. UPCA Scanner In Visual Studio .NET Using Barcode recognizer for VS .NET Control to read, scan read, scan image in Visual Studio .NET applications. 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 realnumber 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 onetoone with the realnumber line By convention, the imaginarynumber line is oriented vertically, as shown in Fig 61 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 Draw Bar Code In Java Using Barcode generator for Java Control to generate, create bar code image in Java applications. Decode USS Code 39 In C#.NET Using Barcode reader for .NET framework Control to read, scan read, scan image in .NET applications. Figure 61 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 Negativeimaginary numbers Positiveimaginary 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 realnumber 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 62 The intersection point between the real and imaginarynumber 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 62 Complex numbers can be depicted as
points on a plane, which is defined by the intersection of perpendicular real and imaginarynumber lines Numbers with Two Parts
to 0 on the realnumber line and j0 on the imaginarynumber line This plane is called the Cartesian complexnumber 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 63 shows nine complex numbers plotted as points on the Cartesian complexnumber 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)

