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barcode in vb.net 2005 Multiplying complex numbers in Software
Multiplying complex numbers Recognizing Quick Response Code In None Using Barcode Control SDK for Software Control to generate, create, read, scan barcode image in Software applications. Drawing QR Code 2d Barcode In None Using Barcode creator for Software Control to generate, create QR Code image in Software applications. You should know how complex numbers are multiplied, to have a full understanding of their behavior. When you multiply these numbers, you only need to treat them as sums of number pairs, that is, as binomials. It s easier to give the general formula than to work with specifics here. The product of (a + jb) and (c jd) is equal to ac jad jbc jjbd. Simplifying, remember that jj 1, so you get the final formula: (a jb)(c jd) = (ac bd) j(ad bc) Reading QRCode In None Using Barcode scanner for Software Control to read, scan read, scan image in Software applications. QR Code Printer In C#.NET Using Barcode creation for .NET framework Control to generate, create Denso QR Bar Code image in .NET framework applications. As with the addition and subtraction of complex numbers, you must be careful with signs (plus and minus). And also, as with addition and subtraction, you can get used to doing these problems with a little practice. Engineers sometimes (but not too often) have to multiply complex numbers. Making QR Code 2d Barcode In .NET Using Barcode creator for ASP.NET Control to generate, create QRCode image in ASP.NET applications. QR Code JIS X 0510 Drawer In Visual Studio .NET Using Barcode creation for .NET framework Control to generate, create Denso QR Bar Code image in .NET framework applications. The complex number plane
QR Code 2d Barcode Generation In Visual Basic .NET Using Barcode maker for .NET Control to generate, create QR Code ISO/IEC18004 image in VS .NET applications. UPC A Drawer In None Using Barcode generation for Software Control to generate, create UPCA image in Software applications. Real and imaginary numbers can be thought of as points on a line. Complex numbers lend themselves to the notion of points on a plane. This plane is made by taking the real and imaginary number lines and placing them together, at right angles, so that they intersect at the zero points, 0 and j0. This is shown in Fig. 152. The result is a Cartesian coordinate plane, just like the ones you use to make graphs of everyday things like bankaccount balance versus time. Encoding Barcode In None Using Barcode drawer for Software Control to generate, create bar code image in Software applications. Print Bar Code In None Using Barcode generator for Software Control to generate, create barcode image in Software applications. Notational neuroses
GTIN  128 Maker In None Using Barcode drawer for Software Control to generate, create GTIN  128 image in Software applications. USS Code 39 Generation In None Using Barcode generation for Software Control to generate, create USS Code 39 image in Software applications. On this plane, a complex number might be represented as a jb (in engineering or physicists notation), or as a bi (in mathematicians notation), or as an ordered pair (a, b). Wait, you ask. Is there a misprint here Why does b go after the j, but in front of the i The answer is as follows: Mathematicians and engineers/physicists just don t think alike, and this is but one of myriad ways in which this is apparent. In other words, it s a matter of UPC Shipping Container Symbol ITF14 Maker In None Using Barcode maker for Software Control to generate, create UCC  14 image in Software applications. Code 3 Of 9 Recognizer In Visual Basic .NET Using Barcode decoder for VS .NET Control to read, scan read, scan image in VS .NET applications. The complex number plane 267
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Creating GS1 DataBar In .NET Using Barcode creator for Visual Studio .NET Control to generate, create GS1 DataBar Truncated image in .NET applications. EAN13 Encoder In .NET Framework Using Barcode generator for ASP.NET Control to generate, create EAN13 Supplement 5 image in ASP.NET applications. notational convention, and that is all. (It s also a somewhat humorous illustration of the different angle that an engineer takes in approaching a problem, as opposed to a mathematician.) Complex number vectors
Complex numbers can also be represented as vectors in the complex plane. This gives each complex number a unique magnitude and direction. The magnitude is the distance of the point a jb from the origin 0 j0. The direction is the angle of the vector, measured counterclockwise from the a axis. This is shown in Fig. 153. Absolute value
The absolute value of a complex number a jb is the length, or magnitude, of its vector in the complex plane, measured from the origin (0,0) to the point (a,b). 268 Impedance and admittance
153 Magnitude and direction of a vector in the complex number plane.
In the case of a pure real number a j0, the absolute value is simply the number itself, a, if it is positive, and a if a is negative. In the case of a pure imaginary number 0 jb, the absolute value is equal to b if b (which is a real number) is positive, and b if b is negative. If the number is neither pure real or pure imaginary, the absolute value must be found by using a formula. First, square both a and b. Then add them. Finally, take the square root. This is the length of the vector a jb. The situation is illustrated in Fig. 154. 154 Calculation of absolute value, or vector length.
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Problem 151 Find the absolute value of the complex number 22 j0. Note that this is a pure real. Actually, it is the same as 22 j0, because j0 = 0. Therefore, the absolute value of this complex number is ( 22) 22. Problem 152 Find the absolute value of 0 j34. This is a pure imaginary number. The value of b in this case is 0 j( 34). Therefore, the absolute value is ( 34) 34. 34, because 0 j34 Problem 153 Find the absolute value of 3 j4. In this number, a 3 and b 4, because 3 j4 can be rewritten as 3 j( 4). Squaring both of these, and adding the results, gives 32 ( 4)2 9 16 25. The square root of 25 is 5; therefore, the absolute value of this complex number is 5. You might notice this 3, 4, 5 relationship and recall the Pythagorean theorem for finding the length of the hypotenuse of a right triangle. The formula for finding the length of a vector in the complexnumber plane comes directly from this theorem. If you don t remember the Pythagorean theorem, don t worry; just remember the formula for the length of a vector.

