# vb.net generate qr code Complex number ratio Suppose that we want to find the ratio (quotient) of two complex numbers in Visual Studio .NET Paint USS Code 39 in Visual Studio .NET Complex number ratio Suppose that we want to find the ratio (quotient) of two complex numbers

Complex number ratio Suppose that we want to find the ratio (quotient) of two complex numbers
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(a + jb) / (c + jd ) Multiplying both the numerator and the denominator by (c jd ), we obtain [(a + jb)(c jd )] / (c + jd )(c jd ) which multiplies out to (ac jad + jbc j 2bd ) / (c 2 jcd + jcd j 2d 2) This expression can be simplified to [(ac + bd ) + j (bc ad )] / (c 2 + d 2) When we separate out the real and imaginary parts, we get [(ac + bd ) / (c 2 + d 2)] + j [(bc ad ) / (c2 + d 2)]
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The square brackets, while technically superfluous, are included to visually set apart the real and imaginary parts of the result We have just derived a general complex-number ratio formula that we can always use: (a + jb)/(c + jd ) = [(ac + bd )/(c + d 2)] + j [(bc ad )/(c 2 + d 2)]
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For this formula to work, the denominator must not be equal to 0 + j0 That means we cannot have both c = 0 and d = 0 If both of these coefficients are 0, then we end up dividing by 0 That operation, unlike the square root of a negative real, remains undefined, at least as far as this book is concerned!
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Complex number raised to positive-integer power If a + jb is a complex number and n is a positive integer, then (a + jb)n is the result of multiplying (a + jb) by itself n times Complex conjugates Suppose we encounter two complex numbers that have the same coefficients, but opposite signs between the real and imaginary parts, as in
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a + jb and a jb We call any two such quantities complex conjugates They have some interesting properties When we add a complex number to its conjugate, we get twice the real coefficient In general, we have (a + jb) + (a jb) = 2a When we multiply a complex number by its conjugate, we get the sum of the squares of the coefficients In general, we have (a + jb)(a jb) = a2 + b2 Complex conjugates are often encountered in engineering They re especially useful in alternatingcurrent (AC) circuit, radio-frequency (RF) antenna, and transmission-line theories
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Sum example Let s find the sum of the two complex numbers 5 + j4 and 2 j3 When we add the real parts, we get
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5+2=7
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When we add the imaginary parts, we get j4 + ( j3) = j1 = j The sum can be expressed directly as (5 + j4) + (2 j3) = 7 + j The parentheses are not technically necessary, but they help to set the individual complexnumber addends apart on the left-hand side of the equation
Difference example To find the difference between 5 + j4 and 2 j3, we first multiply the second complex quantity by 1 That gives us
1 (2 j3) = 2 + j3 Now we can simply add 5 + j4 and 2 + j3 Adding the real parts, we obtain 5 + ( 2) = 3 Adding the imaginary parts gives us j4 + j3 = j 7 The difference can be expressed directly as (5 + j4) (2 j3) = 3 + j 7
Product example Let s multiply the complex numbers 5 + j4 and 2 j3 by each other When we treat them as binomials, the problem works out in a straightforward fashion, but we have to be careful with the signs We get
(5 + j4)(2 j3) = 5 2 + 5 ( j3) + j4 2 + j4 ( j3) = 10 + ( j15) + j8 + j ( j ) 4 3 = 10 + ( j 7) + 12 = 22 j 7 The product can be expressed directly as (5 + j4)(2 j3) = 22 j 7