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In Section A1 complex numbers were de ned in rectangular form. In Fig. A-3, x r cos , y r sin , and the complex number z can be written in trigonometric form as z x jy r cos  j sin  where r is the modulus or absolute value (the notation r jzj is common), given by r angle  tan 1 y=x is the argument of z. p x2 y2 , and the
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Euler s formula, e j cos  j sin , permits another representation of a complex number, called the exponential form: z r cos  jr sin  rej A third form, widely used in circuit analysis, is the polar or Steinmetz form, z r , where  is usually in degrees.
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SUM AND DIFFERENCE OF COMPLEX NUMBERS
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To add two complex numbers, add the real parts and the imaginary parts separately. To subtract two complex numbers, subtract the real parts and the imaginary parts separately. From the practical standpoint, addition and subtraction of complex numbers can be performed conveniently only when both numbers are in the rectangular form.
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EXAMPLE A1 Given z1 5 j2 and z2 3 j8, z1 z2 5 3 j 2 8 2 j10 z2 z1 3 5 j 8 2 8 j6
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The product of two complex numbers when both are in exponential form follows directly from the laws of exponents. z1 z2 r1 e j1 r2 ej2 r1 r2 e j 1 2
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The polar or Steinmetz product is evident from reference to the exponential form. z1 z2 r1 1 r2 2 r1 r2 1 2 The rectangular product can be found by treating the two complex numbers as binomials. z1 z2 x1 jy1 x2 jy2 x1 x2 jx1 y2 jy1 x2 j 2 y1 y2 x1 x2 y1 y2 j x1 y2 y1 x2
EXAMPLE A2 If z1 5e j=3 and z2 2e j=6 , then z1 z2 5e j=3 2e j=6 10e j=6 . EXAMPLE A3 If z1 2 308 and z2 5 458, then z1 z2 2 308 5 458 10 158. EXAMPLE A4 If z1 2 j3 and z2 1 j3, then z1 z2 2 j3 1 j3 7 j9.
DIVISION OF COMPLEX NUMBERS
For two complex numbers in exponential form, the quotient follows directly from the laws of exponents. z1 r1 e j1 r1 j 1 2 e z r2 e j2 r2 Again, the polar or Steinmetz form of division is evident from reference to the exponential form. z1 r1 1 r1 z2 r2 2 r2 1 2
Division of two complex numbers in the rectangular form is performed by multiplying the numerator and denominator by the conjugate of the denominator (see Section A8).   z1 x1 jy1 x2 jy2 x x y1 y2 j y1 x2 y2 x1 x1 x2 y1 y2 y x y2 x1 j 1 2 1 2 2 2 2 2 z2 x2 jy2 x2 jy2 x2 y2 x2 y2 x2 y2 2 2
EXAMPLE A5 Given z1 4e j=3 and z2 2e j=6 , z1 4e j=3 2e j=6 z2 2e j=6 EXAMPLE A6 Given z1 8 308 and z2 2 608, z1 8 308 4 308 z2 2 608 EXAMPLE A7 Given z1 4 j5 and z2 1 j2,   z1 4 j5 1 j2 6 13 j 5 5 z2 1 j2 1 j2
CONJUGATE OF A COMPLEX NUMBER The conjugate of the complex number z x jy is the complex number z x jy. Thus, Re z z z 2 Im z z z 2j jzj p zz
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In the complex plane, the points z and z are mirror images in the axis of reals. In exponential form: z re j , z re j . In polar form: z r , z r . In trigonometric form: z r cos  j sin  , z r cos  j sin  . Conjugation has the following useful properties: i z z ii z1 z2 z z 1 2 iii z1 z2 z z 1 2   z1 z iv 1 z2 z 2
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