Complex number arithmetic

Chapter 2.6 Complex number arithmetic

Lessons In Electric Circuits Volume II – AC Book
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Lessons In Electric Circuits Volume II – AC Book

  • 2.6.COMPLEX NUMBER ARITHMETIC41• Rectangular notation denotes a complex number in terms of its horizontal and vertical dimen-sions. Example: drive 41 miles West, then turn and drive 18 miles South.• In rectangular notation, the first quantity is the “real” component (horizontal dimension ofvector) and the second quantity is the “imaginary” component (vertical dimension of vector).The imaginary component is preceded by a lower-case “j,” sometimes called the j operator.• Both polar and rectangular forms of notation for a complex number can be related graphicallyin the form of a right triangle, with the hypotenuse representing the vector itself (polar form:hypotenuse length = magnitude; angle with respect to horizontal side = angle), the horizontalside representing the rectangular “real” component, and the vertical side representing therectangular “imaginary” component.2.6Complex number arithmeticSince complex numbers are legitimate mathematical entities, just like scalar numbers, they canbe added, subtracted, multiplied, divided, squared, inverted, and such, just like any other kind ofnumber. Some scientific calculators are programmed to directly perform these operations on two ormore complex numbers, but these operations can also be done “by hand.” This section will showyou how the basic operations are performed. It is highly recommended that you equip yourself witha scientific calculator capable of performing arithmetic functions easily on complex numbers. It willmake your study of AC circuit much more pleasant than if you’re forced to do all calculations thelonger way.Addition and subtraction with complex numbers in rectangular form is easy. For addition, simplyadd up the real components of the complex numbers to determine the real component of the sum, andadd up the imaginary components of the complex numbers to determine the imaginary componentof the sum:2 + j54 - j3+6 + j2175 - j3480 - j15+255 - j49-36 + j1020 + j82+-16 + j92When subtracting complex numbers in rectangular form, simply subtract the real component ofthe second complex number from the real component of the first to arrive at the real componentof the difference, and subtract the imaginary component of the second complex number from theimaginary component of the first to arrive the imaginary component of the difference:2 + j5(4 - j3)175 - j34(80 - j15)-36 + j1 0(20 + j82)----2 + j895 - j19-56 - j72For longhand multiplication and division, polar is the favored notation to work with. Whenmultiplying complex numbers in polar form, simply multiply the polar magnitudes of the complexnumbers to determine the polar magnitude of the product, and add the angles of the complexnumbers to determine the angle of the product: