The complex number plane

Chapter The complex number plane

Teach Yourself Electricity and Electronics Third Edition Book
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Teach Yourself Electricity and Electronics Third Edition Book

  • Subtracting complex numbers works similarly. The difference (4 j7)(45j83) is found by multiplying the second complex number by 1 and then adding the re-sult, getting (4 j7)( 1(45j83)(4j7)( 45j83)41j90.The general formula for the sum of two complex numbers (ajb) and (cjd) is(ajb)(cjd)(ac)j(bd)The plus and minus number signs get tricky when working with sums and differ-ences of complex numbers. Just remember that any difference can be treated as a sum:multiply the second number by 1 and then add. You might want to do some exercisesto get yourself acquainted with the way these numbers behave, but in working with en-gineers, you will not often be called upon to wrestle with complex numbers at the levelof “nitty-gritty.’’If you plan to become an engineer, you’ll need to practice adding and subtractingcomplex numbers. But it’s not difficult once you get used to it by doing a few sampleproblems.Multiplying complex numbersYou should know how complex numbers are multiplied, to have a full understanding oftheir behavior. When you multiply these numbers, you only need to treat them as sumsof number pairs, that is, as binomials.It’s easier to give the general formula than to work with specifics here. The productof (a + jb) and (cjd) is equal to acjadjbcjjbd. Simplifying, remember thatjj1, so you get the final formula:(ajb)(cjd) = (acbd)j(adbc)As with the addition and subtraction of complex numbers, you must be careful withsigns (plus and minus). And also, as with addition and subtraction, you can get used todoing these problems with a little practice. Engineers sometimes (but not too often)have to multiply complex numbers.The complex number planeReal and imaginary numbers can be thought of as points on a line. Complex numberslend themselves to the notion of points on a plane. This plane is made by taking the realand imaginary number lines and placing them together, at right angles, so that they in-tersect at the zero points, 0 and j0. This is shown in Fig. 15-2. The result is a Cartesiancoordinate plane, just like the ones you use to make graphs of everyday things likebank-account balance versus time.Notational neurosesOn this plane, a complex number might be represented as ajb (in engineering or physi-cists’ notation), or as abi (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 266 Impedance and admittance