8.6 The Complex Number System


 Alan Rodgers
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1 8.6 The Complex Number System Earler n the chapter, we mentoned that we cannot have a negatve under a square root, snce the square of any postve or negatve number s always postve. In ths secton we want to fnd a way to deal wth an expresson that does have a negatve under a square root. Frst, consder the equaton x 0. Clearly ths equaton has no real number solutons. Therefore, we need to make an entrely new set of numbers to represent these types of values. We start wth the followng defnton Defnton: The magnary unt The magnary unt,, s defned as. Therefore,. Also, we can notce that, and. In fact, the powers of contnue to repeat lke ths 6 So we can easly smplfy any power of snce every th power s just. Also, we can see that we should always end up wth an expresson that has a power of that s at most. Example : Smplfy. a. b. c. 7 Soluton: a. By what we saw above we can smply wrte. Therefore,. as. Snce we have b. Smlar to part a. we want to wrte as havng as many th powers as possble, that way we can have for those values. So can proceed as follows c. So we wll proceed just as we dd n part b. 0 because
2 because Snce we are now dealng wth a negatve under a square root we need to know how to properly smplfy radcals wth ths n mnd. So we have the followng property. Property of negatve square roots c c c c We use ths property to smplfy square roots that contan negatves. Example : Smplfy. a. b. 7 c. 7 Soluton: a. We can smply use the property to smplfy as follows b. Agan, we can remove the negatve from under the radcal by pullng t out as an and then smplfy the resultng expresson. We get 7 7 Note: Textbooks dffer on the poston of the n a problem of ths type. Most put the n the back of the expresson. However, we choose to put t between the coeffcent and the radcal to remove any possble confuson of whether or not the s under the radcal. If we were to wrte as t can appear that the s part of the radcand. c. Agan, we wll pull or the and then contnue to smplfy Snce we do not have lke radcals we cannot combne the remanng terms and thus are fnshed.
3 Now that we have a famlarty wth the magnary unt, we can ntroduce the number system whch t generates. Defnton: Complex Numbers A number of the form a b, where a and b are real numbers and s called a complex number. a s called the real part and b s called the magnary part. A complex number wrtten wth the real part s frst and the magnary part s last s n standard form. We want to be able to perform basc operatons on these complex numbers. Its actually very smple. We smply need to remember that s really a radcal and. Wth ths n mnd, we can smply add, subtract and multply as we dd n the earler part of the chapter. We smply need to make sure that we smplfy all of our powers of. Example : Perform the operatons. Put your answers n standard form. a. 0 7 b. 0 0 c d. e. 0 f. 08 Soluton: a. As we sad, we can smply perform operatons as we dd earler n ths chapter. So that means we need to combne lke radcals. In ths case, the terms contanng would be lke. So we get b. Ths tme we need to start by dstrbutng the negatve, then combne the lke radcals. Ths gves c. Now, to multply complex numbers t s actually easer to just treat them as polynomals and then just smplfy the powers of by rememberng that. So we get Snce we wanted the answers n standard form, we needed to wrte the real part frst followed by the magnary part. d. Agan, we wll multply as f these were polynomals and smplfy the result. Always remember. Ths gves
4 e. In an expresson of ths form, we must always start by removng the negatves from the radcals. If we do not we end up wth a completely dfferent (and therefore ncorrect) result. Once we have the negatves out of the radcals, we can smply multply as before. We get f. Lastly, agan we begn by pullng the negatves out of the radcals. Then we smplfy as before. Ths gves Lastly we need to deal wth how to dvde complex numbers. However, f we smply remember that s a radcal then we can treat the dvson as we dd before. That s, we just use the conjugate. However ths tme we have what s called the complex conjugate and complex conjugates always have a very smple product. Complex Conjugates a b and b a are called complex conjugates. Also, a b a b a b. So rather that multplyng the conjugates out every tme we can smply add the squares of the real and magnary parts to smplfy the process. Example : Perform the operatons. Wrte your answers n standard form. a. 7 b. c. d. 8 Soluton: a. So we can smplfy by multplyng numerator and denomnator by the conjugate of the denomnator as we dd before. Then we just need to smplfy and reduce. Notce that we can just use the formula above for the product of the conjugates. We proceed as follows 7 8 8
5 8 8 b. Agan, we smply multply numerator and denomnator by the complex conjugate of the denomnator and then smplfy and reduce. Notce that the conjugate s n fact the numerator, but we need not be concerned wth that. We smply multply t out as we learned before. We get c. Ths tme we only have one term on the denomnator. Therefore, there s no need to use the conjugate. We smply need to multply by somethng that wll elmnate the. Well, recall. Therefore, we can smply multply by on numerator and denomnator to elmnate the on the denomnator. We get d. Lastly, we need to start by pullng out all the negatves and smplfyng the radcals. Once that s completed, we can multply by the conjugate on the numerator and denomnator. We get 8 8
6 Exercses Smplfy Perform the operatons. Wrte your answers n standard form
7
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