Section 0.0: Comlex Numbers from Precalculus Prerequisites a.k.a. Chater 0 by Carl Stitz, PhD, and Jeff Zeager, PhD, is available under a Creative Commons Attribution-NonCommercial-ShareAlike.0 license. 0, Carl Stitz.
6 Prerequisites 0.0 Comlex Numbers We conclude our Prerequisites chater with a review the set of Comlex Numbers. As you may recall, the comlex numbers fill an algebraic ga left by the real numbers. There is no real number x with x, since for any real number x 0. However, we could formally extract square roots and write x ±. We build the comlex numbers by relabeling the quantity as i, the unfortunately misnamed imaginary unit. The number i, while not a real number, is defined so that it lays along well with real numbers and acts very much like any other radical exression. For instance, (i) 6i, 7i i 4i, ( 7i) + ( + 4i) 5 i, and so forth. The key roerties which distinguish i from the real numbers are listed below. Definition 0.8. The imaginary unit i satisfies the two following roerties:. i. If c is a real number with c 0 then c i c Proerty in Definition 0.8 establishes that i does act as a square root of, and roerty establishes what we mean by the rincial square root of a negative real number. In roerty, it is imortant to remember the restriction on c. For examle, it is erfectly accetable to say 4i 4i() i. However, ( 4) 6 i 4, otherwise, we d get 4 ( 4) i 4i(i) i ( ), which is unaccetable. The moral of this story is that the general roerties of radicals do not aly for even roots of negative quantities. With Definition 0.8 in lace, we are now in osition to define the comlex numbers. Definition 0.9. A comlex number is a number of the form a + bi, where a and b are real numbers and i is the imaginary unit. The set of comlex numbers is denoted C. Comlex numbers include things you d normally exect, like + i and 5 i. However, don t forget that a or b could be zero, which means numbers like i and 6 are also comlex numbers. In other words, don t forget that the comlex numbers include the real numbers, so 0 and are both considered comlex numbers. The arithmetic of comlex numbers is as you would exect. The only things you need to remember are the two roerties in Definition 0.8. The next examle should hel recall how these animals behave. Some Technical Mathematics textbooks label it j. While it carries the adjective imaginary, these numbers have essential real-world imlications. For examle, every electronic device owes its existence to the study of imaginary numbers. Note the use of the indefinite article a. Whatever beast is chosen to be i, i is the other square root of. To use the language of Section 0.., R C.
0.0 Comlex Numbers 7 Examle 0.0.. Perform the indicated oerations.. ( i) ( + 4i). ( i)( + 4i). i 4i 4. 5. ( )( ) 6. (x [ + i])(x [ i]) Solution.. As mentioned earlier, we treat exressions involving i as we would any other radical. We distribute and combine like terms: ( i) ( + 4i) i 4i Distribute 6i Gather like terms Technically, we d have to rewrite our answer 6i as ( ) + ( 6)i to be (in the strictest sense) in the form a + bi. That being said, even edants have their limits, and we ll consider 6i good enough.. Using the Distributive Proerty (a.k.a. F.O.I.L.), we get ( i)( + 4i) ()() + ()(4i) (i)() (i)(4i) F.O.I.L. +4i 6i 8i i 8( ) i i +8 i. How in the world are we suosed to simlify i 4i? Well, we deal with the denominator 4i as we would any other denominator containing two terms, one of which is a square root: we and multily both numerator and denominator by + 4i, the (comlex) conjugate of 4i. Doing so roduces i 4i ( i)( + 4i) ( 4i)( + 4i) Equivalent Fractions +4i 6i 8i 9 6i F.O.I.L. i 8( ) 9 6( ) i i 5 5 5 i 4. We use roerty of Definition 0.8 first, then aly the rules of radicals alicable to real numbers to get i i i 6 6.
8 Prerequisites 5. We adhere to the order of oerations here and erform the multilication before the radical to get ( )( ) 6 6. 6. We can brute force multily using the distributive roerty and see that (x [ + i])(x [ i]) x x[ i] x[ + i] + [ i][ + i] F.O.I.L. x x +ix x ix + i +i 4i Distribute x x + 4( ) Gather like terms x x +5 i This tye of factoring will be revisited in Section.4. In the revious examle, we used the conjugate idea from Section 0.9 to divide two comlex numbers. More generally, the comlex conjugate of a comlex number a + bi is the number a bi. The notation commonly used for comlex conjugation is a bar : a + bi a bi. For examle, +i i and i +i. To find 6, we note that 66+0i 6 0i 6, so 6 6. Similarly, 4i 4i, since 4i 0+4i 0 4i 4i. Note that + 5+ 5, not 5, since + 5+ 5+0i + 5 0i + 5. Here, the conjugation secified by the bar notation involves reversing the sign before i, not before 5. The roerties of the conjugate are summarized in the following theorem. Theorem 0.. Proerties of the Comlex Conjugate: Let z and w be comlex numbers. z z z + w z + w zw z w z n (z) n, for any natural number n z is a real number if and only if z z. Essentially, Theorem 0. says that comlex conjugation works well with addition, multilication and owers. The roofs of these roerties can best be achieved by writing out z a + bi and w c + di for real numbers a, b, c and d. Next, we comute the left and right sides of each equation and verify that they are the same. The roof of the first roerty is a very quick exercise. 4 To rove the second roerty, we comare z + w with z + w. We have z + w a + bi + c + di a bi + c di. To find z + w, we first comute so z + w (a + bi)+(c + di) (a + c)+(b + d)i z + w (a + c)+(b + d)i (a + c) (b + d)i a + c bi di a bi + c di z + w 4 Trust us on this.
0.0 Comlex Numbers 9 As such, we have established z + w z + w. The roof for multilication works similarly. The roof that the conjugate works well with owers can be viewed as a reeated alication of the roduct rule, and is best roved using a technique called Mathematical Induction. 5 The last roerty is a characterization of real numbers. If z is real, then z a +0i, so z a 0i a z. On the other hand, if z z, then a + bi a bi which means b b so b 0. Hence, z a +0i a and is real. We now return to the business of solving quadratic equations. Consider x x + 5 0. The discriminant b 4ac 6 is negative, so we know by Theorem 0.0 there are no real solutions, since the Quadratic Formula would involve the term 6. Comlex numbers, however, are built just for such situations, so we can go ahead and aly the Quadratic Formula to get: x ( ) ± ( ) 4()(5) () ± 6 ± 4i ± i. Examle 0.0.. Find the comlex solutions to the following equations. 6. Solution. x x + x +. t 4 9t +5. z +0. Clearing fractions yields a quadratic equation so we then roceed as in Section 0.7. x x + x + x (x + )(x + ) Multily by (x + ) to clear denominators x x + x +x + F.O.I.L. x x +4x + Gather like terms 0 x +x + Subtract x From here, we aly the Quadratic Formula ± x 4()() () ± 8 ± i 8 ± i ( ± i ) ± i Quadratic Formula Simlify Definition of i Product Rule for Radicals Factor and reduce 5 See Section 9.. 6 Remember, all real numbers are comlex numbers, so comlex solutions means both real and non-real answers.
0 Prerequisites We get two answers: x +i and its conjugate x i. Checking both of these answers reviews all of the salient oints about comlex number arithmetic and is therefore strongly encouraged.. Since we have three terms, and the exonent on one term ( 4 on t 4 ) is exactly twice the exonent on the other ( on t ), we have a Quadratic in Disguise. We roceed accordingly. t 4 9t +5 t 4 9t 5 0 Subtract 9t and 5 (t + )(t 5) 0 Factor t + 0 or t 5 Zero Product Proerty From t + 0 we get t, or t. We extract square roots as follows: r r t ± ±i ±i ±i ± i, where we have rationalized the denominator er convention. From t 5, we get t ± 5. In total, we have four comlex solutions - two real: t ± 5 and two non-real: t ± i.. To find the real solutions to z + 0, we can subtract the from both sides and extract cube roots: z, so z. It turns out there are two more non-real comlex number solutions to this equation. To get at these, we factor: z + 0 (z + )(z z + ) 0 Factor (Sum of Two Cubes) z + 0 or z z +0 From z + 0, we get our real solution z. From z z + 0, we aly the Quadratic Formula to get: z ( ) ± ( ) 4()() () ± ± i Thus we get three solutions to z + 0 - one real: z and two non-real: z ±i. As always, the reader is encouraged to test their algebraic mettle and check these solutions. It is no coincidence that the non-real solutions to the equations in Examle 0.0. aear in comlex conjugate airs. Any time we use the Quadratic Formula to solve an equation with real coefficients, the answers will form a comlex conjugate air owing to the ± in the Quadratic Formula. This leads us to a generalization of Theorem 0.0 which we state on the next age.
0.0 Comlex Numbers Theorem 0.. Discriminant Theorem: Given a Quadratic Equation AX + BX + C 0, where A, B and C are real numbers, let D B 4AC be the discriminant. If D > 0, there are two distinct real number solutions to the equation. If D 0, there is one (reeated) real number solution. Note: Reeated here comes from the fact that both solutions B±0 A reduce to B A. If D < 0, there are two non-real solutions which form a comlex conjugate air. We will have much more to say about comlex solutions to equations in Section.4 and we will revisit Theorem 0. then.
Prerequisites 0.0. Exercises In Exercises - 0, use the given comlex numbers z and w to find and simlify the following. z + w zw z z z w z zz (z). z +i, w 4i. z +i, w i. z i, w +i 4. z 4i, w i 5. z 5i, w +7i 6. z 5+i, w 4+i 7. z i, w +i 8. z i, w i w z 9. z + i, w + i 0. z + i, w i In Exercises - 8, simlify the quantity.. 49. 9. 5 4 4. ( 5)( 4) 5. 9 6 6. ( 9)( 6) 7. ( 9) 8. ( 9) We know that i which means i i i ( ) i i and i 4 i i ( )( ). In Exercises 9-6, use this information to simlify the given ower of i. 9. i 5 0. i 6. i 7. i 8. i 5 4. i 6 5. i 7 6. i 04 In Exercises 7-5, find all comlex solutions. 7. x +64x 8. 5t +t +5t(t + ) 9. y +4y 4 0. w w. y y y. x x x. x 5 x 4. 6. Multily and simlify: x [ i ] 5y 4 + y y 5. z 4 6 x [ + i ]
0.0 Comlex Numbers 0.0. Answers. For z +i and w 4i z + w +7i zw + 8i z 5 + i z i z w 4 i w z + 8 i z i zz (z) 5 i. For z +i and w i z + w zw i z i z i z w +i w z i z i zz (z) i. For z i and w +i z + w +i zw i z z i z w 5 5 i w z +i z i zz (z) 4. For z 4i and w i z + w +i zw 8+8i z 6 z 4 i z w +i w z i z 4i zz 6 (z) 6 5. For z 5i and w +7i z + w 5+i zw 4 + i z 6 0i z 4 + 5 4 i z w 9 5 5 i w z 9 4 + 4 i z +5i zz 4 (z) 6 + 0i
4 Prerequisites 6. For z 5+i and w 4+i z + w +i zw 6i z 4 0i z 5 6 6 i z w 9 0 + 7 0 i w z 9 7 i z 5 i zz 6 (z) 4 + 0i 7. For z i and w +i z + w zw 4 z 4i z 4 + 4 i z w i w z i z +i zz 4 (z) 4i 8. For z i and w i z + w i zw 4 z i z 4 + 4 i z w + i w z i z +i zz 4 (z) +i 9. For z + i and w + i z + w i zw z + i z i z w i w z + i z i zz (z) i 0. For z + i and w i z + w zw z i z i z w i w z i z i zz (z) i. 7i. i. 0 4. 0
0.0 Comlex Numbers 5 5. 6. 7. 8. i 9. i 5 i 4 i i i 0. i 6 i 4 i ( ). i 7 i 4 i ( i) i. i 8 i 4 i 4 i 4 (). i 5 i 4 i ( i) i 4. i 6 i 4 6 i ( ) 5. i 7 i 4 9 i i i 6. i 04 i 4 76 76 7. x ± i 4 0. w ± i 7 5 ± i. x 6. x 6x + 8. t 5, ± i. y ± i 4. y ±i, ± i 9. y ±, ±i. x 0, ± i 5. z ±, ±i