F COMPLEX NUMBERS In this appendi, we review the basic properties of comple numbers. A comple number is a number z of the form z = a + bi where a,b are real numbers and i represents a number whose square is, that is, i = We write i =. The real numbers a and b are called the real part and imaginar part of z, respectivel. We use the notation + i 3i i i imaginar ais 4 3 3 4 i i i 3i FIGURE The comple plane. 3i i i z = 3 + 3 4 z = 3 + i z = 3 + i real ais FIGURE The absolute value z is the distance from z to the origin. + i 3i i i imaginar ais z = a + bi real ais 3 3 4 i i i z = a bi FIGURE 3 The conjugate z is the reflection of z across the real ais. Rez = a, Imz = b For eample, z = 3 i is a comple number with Rez = 3 and Imz =. A comple number is real if its imaginar part is zero. A comple number of the form z = bi is said to be purel imaginar. The set of all comple numbers is denoted b a boldface C. We represent C geometricall as the set of points in the Cartesian plane. The comple number z = a + bi corresponds to the point a, b as in Figures and 3. We call this plane the comple plane or, less commonl, an Argand diagram. The absolute value z also called the modulus is defined as the distance from a, b to the origin: z = a + b For eample, 3 + i = 3 + = 3. The comple conjugate or simpl conjugate" of z = a + bi is the number z defined b z = a bi Figure 3 shows the comple number z = + i and its conjugate z = i. Observe that z is the reflection of z across the real ais. Algebraic operations We manipulate comple numbers using the usual rules of algebra, with one difference: we replace i b whenever it occurs. The commutative, associative, and distributive laws are all valid for comple numbers. Addition and Subtraction These operations are performed b adding or subtracting the real and imaginar parts separatel: a + bi + c + di = a + c + b + di a + bi c + di = a c + b di A
A APPENDIX F COMPLEX NUMBERS For eample, 3 + i + 4i = 3 + + 4i = 4 i πi + 3i = + π 3i Here is the general formula for multipling comple numbers it is not necessar to memorize this formula: a + bic + di = ac bd + ad + bci Multiplication To multipl two comple numbers, use the distributive and associative laws together with the relations For eample: i =, ib = bi b an real number 7i + 3i = 7i + 7i3i = 4i + i }{{} = + 4i distributive law Here is another eample: 3 + i 4i = 3 4i + i 4i = 3 i + i i4i = 3 0i 8i = 3 0i + 8 = 0i To describe division, we state the following properties of the conjugate. The can be verified directl, b writing z = a + bi and w = c + di. THEOREM Properties of the Conjugate and Absolute Value numbers z and w, For an two comple z + w = z + w zw = z w zw = z w If z = a + bi, then zz = z = a + b Division To find the quotient z/w, multipl the numerator and denominator b the conjugate of the denominator: z w = zw ww = zw w EXAMPLE Dividing Comple Numbers Write + i 3 4i in the form a + bi. Solution Multipl numerator and denominator b the conjugate 3 + 4i and use : + i + i 3 + 4i + i3 + 4i = = 3 4i 3 4i 3 + 4i 3 + 4 = 4 0 + 60 + i = + 7i = + 3i
APPENDIX F COMPLEX NUMBERS A3 Ever non-zero comple number z has a reciprocal, namel z = z z since z = zz zz zz zz = EXAMPLE Finding the Reciprocal Write the reciprocal of z = + i in the form a + bi. Solution z = + i = i + i i = i + = 6 6 i Roots of Polnomials Mathematicians were first confronted with comple numbers when the began to investigate roots of polnomials. Recall that if a>0, then a has two real square roots ± a. Thus, the roots of the equation a = 0 are =± a. B convention, a denotes the positive square root. The negative number a also has two square roots, namel, the purel imaginar numbers ± ai: ± ai = ± a i = a = a Said another wa, the equation + a = 0 has two imaginar roots ± ai. For eample, the roots of + = 0 are =± =± i Caution: the relation a b = ab is onl valid if one or both a or b is positive. If a<0 and b<0, this relation fails to hold it is off b a minus sign: 3 = i 3i = 6 3 = 6 B convention, when a>0, a denotes the square root ai. More generall, the roots of a quadratic equation a + b + c = 0 are described b the famous quadratic formula = b ± b 4ac a = b ± D a where D = b 4ac is the discriminant of the equation. The roots are real if D 0 and comple if D<0. For eample, the equation + 3 + = 0 has comple roots: = 3 ± 3 4 = 3 ± = 3 ± i The quadratic formula was known in antiquit. Equivalent forms of the formula appear in Bablonian cla tablets dating to 800 BC. However, mathematicians did not immediatel accept the notion of a comple number. The focused on the real roots and ignored the comple roots. It was not until the 00 s that similar formulas for cubic degree three and quartic degree four polnomials were discovered. These formulas were
A4 APPENDIX F COMPLEX NUMBERS not onl much more complicated than the quadratic formula, the had a ver surprising feature which is illustrated b the cubic equation 3 3 = 0. According to cubic formula, one root of this equation is /3 = + /3 + 3 +.8793 3 This root is a real number, but it is epressed b a formula that involves the comple number 3. So we find, surprisingl, that even if we wanted to restrict ourselves to real roots of equations, we would still need comple numbers to write down those roots. Since there was no escape, mathematicians came to accept that comple numbers are legitimate mathematical entities that can be manipulated much like real numbers. The figure prominentl in the mathematics of the 8th centur and had been full integrated into mathematics b the earl 800 s. However, the terms real" and imaginar" are a reminder of the time when comple numbers were regarded with suspicion. The formulas discovered in the 6th centur showed that quadratic polnomials have two roots, cubic polnomials have three roots, and quartic polnomials have four roots. The suspicion grew that a polnomial of degree n has n comple roots. This was eventuall proved b Gauss in 799 at the age of. The precise form of this result is called the Fundamental Theorem of Algebra. Since some roots of a polnomial ma be multiple roots double roots, triple roots, etc, we cannot claim that a polnomial of degree n has n distinct roots. Rather, it has n roots where each root is counted with its multiplicit". THEOREM Fundamental Theorem of Algebra Ever polnomial f degree n with comple coefficients has n comple roots. More precisel, if f= a n n + +a + a 0 a n = 0 then there eist n comple numbers α,..., α n unique up to order such that f= a n α α α n The numbers α i are the roots of f. The multiplicit of a root α is defined as the number of times α occurs in the list α,α,...,α n. Polar Representation b = r sin r = z a = r cos z = a + bi FIGURE 4 Polar coordinates of a comple number A comple number z = a + bi corresponds to the point a, b in the comple plane. We ma describe z in terms of the polar coordinates r, θ of a, b Figure 4. The radial coordinate is the distance from the origin, and thus r = z The angular coordinate θ, called the argument of z, is denoted argz. As we see in Figure 4, z = a + bi has the polar representation where θ = argz. The argument θ = argz satisfies tan θ = b a z = z cos θ + i sin θ if a = 0 However, arg z is not a uniquel defined angle. Since we are free to increase θ b an integer multiple of π, the angles θ + πk for k an integer all serve as arguments for z.
APPENDIX F COMPLEX NUMBERS A We must keep in mind, therefore, that argz reall refers not to a single angle but rather to a whole set of angles that differ b multiples of π. A common convention is to choose θ in the interval π, π]. Note also: If z = a is real, then argz = 0ifa>0 and argz = π if a<0. If z = bi is purel imaginar, then argz = π/ ifb>0 and argz = π/ if b<0. EXAMPLE 3 Find the polar representation of z = 4 + i. Proof The radial coordinate is r = z = 4 + = 9. The angular coordinate is The polar representation is θ = tan 4 0.896 z = 4 + i = 9cos 0.896 + i sin 0.896 The net theorem describes several properties of the argument. Especiall important is Eq., epressing the behavior under multiplication: the argument of a product is the sum of the arguments. The argument satisfies the following prop- THEOREM 3 Properties of the Argument erties: argzw = argz + argw arg z = argz argw 3 w argz = argz 4 argz = 0 if z is a real number Proof Let θ = argz and θ = argw. We compute zw using the polar representation. Note that the addition formulas for cosine and sine are used in the last two lines: zw = z cos θ + i sin θ w cos θ + i sin θ = z w cos θ + i sin θ cos θ + i sin θ = z w cos θ cos θ sin θ sin θ + icos θ sin θ + cos θ sin θ = z w cosθ + θ + sinθ + θ Therefore, argzw = θ + θ which is equal to argz + argw as required. Net, observe that since zz =, we have argzz = argz + argz = arg = 0 This implies Eq. 4. Finall, we ma appl and 4 to prove 3: arg z w = argzw = argz + argw = argz argw
A6 APPENDIX F COMPLEX NUMBERS The polar representation gives us a geometric interpretation of multiplication of comple numbers Figure. If z and z, have arguments θ and θ, then the product z z is the comple number of argument θ + θ that lies on the circle of radius z z. z FIGURE Geometric picture of multiplication of comple numbers. z z z z + z The set of comple numbers of absolute value one make up the unit circle in the comple plane Figure 6A. Indeed, if z =, then z has radial component. Comple numbers of absolute value one have polar representation z = cos θ + i sin θ To multipl two comple numbers of absolute value one, we need onl add their arguments Figure 6B. z = e i = cos + i sin z = z z + z z FIGURE 6 Comple numbers of absolute value one lie on the unit circle. A B Euler s Formula and the Comple Eponential The formula argzw = argz + argw has the same form as the logarithm law lnzw = ln z + ln w This is the first hint that comple numbers and eponentials are related. Euler discovered a deep and important connection between comple numbers and the eponential function. As we know, e is defined for ever real number. Euler realized that e is also meaningful if is a comple number. For = iθ, Euler s formula reads e iθ = cos θ + i sin θ 6 The comple number on the right-hand side has absolute value one. So according to Euler s formula, we obtain all comple numbers of absolute value one as values of e for = iθ. Furthermore, if z = e iθ, then argz = θ.
APPENDIX F COMPLEX NUMBERS A7 We ma take Euler s formula as a definition of the previousl undefined quantit e iθ. If so, we must justif this definition b showing that imaginar eponentials obe the law of eponents: e iθ e iθ = e iθ +θ 7 This follows directl from. Indeed, if z = e iθ and z = e iθ, then and z z = z z = argz z = argz + argz = θ + θ We see that z z is the comple number of absolute value one with argument θ + θ. Thus z z = e iθ +θ as required. A more direct and perhaps more satisfactor approach is to define e z for an comple number z b the power series e z = n=0 z n n! = + z + z! + z3 3! + This epansion is valid if z is real. With some care, we can verif that it converges if z is comple and satisfies e z+w = e z e w Setting z = iθ and separating e iθ into real and imaginar parts, we obtain e iθ = + iθ + iθ! = + iθ θ =! i iθ3 3! + iθ3 3! + iθ4 4! + iθ! + θ 4 4! + i θ! θ 6 θ! + θ 4 4! θ 6 6! + + i = cos θ + i sin θ + iθ6 6! 6! θ θ 3 3! + θ! In the last step we use the Talor series of cos θ and sin θ. For z = a + bi, we have e z = e a+bi = e a e ib = e a cos b + i sin b As a first application of Euler s formula, we show that the sine and cosine functions can be epressed in terms of imaginar eponentials. EXAMPLE 4 Verif the formulas cos θ = eiθ + e iθ, sin θ = eiθ e iθ i Proof Since cos θ = cos θ and sin θ = sin θ, we find that Euler s formula applied with θ ields: e iθ = cos θ+ i sin θ = cos θ i sin θ
A8 APPENDIX F COMPLEX NUMBERS Therefore e iθ + e iθ = cos θ + i sin θ+ cos θ i sin θ = cos θ The first formula follows. The second formula follows similarl. Eq. 7 has the following generalization to a product with n factors e iθ e iθ e iθ n = e iθ +θ + +θ n Aformal proof involves induction. This relation is true for n =. If it is valid for n = k, then it also valid for n = k b the following computation: e iθ e iθ e iθ n = e iθ e iθ e iθ n e iθ n = e iθ +θ +θ n e iθ n = e iθ +θ +θ n +θ n Taking θ = θ = =θ n, we obtain DeMoivre s Formula: This formula is often written in the form e iθ n = e inθ 8 cos θ + i sin θ n = cos nθ + i sin nθ 9 We ma use Euler s Formula to describe the roots of a comple number. Recall that if is real, then e /n = e /n. But notice that e /n is a positive nth root of e. There ma eist other nth roots. For eample, if n is even, then e /n is also an nth root. When we allow comple roots, we find that ever non-zero comple number has n distinct comple nth roots. To prove this, recall that if θ = argz, then for ever integer k, Therefore, z = z e iθ+πk z /n = z /n e i θ+πk n = z /n cos θ + πk n + i sin θ + πk n We obtain a distinct root for each of the values k = 0,,..., n. Graphicall, these roots are spaced equall along the circle of radius z /n see Figure 7. z + π + 4π FIGURE 7 The five roots z / of z lie on the circle of radius z / with angular spacing π/ + 6π 8π + Radius = z /
APPENDIX F COMPLEX NUMBERS A9 EXAMPLE Find the cube roots of 7. Solution Since z = 7 is real, θ = arg7 = 0, and the cube roots of 7 are the three comple numbers 7 /3 cos πk 3 Eplicitl, the three cube roots of 7 are: πk + i sin 3 k = 0,, k = 0: 7 /3 the positive cube root k = : 7 /3 cos π 3 + i sin π 3 = 7/3 + 3 i k = : 7 /3 cos 4π 3 + i sin 4π 3 = 7/3 3 i Euler s Formula and Calculus In the chapter on second order differential equations, we use Euler s formula to find solutions of constant coefficient equations. In this subsection, we provide the background for this application. Given a comple number λ = a + bi, we define the comple eponential function: ft= e λt = e a+bit = e ta cos tb + i sin tb We shall take the domain of ftto be the set of all real numbers R in a more advanced treatment, we would consider the domain of all comple numbers. The values of ft are comple numbers, so the range of ftis contained in C. If λ = a is a real number, then the following formula holds b the Chain Rule: d dt eλt = λe λt This formula remains valid for all values of λ. Before we can verif this fact, we must define what is meant b the derivative f t. Write ftas a sum of real and imaginar parts: ft= } e ta cos {{ tb } +i e } ta {{ sin tb } = f t + if t f t f t The derivative is then defined b differentiating the real and imaginar parts separatel: f t = f t + if t Equivalentl, we ma define the derivative as a limit of difference quotients: f t = lim h 0 ft + h ft h This limit is a limit of comple values. B definition, this means that the following difference tends to zero in absolute value as h 0: f ft + h ft t 0 h
A0 APPENDIX F COMPLEX NUMBERS THEOREM 4 For all comple numbers λ, d dt eλt = λe λt Proof If λ = a + bi, then λe λt = a + bie ta cos tb + ie ta sin tb = ae ta cos tb be ta sin tb + i be ta sin tb + ae ta sin tb = d dt te ta cos tb + i d e ta sin tb dt = f t + f t = d dt eλt In our stud of second order differential equations, we use the operator notation D = d for the derivative. In this notation, Theorem 4 states: dt De λt = λe λt Appling Theorem 4 twice, we obtain D e λt = λ e λt More generall, if we appl the operator PD = ad + bd + c to the comple eponential function = e λt, we obtain PDe λt = aλ + bλ + ce λt = P λe λt Eercises. Find the conjugate z and absolute value z of the comple numbers a 4 + i b 4 + i c 9i d i. Plot the comple numbers and their conjugates in the comple plane: + i, 3 i, 4 + i. In Eercises 3 -, write the comple number in the form a + bi. 3. 3 + i4 i 4. 3 i 7i. 76i 6. + i + i 4 3 i 9.. i + i 0 i 0.. 4 + i + i i In Eercises 3-0, write the polar epression re iθ in the form a + bi. 3. e π 3 i 4. e π 3 i. e + π i 6. e 4+i 7. e + π 4 i 8. e ln 9 πi 7. + i 8. 3 + i 9. 3e 3i e 6i 0. e +3i e i
APPENDIX F COMPLEX NUMBERS A In Eercises - 4, write the comple number in polar form re iθ.. i. 3 + i 3. 3 + 4 3 i 4. i In Eercises - 8, calculate the argument of the comple number.. i 6. i 7. 3 i 8. 4e 6+ π 3 i 9. What are the arguments of the four fourth roots of. 30. What are the arguments of the five fifth roots of. 3. Find the three cube roots of. 3. Find the three cube roots of i and i. 33. Find the four fourth roots of. 34. Find the si sith roots of 9. 3. Appl De Moivre s theorem with n = 3 to prove the identities a cos 3θ = cos 3 θ 3 cos θ sin θ b sin 3θ = 3 cos θ sin θ sin 3 θ 36. If a>0, then ln a does not eist as a real number. Show that negative real numbers have comple logarithms b proving ln a = ln a + πi. 37. Let z = a + bi and w = c + di be comple numbers. Verif that z + w = z + w zw = zw 38. Give an eample where the equalit a b = ab does not hold. Under what conditions on real numbers a and b does it hold? 39. Use Euler s formula to verif the identities cosh = cosi, sinh = i sini 40. Calculate f 0 and f where ft= e +3it.