Adding Some Perspective to De Moivre's Theorem: Visualising The

Adding some perspective to de Moivre’s theorem: Visualising the n-th roots of unity

Nicholas S . Bardell RMIT University nick .bardell@rmit .edu .au

Introduction

n n-th root of unity, where n is a positive integer (i.e., n = 1,2,3,….), is a Anumber z satisfying the equation

zn = 1 (1)

Traditionally, z is assumed to be a complex number and the roots are usually determined by using de Moivre’s theorem adapted for fractional indices. The roots are represented in the Argand plane by points that lie equally pitched around a circle of unit radius. The n-th roots of unity always include the real number 1, and also include the real number –1 if n is even. The non-real n-th roots of unity always form complex conjugate pairs. This topic is taught to students studying a mathematics specialism (ACARA, n.d., Unit 3, Topic 1: Complex Numbers) as an application of de Moivre’s theorem with the understanding that the roots occur in the complex domain. Meanwhile, in the Cartesian plane, a closely related topic deals with the solution of polynomials (ACARA, n.d., Unit 2, Topic 3: Real and Complex Numbers). Consider the most general case in which

n n–1 2 y = anx + an–1x + … + a2x + a1x + a0.

The roots are found by setting y = 0 and solving for x, which can be carried out easily when n = 1 and n = 2 but becomes significantly more challenging for higher order values of n ≥ 3. However, there is one special case which is amenable to solution for any value of n, namely the one in which all the

coefficients equal zero excepta n = 1 and a0 = –1; then, the following reduced monic polynomial results:

xn – 1 = 0 (2) Australian Senior Mathematics Journal vol. 29 no. 2

40 Adding some perspective to de Moivre’s theorem Australian Senior Mathematics Journal vol. 29 no. 2 41 (3) -roots -roots . to find 1 -th roots 1 n − = – 1. – 1 and i = 8 which will be sufficient = 0, 1, 2, … n into Equation (3) to determine = 0, 1, 2, … n where k

= 5 and n ⎞ ⎠ k π n 2 ⎛ ⎝ = 3, n ) where k n i s i + in the Cartesian plane and the resulting n plane and the – 1 in the Cartesian n ⎞ ⎠ k π -th roots of unity, although the location of these n although the location of unity, -th roots sin (2pk n = x 2 ⎛ ⎝ -multiple solutions is accounted for by the inclusion of the s ) + i . o c = 1 n z . Now apply de Moivre’s theorem using the fractional index . Now apply de Moivre’s k = cos (2pk z -th roots of z The aim of this paper is to demonstrate visually the connection between the is to demonstrate visually the The aim of this paper , in polar form thus: synonymous problems, the connection between them is rarely made. the connection between them synonymous problems, and illustrates not only the location of the roots in relation to the original and illustrates not only the location a natural extension of the surface techniques first presented by visualisation de Moivre’s method de Moivre’s Students studying a mathematics specialism as a part of the Victorian VCE Students studying a mathematics for Equation (2). However, although both topics deserve to be taught as However, for Equation (2). discourse by Bardell (2014) for further details). In order to find the n discourse by Bardell (2014) for further details). In order to find the integer of unity, the value of unity itself has first to be expressed as a complex number, itself has first to be expressed as a complex number, the value of unity of unity, conjugate roots in the Cartesian plane at one of the turning points of a cubic the Cartesian plane at one of the conjugate roots in essentially the same, since Equations (1) and (2) are equation.) In fact, equation but also shows why they occur with conjugate pairings. Examples equation but also shows why they reduced polynomial y reduced polynomial the possibility of complex solutions should most definitely be entertained complex solutions should most the possibility of the reader will find a three-dimensional surfacethe reader will find a three-dimensional representation of Equation emerges. The approach adopted here is to illustrate the general pattern that the n the roots of specific cases. relative to the Cartesian plane is commonly misunderstood (see Stroud (1986, (see Stroud (1986, misunderstood is commonly the Cartesian plane relative to will be provided for the cases n which invariably appear in the Argand plane. There is no contradiction here: contradiction is no plane. There Argand the in appear invariably which Bardell (2012) for quadratic equations. Programme 2, TheoryProgramme places the complex he (incorrectly) of Equations); Moivre’s theorem and its applications to complex numbers (see the short (see numbers complex to its applications and theorem Moivre’s

It only remains to substitute chosen values of n chosen values to substitute It only remains z This also leads to the n This also The possibility of n 1997) or Queensland QCE (Mathematics C, 2009) will be familiar with de 1997) or Queensland QCE (Mathematics (Specialist Mathematics, 2010), HSC in NSW (Mathematics Extension in NSW, in NSW (Mathematics Extension in NSW, (Specialist Mathematics, 2010), HSC (2) provides the full link between both the Cartesian and Argand planes, (2) provides the full link between 42 Australian Senior Mathematics Journal vol. 29 no. 2 Bardell The cube roots ofunity:n=3 The cuberoots (Argand) plane they appear equally spaced at 120° intervals aroundacircleof (Argand) planetheyappearequallyspacedat120°intervals An indicationisgiveninFigure2,basedontheapriori knowledgethatthereal When k

By usingdeMoivre’s TheoremfromEquation(3), when k rather could—andpossibly should—includecomplexvalues.Whilstthis is to unit radiuscentredattheoriginO(seeFigure1). by Stroud(1986). means is another matter altogether! Note: The turning point of this curve means isanothermatteraltogether!Note:The turning pointofthiscurve matter ofsomeconjectureexactlywheretheremaining tworootsarelocated. equation intheCartesianplaneisshownFigure2.Althoughthreeroots implicitly obvious,since no restrictionwaseverplacedonx is foundbysetting part ofbothcomplexconjugaterootsis–0.5.Making senseofwhatthisreally given thatthecomplexconjugaterootsoccurat aturningpoint,asinferred and alwaystooccurasa conjugatepair, thisimplies thatthevaluesassigned and whenk are expected, only one real root is visible in this plot, occurring at x These thenarethethreerootsofunity. Iftheseareplottedinthecomplex In thisparticularcase,sincethetwootherrootsare knowntobecomplex Now considerthecorrespondingpolynomialy inEquation(2)donot have tobelimitedthesetofrealnumbersbut =1 =0

=2 Figure 1.PlotintheArgand planeshowingthethree cuberoots of1.

3 z d d = x y =3x z 3 1 = c 2 =0,whichgivesx o s ⎝ ⎛ 2 3 3 3 π z z k = = ⎠ ⎞ 3 − − + z 2 2 1 1 i = s − + i 1 n ⎝ ⎛ 2 2 3 3 2 i i 3 π =0.Henceitisbynomeansa k ⎠ ⎞ =x , k =0,1,2. 3 –1.Asimpleplotofthis ory attheoutset, x = 1. It is a Adding some perspective to de Moivre’s theorem Australian Senior Mathematics Journal vol. 29 no. 2 43 (5a) (4a) (4b) can and G or B must also be – 1. 3 is allowed to take the – 1 3 – 1 3 + iH) (5b) – iH 3 2 – 1 2 – H = (G and so forth. Since y H , then it is possible to equate the real the equate to possible is it then iB, 2 – 3GH + – 3GH H 3 = –i A 2 3 = 3G B = G A is not for the faint-hearted! is not for + 3iG 3 + iH, then – 1 becomes y = –1, i 3 2 = G y = x y Figure 2. Plot in the Cartesian 2. Figure plane of the equation y = x Nevertheless, if this line of reasoning is pursued and x Nevertheless, if this line of reasoning Equations (5a) and (5b) describe the generalised complex form of Equations (5a) and (5b) describe the generalised complex simplifying the complex form of this equation, viz. actual classification of this, and the higher-order surfaces and the higher-order actual classification of this, presented herein, and remembering that i generally complex value G generally complex values of x complex values generally generally complex, say of the form the of say complex, generally A is beyond the scope of this paper) in which the ordinate value it is rarely presented as an explicit proposition. Perhaps one reason why this one reason Perhaps explicit proposition. presented as an it is rarely a plot using one of expedience—constructing considered is simply is seldom explained previously), there is much profit to be gained from expanding and explained previously), there is much be plotted over a grid of points in the Argand plane defined by the H be plotted over a grid of points in the Argand plane defined by Equation (4a). Each equation represents a three-dimensional surface (the Equating Im parts: Equating Re parts: Rather than solve this equation directly using de Moivre’s theorem (as theorem (as directly using de Moivre’s Rather than solve this equation

The expansion of Equation (4a) is easily accomplished using Pascal’s triangle The expansion of Equation (4a) is easily accomplished using Pascal’s (Re) and imaginary (Im) parts of both sides of Equation (4b). 44 Australian Senior Mathematics Journal vol. 29 no. 2 Bardell H G Re(y Equation (5b)forarangeofcomplexx visualised. which surface is being investigated. It is understood that the which surface location and nature of the roots will be defined where the two surfaces for location andnatureoftherootswillbedefinedwheretwosurfaces the Re(x that bothRe(y the verticalaxisrepresentseitherRe(y the followingACARA(n.d.,Rationale)statedaim:“todevelopstudents’ using colloquially as‘monkeysaddles’ onaccountthatamonkeysittingastride the capacity tochooseandusetechnologyappropriately”. computer algebrasystemsavailabletoschools,andfullycommensuratewith plane positionedatzeroaltitude. planes representtheCartesianplaneforeachsurface. axes. By calculating both individual surfaces definedbyEquation(5a)and axes. Bycalculatingbothindividualsurfaces Figure 3(a). Surface A: The real part ofthecomplex A:TherealFigure part 3(a).Surface Figure AatH =0, 4.Sectiontakenthrough Surface and revealing thetraceoforiginalcubicequation

≤ 2,areshowninFigure3(a) and3(b)respectively. areknown Bothsurfaces All thethree-dimensionalplotspresentedinthispaperwereconstructed The three-dimensional surfaces A The three-dimensionalsurfaces Clearly, fromthedefinitionofaroottooccurwheny In the three-dimensional surfaces thatfollow,In thethree-dimensionalsurfaces thehorizontalplanecontains ) (≡ Mathcad (2007), which is one of many VCE/QCE/HSC-approved H ) (≡ values, the true and most general form of Equation (4a) can be representation ofy=x A ) andIm(y G ) andIm(y ) andIm(x y =x 3 –1 ) (≡ 3 ) (≡ ) mustsimultaneouslybezero.Inotherwords,the –1. B ) haveacommonintersectionwithhorizontal H ) axes,thusformingtheArgandplane,whilst andB ) (≡ values,i.e.,arangeofcorresponding Figure 3(b). Surface B: The imaginary part ofthe part B:Theimaginary Figure 3(b).Surface Figure 5. The intersections of Surfaces A =B0, Figure 5.TheintersectionsofSurfaces revealing thelocationofthree roots of A , plottedovertherange –2≤ complex representation ofy=x ) orIm(y y =x ) (≡ 3 –1. =0,thisimplies B ) dependingon GA and 3 –1. GB G

, Adding some perspective to de Moivre’s theorem Australian Senior Mathematics Journal vol. 29 no. 2 45 = 3, = 0 – 1, and – 1. This 3 3 = x = G -axis. This means the -axis. This means is also the symmetryHB is also the . , has a place for its two legs and and legs two its for place a has , = 0. These are clearly shown in B ) plane is in fact the Cartesian the in fact is plane GB) , which is its plane of symmetry, is is its plane of symmetry, , which = B ) have common intersections with a ) have common intersections with = 0; an immediate association with B = B ) must simultaneously be zero. In other ) must simultaneously be zero. ) (≡ (or GA -axis of surfaceof -axis H ) and Im(y = 0 in Equation (5), the following results: A = 0 in Equation The dashed line shows De Moivre’s circle superimposed. circle The dashed line shows De Moivre’s , or the or , . ) (≡ A) and Im(y -axis. This observation holds for all reduced polynomials of the – 1. Figure 6. Plan view of Figure 5 showing the location of the cube roots of unity. of unity. 5 showing the location of the cube roots 6. Plan view of Figure Figure n has no influence at all on the Cartesian plane, which explains why at all on the Cartesian plane, has no influence B = x Upon setting H With reference to Figure 5, the definition of a root to occur when y With reference to Figure 5, the This is confirmed from a plan view of these surface intersections, as -axis of surfaceof -axis A = 0. In other words, the trace of the purely real cubic polynomial, as first as polynomial, cubic real purely the of trace the words, other In 0. = surfaces for Re(y should be observed when the surfaces A surface shown in Figure 6. The unit circle centred at the origin is superimposed, and shown in Figure 6. The unit circle centred at the origin is superimposed, also explains why it is reasonable to expect any real root(s) to lie somewhere real root(s) to any to expect reasonable is why it explains also along the G form y implies that both Re(y plane. It is observed points represented by B GA plane the locus of that in the plane in Figure 2, is found in the GA plane Cartesian x–y plane in Figure presented in the Moivre’s solution shown in Figure 1 is hence made. The ability to visualise de Moivre’s passes through each point where A plane of surface HB plane of to the principal perpendicular B plane of surface B much insight to be gained. For instance, by extending the solution space of much insight to be gained. For instance, by extending the solution three points plotted in the Argand diagram in Figure 1. three points plotted in the Argand three roots will be expected, and hence three unique points of intersection three roots will be expected, and the single curve shown in Figure 2 provides a full description of y the connection between the two approaches is now established and allows the connection between the two approaches is now established plane of surface principal GA plane of tail. The A words, the location and nature of the roots will be defined where the two words, the location and nature horizontal plane positioned at zero altitude. For the present case with n horizontal plane positioned at zero Figure 5. These points of intersection are, of course, none other than the Figure 5. These points of intersection = 0 is in fact a straight ‘nodal’ line coincident with the G ‘nodal’ line coincident with = 0 is in fact a straight G B (see Figure 4). This confirms the confirms This 4). Figure (see 46 Australian Senior Mathematics Journal vol. 29 no. 2 Bardell The fifth roots ofunity:n=5: The fifth (Argand) plane they appear equally spaced at 72° intervals aroundacircleof (Argand) planetheyappearequallyspacedat72°intervals The surfaces A The surfaces The polynomial When k x–y plotintheCartesianplaneismerelyatwo-dimensional‘slice’ofmuch

By usingdeMoivre’s theoremfromEquation(3), Equating Imparts: Expanding andsimplifyingthecomplexformofthisequationgives: Equation (2)tonowincludecomplexnumbers,itisevidentthattheoriginal Equating Reparts: A planesGAforsurface with symmetry when k when k when k now exhibits five primary ‘lobes’equi-spacedaroundthe origin,again now exhibitsfiveprimary real rootappearsatx the original Cartesian to occurinpairsequi-distantfromtheG unit radiuscentredattheoriginO.Again,becausen clear thatthetwo,somewhattentativelyplaced,complexconjugaterootsare each otheratrightangles,itisevidentwhythecomplexconjugaterootshave Also,byvirtueofthesymmetry more generalthree-dimensionalsurface. indeed locatedatx present in both surfaces present in both surfaces imaginary (Im)partsofbothsidesEquation (6b): imaginary front ofthegraphasitiscurrentlyplottedintwo-dimensions. and whenk surfaces aresuperimposed,and a zeroplaneintroduced,fiveuniquepoints surfaces These thenarethefifthrootsofunity. Whenplottedinthecomplex Surfaces Surfaces =2, =1, =3, =0, =4, y A =G andB 5 andBareagainfoundfromequatingthereal (Re)and z 5 = +5iG z =–0.5,butoccuradistance0.866unitsbehindandin y 1 5 =1. =x are plotted in Figures 7(a) and 7(b). Each surface areplottedinFigures 7(a) and7(b).Eachsurface = x–y plane! Returning to Figure 2, it should now be 4 A c A H 5 o –1becomesy and =G B s –10G 5 5 5 5 5 ⎝ ⎛ =5G z z z z z 2 =0.309+0.951i =1 =0.309–0.951i =–0.809–0.588i =–0.809+0.588i 5 π 5 B –10G k , and the way these symmetry planes intersect , and the way these symmetry ⎠ ⎞ 4 3 H H + –10G 2 i –10iG s 3 i H n 2 ⎝ ⎛ for surface B andHBforsurface =(G +5GH 2 2 -axis, i.e.,behindandinfrontof π 5 H 2 k H 3 +iH) ⎠ ⎞ +H 3 , +5GH 4 k –1 = 5 (7b) 5 0 –1. , isodd,onlyonesingle 1 4 , +iH 2 , 3 , 4 5 –1 . . Whenthese (6b) (7a) (6a) Adding some perspective to de Moivre’s theorem Australian Senior Mathematics Journal vol. 29 no. 2 47 – 1. 5 complex representation of y = x complex representation Figure 7(b). SurfaceFigure B: The imaginary part of the . The roots are indicated by black dots. are – 1. The roots 5 – 1. 5 = 0 through surface quintic A will reveal the original = 0 is shown in Figure 9, with the unit circle superimposed superimposed with the unit circle in Figure 9, = 0 is shown The dashed line shows De Moivre’s circle superimposed. circle The dashed line shows De Moivre’s the five roots of y = x the five = B A Figure 8. The intersections of Surfaces 8. the location of Figure A = B = 0, revealing Figure 9. Plan view of Figure 8 showing the location of the fifth roots of unity. of unity. roots 8 showing the location of the fifth 9. Plan view of Figure Figure representation of y = x representation

Figure 7(a). Surface 7(a). partFigure A: The real of the complex a section taken at H a section taken intersections intersections equation as it would appear in the Cartesian x–y plane. equation as it would of intersection are found, as shown in Figure 8. A true plan view of the surfacethe of view plan true A 8. in Figure shown as found, are intersection of to indicate the solution found from using de Moivre’s theorem. Finally, Finally, theorem. de Moivre’s found from using the solution to indicate 48 Australian Senior Mathematics Journal vol. 29 no. 2 Bardell –1, –0.707–0.707i The eighth roots ofunity:n=8 The eighthroots The surfaces A The surfaces The polynomial These eightrootsappearinorderasfollows:1,0.707+i y =G

By usingdeMoivre’s theoremfromEquation(3), Equating Imparts: Equating Reparts: Expanding andsimplifyingthecomplexformofthisequationgives: view of the surface intersectionsA view ofthesurface to a sensible value). Each surface consists of eight primary ‘lobes’ equi-spaced ‘lobes’equi-spaced consistsofeightprimary to asensiblevalue).Eachsurface theorem. Finally, asectiontakenatH A these surfaces 11. A true plan unique pointsofintersectionarefound,asshowninFigure 11.Atrueplan centred attheoriginO.Now, becausen circle superimposedtoindicatethesolutionfoundfromusingdeMoivre’s plane. original octavicequationasitwouldappearinthe Cartesianx–yplane. imaginary (Im)partsofbothsidesEquation(8b): imaginary plane they appear equally spaced at 45° intervals around a circle of unit radius plane they appear equally spaced at 45° intervals around the origin, where a central plateau is evident. Again, the principal around theorigin,whereacentralplateauisevident.Again,principal for surface A planesarefoundtobeGAforsurface symmetry H

Figure 10(a). Surface A: The real part ofthe A: TherealFigure part 10(a).Surface ≤ Surfaces Surfaces 1.5. (This curtailed range has been necessary to restrict the vertical scale torestrictthe verticalscale 1.5.(Thiscurtailedrangehasbeennecessary 8 +8iG complex representation ofy=x 7 H–28G A andB 8 andB andB z 6 = , –i A H are plottedinFigure10(a)and10(b)overtherange–1.5 ≤ are z B = G 2 y 8 1 –56iG , 0.707–i =8G =x are superimposed, and a zero plane introduced, eight aresuperimposed,andazeroplaneintroduced,eight = areagainfoundfromequatingthereal(Re)and 8 c –28G o 8 –1becomesy s 7 5 H ⎝ ⎛ H 8 2 –1. –56G 3 +70G 8 π 6 H k ⎠ ⎞ 2 =B +70G + 4 5 i H . Whenplottedinthecomplex(Argand) H s 0 is shown in Figure 12, with the unit =0 isshowninFigure12,withtheunit i 4 n 3 = 0 through surface A =0throughsurface +56iG +56G iseven,realrootsappearatx ⎝ ⎛ Figure 10(b). Surface B: The imaginary part of part B: Theimaginary Figure 10(b).Surface 4 =(G H 2 8 π 4 the complexrepresentation ofy=x k –28G 3 H ⎠ ⎞ 3 +iH) H , 5 –28G k 5 for surface B andGBforsurface –8GH = 2 H 0 8 –1 , 2 6 1 H +H , 6 2 –8iGH 7 , (9b) … 8 –1 , i 7 , –0.707 +0.707i . will reveal the willrevealthe 7 +H 8 –1 (8b) 8 –1. =±1. . When . When (9a) (8a)

G , , Adding some perspective to de Moivre’s theorem Australian Senior Mathematics Journal vol. 29 no. 2 49 , these ) and the and ) . The roots are indicated by black dots. are – 1. The roots 8 – 1 presented in the Cartesian x–y plane, and its n = x The dashed line shows De Moivre’s circle superimposed. circle The dashed line shows De Moivre’s the eight roots of y = x the eight roots Figure 11. The intersections of Surfaces 11. The intersections of the location of Figure B = 0, revealing A = Figure 12. Plan view of Figure 11 showing the location of the eighth roots of unity. of unity. eighth roots 11 showing the location of the Plan view of Figure 12. Figure ) axes respectively. It has been noted that for any given value of n ) axes respectively. simple polynomial y shown. This explains why the complex roots have to occur as conjugate pairs, shown. This explains why the complex roots have to occur as conjugate appearing the same distance in front and behind the Cartesian plane. applies to any other polynomial and its roots—regardless of their nature—and applies to any other polynomial and its roots—regardless of their nature—and Conclusions it is hoped the surfacethe is hoped it have proved helpful in herein presented visualisations principal symmetry planes are always arranged at right angles to one another. symmetryprincipal another. one to angles right at arranged always are planes patterns thus formed, are observedto result directly from the symmetry forms establishing this connection. All the real and imaginary surfaces presented roots, which are invariably presented in the complex Argand plane. This link roots, which are invariably presented in the complex Argand plane. here have exhibited strong symmetrystrong exhibited have here Re(x the about properties Im(x This paper has provided the full and complete visual connection between the This paper has provided the full and complete visual connection The nature of the surfaceThe nature of the intricate and the zero plane, with the intersections The approach adopted here should help students and teachers alike Bardell appreciate the rich interplay that exists between complex numbers and polynomial equations. The explanations provided should also help stimulate students’ interest in complex numbers and assist in developing their ability to visualise such concepts in three dimensions, especially the orthogonal relationship between the Cartesian and Argand planes. It is anticipated this subject matter could easily form the basis of a classroom-based learning exercise in three-dimensional graphics using computer algebra systems that would amply satisfy some of the ACARA (n.d., Rationale) stated aims of Specialist Mathematics, namely “to develop students’ understanding of concepts and techniques drawn from complex numbers, ability to solve applied problems using concepts and techniques drawn from complex numbers and capacity to choose and use technology appropriately”.

References

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