Hyperbolic Numbers Revisited

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1 Hyperolic Numers Revisited Grret Soczyk Universidd de ls Américs-Puel Deprtmento de Físico-Mtemátics Puel, Pue., México Decemer 17, 2017 Astrct In 1995, my rticle The Hyperolic Numer plne ws pulished in the College Mthemtics Journl. The unit four rnched hyperol in the hyperolic numer plne plys the sme role for the hyperolic trigonometric functions s the unit circle in the complex numer plne plys for the trigonometric functions. Here we unify these different numer plnes into much more powerful numer system tht is esy to lern, nd which provides fun stepping stone to higher mthemtics. 0 Introduction The development of the rel nd complex numer systems ws truly multimillennil effort undertken y key plyers from mny civiliztions with different pressing needs. Sheep hd to e ccounted for, records needed to e kept for incresingly complicted trding reltionships, legs to rech the lofty drems of understnding the motion of the plnets nd strs hd to e invented, nd the more sinister demnds for ever more sophisticted wepons of mss killing nd destruction hd to e met. A eutiful ccounting of this story is relted y Tois Dnzig in his ook, Numer the Lnguge of Science [1]. The reder my hve the impression tht with the development of the rel nd complex numer systems, the jo hd een completed nd tht these numer systems form the sis of ll higher mthemtics. On the contrry, we show tht there is very simple extension of the rel nd complex numer systems tht is esy to lern, nd gives insight into higher mthemtics tht is usully considered in the domin of the experts nd out of rech of the eginning student. Indeed, there re mny geometric extensions of the numer system tht should e incorported into the concept of numer itself [2, 3]. In nother pper [4], ment to demonstrte the power nd nturlness of the geometric extension of numer, the uthors show how the fmous Pythgoren theorem for right-tringle serves s n elementry model for constructing the geometric multipliction of vectors. 1

2 1 Rel new numers The rel numer system R is generlly represented on the rel numer line. Wht mkes the rel numer system so powerful is the ility to oth dd nd multiply rel numers to get other rel numers. We ssume tht the reder is fmilir with the ddition nd multipliction of rel numers, nd ll of the sic properties of these opertions. We review only the properties tht re most pertinent to wht we wish to do here. For r, s, t R, R1 rs = sr Commuttive lw of multipliction. R2 rs + t = rs + rt Distriutive lw of multipliction over ddition. R3 rst = rst Associtive lw of multipliction. R4 rs = 0 r = 0 nd/or s = 0. We now introduce two new numers nd not in R. To emphsize tht nd re not rel numers, we write / R nd / R. To mke the extended numer system N := R, fully functionl, nd comptile with R, we extend the opertions of ddition nd multipliction to include the new numers,. This is ccomplished y ssuming tht the extended numers in N oey exctly the sme rules of ddition nd multipliction s do the numers in R, with the exception tht. Regrding the the new numers nd, they stisfy the following two specil properties: N1 2 = 0 = 2 The new numers nd re nilpotents. N2 + = 1 The sum of nd is 1. In ddition, we ssume tht numers in N commute with rel numers in R nd tht the ssocitive nd distriutive properties R2 nd R3 ove, remin vlid for our new numers. Becuse of the geometric interprettion tht we give these new numers, we cll them g-numers. The g-numers in the tle 1 re clled the stndrd cnonicl sis of N = R, over the rel numers. Clerly the non-zero g-numers nd cnnot e rel numers ecuse 2 = 0 = 2, since there re no non-zero rel numers with this property. Any g-numer g such tht g 2 = 0 is sid to e nilpotent. Also, the products nd cnnot e rel numers since. Never-the-less the property N2 tells us tht the sum + = 1 R, providing direct reltionship etween the extended g-numers in N nd the rel numers, nd showing tht R N = R,. Ech g-numer g N is uniquely specified y four rel numers, which we g11 g write in tulr form [g] := 12. Thus, g 21 g 22 g11 g g = g 11 + g 12 + g 21 + g 22 [g] := 12 g 21 g 22, 2 2

3 where g 11, g 12, g 21, g 22 R. 1 The Multipliction Tle 1 for g-numers is esily derived from the ssumed properties N1 nd N2, nd the ssocitive lw. Since hlf of its entries re zeros, it is esily rememered. Tle 1: Multipliction tle To show tht =, we use oth properties N1 nd N2, nd in prticulr N2 to sustitute in 1 for, getting = = 1 = 2 =, nd similrly, =. The sme sustitution works for showing tht 2 = 1 = + 2 =, nd similrly tht 2 =. Any non-zero g-numer g with the property tht g 2 = g is sid to e n idempotent, so nd re idempotents, nd since + = 1, nd = 0 =, they re sid to prtition unity nd to e mutully nnihilting. We cn now esily derive the generl rule for the ddition nd multipliction of two g-numers f, g N. In ddition to g, lredy defined, let f11 f f = f 11 + f 12 + f 21 + f 22 [f] := 12. f 21 f 22 Clculting f + g nd fg, we find tht f + g = f 11 + g 11 + f 12 + g 12 + f 21 + g 21 + f 22 + g 22 f11 + g [f + g] = 11 f 12 + g 12 f 21 + g 21 f 22 + g 22 for ddition, nd fg = f 11 g 11 + f 12 g 21 + f 11 g 12 + f 12 g 22 +f 21 g 11 + f 22 g 21 + f 21 g 12 + f 22 g 22 1 We include n Appendix on the reltionship of our new numer system N to rel 2 2 mtrices. 3

4 f11 g [fg] = 11 + f 12 g 21 f 11 g 12 + f 12 g 22 f 21 g 11 + f 22 g 21 f 21 g 12 + f 22 g 22 for multipliction. Reders who know how to dd nd multiply mtrices will e surprised to find tht [f + g] = [f] + [g] nd [fg] = [f][g]. We leve further discussion of this reltionship to the Appendix. 2 Conjugtions nd inverses To complete our new numer system, we define three powerful conjugtion opertors on N. First note tht ech g-numer g N, with respect to the cnonicl sis 1, is the sum of two prts, g = g o + g e, n odd prt g o nd n even prt g e, where g o := g 12 + g 21, nd g e := g 11 + g 22, 3 respectively. We define the reverse g of g, with respect to the stndrd cnonicl sis 1, y g := g o + g e = g o + g e = g o + g e, 4 where g o := g o nd g e = g 11 + g 22. The reverse opertion reverses the order of the multipliction of nd, i.e., =, leving the odd prt g o unffected. It follows tht for f, g N, f + g = f + g nd fg = g f. The inversion g of g, with respect to the stndrd cnonicl sis 1, is defined y g := g o + g e = g o + g e = g o + g e, 5 where g o := g 12 g 21 = g o nd g e = g e. The inverse opertion chnges the sign of oth nd, i.e., = nd =, leving g e unffected. Clerly nd for f, g N, g o = 1 2 g g, nd g e = 1 2 g + g, f + g = f + g nd fg = f g. Comining the opertions of reverse nd inversion gives the third mixed conjugtion. For g N, the mixed conjugtion g of g, with respect to the stndrd cnonicl sis 1, is defined y, g := g = g o + g e = g o + g e. 6 4

5 The mixed conjugtion of the sum nd product of f, g N, stisfies nd where nd f + g = f + g = f o + g o + f e + g e, fg = g f = g o + g e f o + f e = fg o + fg e, fg o = g o f e + g ef o fg e = g o f o + g ef e. Using the mixed conjugtion, we find tht for g N, trg := g + g = g e + g e = g 11 + g 22 R, 7 clled the trce of g. Also, using tht n even g-numer times n odd g-numer is odd, we clculte since nd det g := gg = g o + g e g o + g e = g o g e g o g e + g e g e g o g o = g e g e g o g o = g 11 g 22 g 12 g 21 R, 8 g e g e = g 11 + g 22 g 11 + g 22 = g 11 g 22 g o g o = g 12 + g 21 2 = g 12 g 21 + = g 12 g 21. For g N, det g is clled its determinnt. Given g-numer g N, when is there f N such tht gf = fg = 1? When such n f exists, we sy tht g 1 := f is the multiplictive inverse of g. Since g g g g = gg gg = 1, it immeditely follows tht g 1 := g gg = g g g = g o + g e gg provided tht gg 0. Whenever g-numer hs the property tht det g 0, we sy tht g is non-sigulr. If gg = 0, we sy tht g is singulr. Given g-numers f, g N, the product fg cn e decomposed into even nd odd prts. We hve fg = fg o + fg e where fg o = f o g e + f e g o nd fg e = f o g o + f e g e 5

6 The product of two g-numers fg cn lso e decomposed into the sum of symmetric prt f g nd skew-symmetric prt f g. We hve fg = 1 2 fg + gf + 1 fg gf = f g + f g, 9 2 where f g := 1 2 fg + gf nd f g := 1 2 fg gf. We give here couple of useful vector nlysis-like identities stisfied y the symmetric nd skewsymmetric products of three odd g-numers f o, g o, h o N. nd f o g o h o = f o g o h o f o h o g o, 10 f o g o h = 0 = f g h. 11 For the odd g-numers, f = nd g =, we find tht = = 1 2 nd = 1. 2 Squring, gives so tht 2 = 2. 2 = = Geometry of N = 1 4, Much conceptul clrity is gined when it is possile to pictorilly represent fundmentl concepts. To pictoril represent properties of g-numers g = g 11 + g 12 + g 21 + g 22 = g o + g e N, we picture the odd nd even prts of g, seprtely, in the odd g-numer plne N o, nd in the even g-numer plne N e, pictured in Figure 1. Becuse nd re nilpotents, they re pictured on the 2-dimensionl light-cone in N o, [5]. Similrly, since nd re singulr idempotents, they re lso pictured on the light-cone in N e. The even g-numer plne N e hs een studied under different guises. In [6], it is referred to s the hyperolic numer plne ecuse of the prominence of the four rnched hyperol. Wheres here we hve plced the idempotents nd long the symtotes, in [6] the coordinte x-xis is the rel line, nd the coordinte y-xis is u = 2, where u 2 = 1. This difference is no more thn chnge of sis, see Figure 1. There re only two idempotents 0 nd 1 in the rel numer system R. Idempotents in N hve much richer structure. If P 2 = P is n idempotent, then P 2 = P, P 2 = P nd P 2 = P re lso idempotents, so once 6

7 1 u=2 1 f=- N odd - N even e= = Figure 1: The odd numer plne N o. The even numer plne N e. cnonicl form is found for P we hve lso found cnonicl forms for P, P nd P. We chrcterize the idempotent p11 p P = p 11 + p 12 + p 21 + p p 21 p 22 in the following Lemm 1 Suppose the mtrix [P ] of P N stisfies 1 p 221 p 22 p22 [P ] := 21, 12 p 21 p 22 then P is n idempotent with two degrees of freedom p 21, p 22 R. Proof: Since the mtrix of P is given, it cn e directly verified tht P 2 = P. Insted, we will show tht the ssumption tht P 2 = P leds to the given mtrix [P ] of P s solution. For P = P o + P e, we clculte where P 2 = P o + P e 2 = P o P e + P e P o + P 2 o + P 2 e = P o + P e = P, P o = P o P e + P e P o nd P e = P 2 o + P 2 e. For P o = p 12 + p 21 nd P e = p 11 + p 22, we clculte P 2 o = p 12 + p 21 2 = p 12 p 21 + = p 12 p 21, 7

8 nd similrly, P 2 e = p 11 + p 22 2 = p p 2 22, P o P e = p 12 p 22 + p 11 p 21 nd P e P o = p 11 p 12 + p 21 p 22. Sustituting these vlues into the equtions P o P e + P e P o P o = 0 nd P 2 o + P 2 r P e = 0 leds to the respective equtions nd p 12 p 11 + p p 21 p 11 + p 22 1 = 0 p 2 11 p 11 + p 12 p 21 + p 2 22 p 22 + p 12 p 21 = 0. Equting the coefficients of the cnonicl sis elements to zero gives system of four liner equtions in four unknowns, one of whose solutions is the mtrix [P ] given in the Lemm. Other solutions re otined y noting tht P, P, nd P re ll idempotents. 4 N o Figure 2: Ech constnt p 22 defines point in the even numer plne N e not shown nd hyperol in the odd numer plne N o. If for [P ], given in 12, we consider p 22 to e constnt, the resulting eqution is fmily of points in N e, defined y p 11 = 1 p 22, nd fmily of hyperols in N o, defined y p 12 = p221 p22 p 21 shown in Figure 2. Theorem 1 Given non-zero singulr idempotent P N. 8

9 i There exists cnonicl sis AB A B := B BA 13 with the properties tht P = AB, A 2 = 0 = B 2 nd A B = 1 2. ii For ech cnonicl sis 13, there exists g N such tht g 1 AB A g =. B BA For the idempotent P. with mtrix the [P ] given in Lemm 1, the mtrix of g is g12p 21 p [g] := 22 g 12, with det[g] = g12g21 p 22. g 21 g211 p22 p 21 Proof: i For the non-zero singulr idempotent P = P o + P e, with mtrix 12 given in Lemm 1, we find non-singulr g-numer g = g o + g e such tht g 1 g = P, or equivlently, g o = g o P e + g e P o nd g e = g e P e + g o P o. Clculting, nd g o = g 12 + g 21 = g 12, g o P e + g e P o = g 12 + g 21 p 11 + p 22 + g 11 + g 22 p 12 + p 21 nd = g 12 p 22 + g 21 p 11 + g 11 p 12 + g 22 p 21 = g 12 p 22 + g 11 p 12 + g 21 p 11 + g 22 p 21. Similrly, we clculte oth sides of g e = g e P e + g o P o, getting g e = g 11 + g 22 = g 11, g e P e + g o P o = g 11 + g 22 p 11 + p 22 + g 12 + g 21 p 12 + p 21 = g 11 p 11 + g 22 p 22 + g 12 p 21 + g 21 p 12 = g 11 p 11 + g 12 p 21 + g 22 p 22 + g 21 p 12. It follows tht g o P e + g e P o g o = 0, nd g e P e + g o P o g e = 0, giving the reltionships g 12 p 22 + g 11 p 12 g 12 + g 21 p 11 + g 22 p 21 = 0 9

10 nd g 11 p 11 + g 12 p 21 g 11 + g 22 p 22 + g 21 p 12 = 0. Setting the coefficients of the cnonicl sis = 0, nd imposing the conditions from Lemm 1 for the idempotent P, gives the solution g12p 21 p [g] := 22 g 12, 14 g 21 g211 p22 p 21 with det[g] = g12g21 p 22. The cnonicl sis 1, with the required properties, is then esily constructed from g, getting g 1 AB A g =, B BA where A = g 1 g, B = g 1 g. ii Notice in the solution for g N in prt i, the two prmeters g 12 nd g 21 re free. These extr prmeters cn e used to djust the scle of the nilpotents A nd B. 4 Structure of geometric numer The fct tht the stndrd cnonicl sis 1 consists only of g-numers which re idempotents or nilpotents, suggests tht these g-numers re of fundmentl importnce. With this in mind, we mke the following Definition 1 Given geometric numer f N. The chrcteristic polynomil of f is ϕ f x := x 2 trfx + det f = x λ 1 x λ The roots λ 1, λ 2 of this polynomil re sid to e the eigenvlues of f. The structure of geometric numer f is completely determined y its chrcteristic polynomil ϕ f x, [7, 8, 9]. The eigenvlues of f cn e either rel or complex numers. Discussion of g-numers f with complex eigenvlues is deferred to the Appendix. We hve the following Theorem 2 Given g-numer f N with rel eigenvlues λ 1, λ 2 R. Then f is either type i or type ii given elow. i When λ 1 λ 2, then f = λ 1 s 1 + λ 2 s 2, where s 1, s 2 N re idempotents stisfying the conditions s 1 + s 2 = 1, nd s 1 s 2 = s 2 s 1 = 0. When ϕ f x = x λ 2 nd f λ = 0, then f = λ. 10

11 ii When ϕ f x = x λ 2 ut f λ 0, then f = λ + m, where m N is non-zero nilpotent. Proof: We first note tht every f N trivilly stisfies its chrcteristic polynomil, tht is i For type i, when λ 1 λ 2. Define ϕ f f = f 2 f + ff + f f = s 1 := f λ 2 λ 1 λ 2, nd s 2 := f λ 1 λ 2 λ 1. To see tht s 1 nd s 2 re idempotents, we clculte for s 1 since y 16, f s 2 λ2 2 f λ1 + λ 1 λ 2 f λ2 1 = = λ 1 λ 2 λ 1 λ 2 λ 1 λ 2 f λ1 f λ 2 + λ 1 λ 2 f λ 2 1 = = s 1, λ 1 λ 2 λ 1 λ 2 ϕ f f = f λ 1 f λ 2 = 0. Tht s 2 is n idempotent is similrly estlished. It lso trivilly follows tht s 1 s 2 = 0 = s 2 s 1, s 1 + s 2 = 1, nd the cse when f λ = 0. ii This is the cse when the chrcteristic polynomil ϕ f x = x λ 2, nd f λ 0. Since y 16, ϕ f f = f λ 2 = 0, we simply let the nilpotent m = f λ, nd we re done. Given tht f is type i, so tht f = λ 1 s 1 + λ 2 s 2, y multiplying oth sides of this eqution on the right y s 1 nd s 2, successively, we get fs 1 = λ 1 s 1, nd fs 2 = λ 2 s 2, 17 respectively. We sy tht s 1 nd s 2 re eigenpotents for the respective eigenvlues λ 1 nd λ 2. When f = λ+m for type ii, multiplying on the right y m gives fm = λm. In this cse, we sy tht m is n eigen-nilpotent of f. It is interesting tht for type i f N, tht the eigenpotents re idempotents, wheres for type ii f, the eigen-nilpotent is nilpotent. We explore this sitution further s follows. Applying prt i of Theorem 1, to the nonzero singulr idempotent P = s 1, we cn find non-singulr g N nd construct cnonicl sis 1 such tht B = g 1 AB A g = B BA nd where s 1 = AB, s 2 = BA nd A 2 = 0 = B 2, so tht f = λ 1 AB + λ 2 BA., 11

12 Multiplying oth sides this eqution on the right y A, nd then y B, we get fa = λ 1 ABA = λ 1 A nd fb = λ 2 BAB = λ 2 B, 18 respectively. Equtions 17 nd 18 re equivlent, however, since we cn esily get ck the first equtions from the second. In the cnonicl sis B, the mtrix of f is λ1 0 [f] =. 0 λ 2 Let us lso find cnonicl sis for type ii f = λ + m, where [f] tkes its simplest form. In this cse, we find non-singulr g N such tht g = gm. 19 Aside from the stndrd cnonicl nilpotents like nd, clss of quite generl nilpotents m hve the form s s [m] = 2 /t, for s, t R. t s For nilpotent m of this form, g defined y t [g] := s 0 t t s stisfies the property 19. Now define the nilpotent n y specifying 0 0 [n] := t/s 2. 0 The nilpotent n is sid to e comptile with m, ecuse together they define the cnonicl sis of N, given y AB A B m,n := = g 1 g, B BA where A := g 1 g = m nd B := g 1 g = n. With respect to B m,n, f is specified y λ 1 [f] :=. 0 λ 12

13 5 Appendix: Geometric lgers nd mtrices The purpose of this Appendix is to give reders fmilir with mtrix lger insight into how mtrix nd geometric lgers compliment ech other, [10, 11]. Complex eigenvlues re delt with y extending the rel g-numers N to the complex g-numers N C. The rel nd complex g-numers N nd N C re lgericlly isomorphic to the Clifford geometric lgers G 1,1 nd G 1,2, respectively. In terms of its mtrix [f], since = nd [f] = [f], f = [f] = [f ] T f11 f = 12 = f f 21 f f 12 + f 21 + f Furthermore, f = [f] T, nd f = [f] T, where [f] T is the trnspose of the mtrix [f]. If g = fg = [f] 0 = [f] [g] 0 [g] [g] = [f][g],, then showing tht mtrix multipliction of g-numers is preserved. The eqution 20 cn e directly solved for the mtrix [f] of f. Multiplying eqution 20 on the left nd right y nd, respectively, gives the eqution f = [f] 0 0 = [f] = [f]. 0 0 Similrly, multiplying eqution 20 on the left nd right y nd, respectively, gives f = = 0 0 [f] [f] 0 = [f]. 0 13

14 Adding these two equtions together give the desired result [f] = f + f f + f f + f =. 21 f + f f + f The geometric lger G 1,1 is defined y G 1,1 := Re, f where e 2 = 1 = f 2, nd ef = fe. The geometric lger G 1,1 is the rel numer system R extended to include the new nticommuting squre roots e, f of ±1, respectively. Since e f = it follows tht = e f, 1 1 N = R, = Re, f = G 1,1, re different nmes for the sme thing. We hve defined N = R, to e the rel numer system extended to include the new elements,. Becuse of the prolem of complex eigenvlues, we extend N to N C y llowing the mtrix [g] c to consists of complex numers. Thus g = [g] c, where [g] c is 2 2 mtrix over the complex numers C. Of course, we mke the dditionl dhoc ssumption tht complex numers commute with nd, nd therefore with ll the g-numers in N. On the level of geometric lgers, we cn give the complex g-numers comprehensive geometric interprettion. In either of the geometric lgers G 1,2 or G 3, N C = G 1,2 := Re 1, f 1, f 2 = Re 1, e 2, e 3 = G 3, where {e 1, e 2, e 3 }, nd {f 1, f 2 }, re new nti-commuting squre roots of ±1, respectively. A complex g-numer f G 1,2 hs the form f = f 11 + f 12 + f 21 + f 22, where f jk C = G 0+3 1,2. This mens tht the complex sclrs f jk re of the form f jk = x jk + iy jk, where i := e 1 f 1 f 2 for x jk, y jk G 0+3 1,2. In N C, the definitions 3 of the even n odd prts of complex g-numer must e crefully re-exmined [3]. 14

15 References [1] T. Dntzig, Numer: The Lnguge of Science, 4th edn. Free Press, New York [2] G. Soczyk, Geometriztion of the Rel Numer System, July, [3] G. Soczyk, New Foundtions in Mthemtics: The Geometric Concept of Numer, Birkhäuser, New York [4] J.A. Juárez González, S. Rmos Rmirez, G. Soczyk, From Vectors to Geometric Alger, 2017 to pper. [5] See wikipedi: [6] G. Soczyk, Hyperolic Numer Plne, The College Mthemtics Journl, Vol. 26, No. 4, pp , Septemer [7] G. Soczyk, The Generlized Spectrl Decomposition of Liner Opertor, The College Mthemtics Journl, 28: [8] G. Soczyk, The missing spectrl sis in lger nd numer theory, The Americn Mthemticl Monthly 108 April 2001, pp [9] G. Soczyk, Spectrl Integrl Domins in the Clssroom, Aportciones Mtemtics, Serie Comunicciones Vol. 20, Sociedd Mtemátic Mexicn 1997 p [10] G. Soczyk, Geometric Mtrix Alger, Liner Alger nd its Applictions, [11] G. Soczyk, Conforml Mppings in Geometric Alger, Notices of the AMS, Volume 59, Numer 2, p ,