Complex Number Theory without Imaginary Number (i)

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Ope Access Library Joural Complex Number Theory without Imagiary Number (i Deepak Bhalchadra Gode Directorate of Cesus Operatios, Mumbai, Idia Email: deepakm_4@rediffmail.

1 Ope Access Library Joural Complex Number Theory without Imagiary Number (i Deepak Bhalchadra Gode Directorate of Cesus Operatios, Mumbai, Idia Received 6 July 04; revised 0 September 04; accepted October 04 Copyright 04 by author ad OALib. This work is licesed uder the Creative Commos Attributio Iteratioal Licese (CC BY. Abstract I this paper ew method is itroduced for complex umbers; this method does ot iclude imagiary umber i but produces the same results that occur i Additio, Subtractio, Multiplicatio & Divisio of complex umbers, also proof of Eluer Formula e iθ ad De Moivre theorem without usig imagiary umber i. Furthermore placig the light o the square root of a egative umber, the square root of a egative umber is equal to the square root of the same positive real umber but with a agle of 90 degree to real lie. The itetio is that there s othig mystical about imagiary umber i. The square root of mius oe is just as real as ay other umber. Complex umbers exists without imagiary umber i. This paper is limited to the results which are already established. Keywords Additio, Subtractio, Multiplicatio, Divisio, Eluer Formula, De Moivre Theorem, Square Root of Negative Number Subject Areas: Mathematical Aalysis, Mathematical Logic, Foudatio of Mathematics. Itroductio A algebra of real umber is ot eough to solve the equatios like X + = 0. To avoid this, cocept of complex umber was itroduced; it is a combiatio of real umbers ad imagiary umber i, where i is equal to square root of mius oe. Sice the it was used to solve the equatio which has o solutio i real umber theory ad also established its use i differet disciplies like electroics, quatum theory etc. But sice has o exact value & meaig, it is a fact that a ijudicious use of this symbol ofte leads to mutually cotradictory ad absurd coclusios ([], p.. It is also observed that i graphical represetatio of complex umber the poit i show o Y axis has exactly the same altitude or legth as umber or of X axis has from origi. This implies that value of i = is a How to cite this paper: Gode, D.B. (04 Complex Number Theory without Imagiary Number (i. Ope Access Library Joural, : e856.

2 cotradictio, sice we do t kow the exact value of square root of (i.e. i =. So it is better to have a umber system which does ot ivolve i but gives all the beefits of complex umber system. Herewith complex umber without imagiary umber is beig itroduced; it is a combiatio of real umbers & trigoometric fuctios i.e. cos(θ, si(θ. Ay umber i complex umber theory without i is represeted by acos( θ + bsi ( θ where a & b are real umbers ad θ = ta ( ab. It has directio ad altitude/modulus same as complex umber but without i. I graphical represetatio, a cos( θ represets X axis ad bsi ( θ represets Y axis. This is a simple graph, X axis represets combiatio of real umber a ad cos(θ, whereas, Y axis represets combiatio of real umber b ad si(θ where θ = ta ( ab. Poits show i Figure are explaied below Poit o graph Here real umber a equal to Here real umber b equal to Agle θ equal to ta (a/b. Correspodig complex umber without imagiary umber i is (0, 0 90 deg si(90 (, 45 deg cos(45 + si(45 (, deg cos( si( (5, deg 5cos(0 (, deg cos(4.449 si(4.449 Note: Here owards short form CNWOI is used i place of complex umber without imagiary umber i; The value of agle θ is take i degrees; The cos(θ & si(θ ca have agle from 0 to 60 degrees; 4 The ta(θ ca have agle from 80 to 80 degrees. Every CNWOI is iterliked with its members i.e. the real umbers a & b, the agle θ ad the legth R from origi (i.e. radius or modulus. Let a + ib be ay complex umber the its equal CNWOI is acos( θ + bsi ( θ where θ = ta ( ab Radius or Modulus R: R is calculated by puttig values of cos & si for θ ta ( ab i CNWOI i.e. ( θ si ( θ R = acos + b 0 0 Example: For a poit (0, i above Figure, θ = 90 deg so R = acos θ0 + bsi θ0 = 0 cos 90 + si 90 R = 0+ = This R is same as modulus of complex umber i.e. R = a + b Calculated R for poits show i Figure is give i table Poit o graph Here real umber a equal to. Here real umber b equal to Ad agle θ equal to = = θ (some agle 0 R is equal to (0, 0 90 deg (, 45 deg.444 (, deg (5, deg 5 (, deg.6068 Coefficiets a & b: If R ad θ 0 are give the coefficiets a & b are equal to a = Rcos( θ0 & b = Rsi ( θ0 ([], pp. & ad [], p.. Example: Suppose R ad θ 0 are.6068 ad deg respectively the a =.6068cos = = b =.6068si = = OALibJ DOI:0.46/oalib October 04 Volume e856

3 Figure. Graph of complex umber without imagiary umber i. This is how R, θ, a & b are iterliked to each other.. Additio of CNWOI There are two CNWOI The the additio is + = ad ccosθ + dsiθ where θ = ta ( cd acosθ bsiθ where θ ta ab. Subtractio of CNWOI There are two CNWOI ( a+ c cosθ+ ( b+ d si θ where θ = ta (( a+ c ( b+ d + = ad ccosθ + dsiθ where θ = ta ( cd acosθ bsiθ where θ ta ab The the subtractio is 4. Multiplicatio of Two CNWOI There are two CNWOI ( a c cosθ+ ( b d si θ where θ = ta (( ac ( b d + = ad ccosθ + dsiθ where θ = ta ( cd acosθ bsiθ where θ ta ab Puttig agle of θ i first umber will gate a umber which is equal to a + b let s say this umber is R, similarly puttig agle of θ i secod umber will gate a umber which is equal to c + d let s say this umber R the multiplicatio of this umber is equal to RR cos( θ+ θ cos θ + RR si ( θ+ θ siθ where θ = ta ( RR cos( θ+ θ RR si ( θ+ θ Related multiplicatio rule for two complex umbers by De Moivre is zz = rr ( cos( θ + θ + i si ( θ + θ ([], p. 4 & [], p.. Example + = here agle θ = ad R =.6068 for first umber ad sice it is a product of same umber (i.e. square of a umber agle ad R are same, so product is equal to Square of a umber cosθ siθ where θ ta ( 0,, 0 5, 0 b cos(θ , a si (θ 4 θ θ θ cos cos si si θ cos si, OALibJ DOI:0.46/oalib October 04 Volume e856

4 This is equal to + = cosθ 4siθwhere θ ta 4 This is idetical to square of complex umber ( + i 5. Divisio of CNWOI There are two CNWOI. + = ad ccosθ + dsiθ where θ = ta ( cd acosθ bsiθ where θ ta ab The for divisio of acosθ+ bsiθ ccosθ + dsi θ. First fid R ad agle θ for both umbers as did i multiplicatio calculatio, the the divisio is equal to ( R R cos( θ θ cos θ + ( R R si ( θθ si θ where θ = ta (( R R cos( θθ ( R R si ( θθ Related divisio rule for two complex umber by De Moivre is z z = r r ( cos( θ θ + i si ( θ θ 4 & [], p. 4.. ([], Example Divisio of two CNWOI cosθ+ si θ where θ = ta So agle θ = ad R =.6068 for first umber cosθ + siθ where θ = ta So agle θ = ad R =.6055 for secod umber. cosθ + siθ divided by cosθ + siθ is equal to Now ( R R cos( θ θ cosθ + ( R R si ( θθ siθ cos( cosθ si ( cos( cos θ si ( si θ [ ] cosθ + [ ] siθ si θ cosθ siθ This is equal to 0.6 ere ta 585cosθ siθ wh θ = This is idetical to the divisio of above complex umbers usig imagiary umber i, i.e. equal to 8 + i. 6. Proof of Eluer Formula i θ e cos( θ isi( θ = + []-[4] Let a + bi is ay complex umber Its equivalet complex umber without imagiary umber i (CNWOI is Now ( θ + ( θ θ = acos bsi where ta ab acos( θ + bsi( θ acos θ bsi( θ e = e e sice we have parted it ito two complex umbers, there are two cases each for these two parted CNWOI For first CNWOI θ may be 0 deg or 80 deg deped o the +ve or ve value of a ad its R is equal to ab- OALibJ DOI:0.46/oalib October 04 Volume e856

5 solute value of a. Similarly for secod CNWOI θ may be 90 deg or 70 deg deped o the +ve or ve value of b ad its R is equal to absolute value of b, solvig these cases oe by oe we get For first CNWOI if θ is equal to 0 deg i.e. a is ay positive umber o X axis, the a cos( 0 { a ( a ( a ( a 4 } e = + cos 0! + cos 0! + cos 0! + cos 0 4! + Usig product rule of CNWOI for terms with power greater tha oe ad takig R is equal to absolute value of a we get { { } a cos( 0 cos( θ + } a si 0 si θ 4 4 a cos( 4 0 cos( θ + si ( 4 0 si } a θ! 4! + } { 4 acos( 0! a cos( θ! a cos( θ! a cos( θ 4! } a cos( 0 = + a + a θ + a θ e cos 0! cos 0 cos si 0 si! Ad usig additio rule of CNWOI So if a is ay positive umber o X axis = a 4 + a! + a! + a! + a 4! + cos α = e cos α [ e] acos( θ a = e cos ( α It is same as complex umber with imagiary umber i. Now if θ is equal to 80 deg i.e. a is ay egative umber o X axis the ( ( ( acos( 80 4 e = acos 80! + acos 80! + acos 80! + acos 80 4! + Usig product rule of CNWOI for terms with power greater tha oe ad talkig R is equal to absolute value of a we get { { } a cos( 80 cos( θ + } a si 80 si θ 4 4 a cos( 4 80 cos( θ + si ( 4 80 si } a θ! 4! + } { 4 acos( 80! a cos( θ! a cos( θ! a cos( θ 4! } acos( 80 = a + a θ + a θ e cos 80! cos 80 cos si 80 si! Usig additio rule of CNWOI = So if a is ay egative umber o X axis a a a a a α α 4! +!! + 4! + cos = e cos a cos( θ = a e e cos It is same as complex umber with imagiary umber i. Now for secod CNWOI if θ is equal to 90 deg i.e. b is ay positive umber o Y axis the bsi( 90 ( α ( ( ( 4 e = + bsi 90! + bsi 90! + bsi 90! + bsi 90 4! + Usig product rule of CNWOI for terms with power greater tha oe ad talkig R is equal to absolute value of b we get OALibJ DOI:0.46/oalib October 04 Volume e856

6 { { } b cos( 90 cos( θ + } b si 90 si θ 4 4 b cos( 4 90 cos( θ + b si ( 4 90 si } θ! 4! + } { b { b ( θ b ( θ } + { b cos( 70 cos( θ + } b si 70 si θ! 4 4 b cos( 60 cos( θ + b si ( 60 si ( θ } 4! + } bsi( 90 = + b + b θ + b θ e si 90! cos 90 cos si 90 si! bsi( 90 e = + si 90! + cos 80 cos + si 80 si! 4 ( θ ( θ ( θ = + bsi 90! b cos! b si! + b cos 4! + Usig additio rule of CNWOI ( b ( α ( b ( α ( α ( α = b! b 4! cos b! b! si = cos cos + si si So if b is ay positive umber o Y axis D. B. Gode bsi( θ [ e] = cos ( b cos ( α + si ( b si ( α ib It is same as complex umber with imagiary umber, that is [ e] = cos( b + isi ( b. Remember that a complex umber with i is equal to a + ib ad its equivalet CNWOI is a ( α + b ( α Now if θ is equal to 70 deg i.e. b is ay egative umber o y axis the ( ( ( cos si. bsi( 70 4 e = bsi 70! + bsi 70! + bsi 70! + bsi 70 4! + Usig product rule of CNWOI for terms with power greater tha oe ad talkig R is equal to absolute value of b we get { { } b cos( 70 cos( θ + } b si 70 si θ 4 4 b cos( 4 70 cos( θ + b si ( 4 70 si ( θ }! 4! + } { b { b ( θ b si ( θ }! b cos( 80 cos( θ + } b si 80 si θ! 4 4 b cos( 080 cos( θ + } b si 080 si θ 4! + } { b { b ( θ b ( θ } b cos( 90 cos( θ + b si ( 90 si ( θ }! 4 4 b cos( 60 cos( θ + si ( 60 si } b θ 4! + } bsi( 70 = b + b θ + b θ e si 70! cos 70 cos si 70 si! bsi( 70 e = si 70! + cos 540 cos + si 540 bsi( 70 e = si 70! + cos 80 cos + si 80 si! Usig additio rule of CNWOI 4 ( θ ( θ ( θ = bsi 70! b cos! + b si! + b cos 4! + ( b ( α ( b ( α ( α ( α = b + b + + b + b + 4! 4! cos!! si ( α ( α b + b + b b + 4! 4! cos!! si = cos cos si si OALibJ DOI:0.46/oalib October 04 Volume e856

7 So if b is ay egative umber o Y axis bsi( θ ( b ( α ( b ( α e = cos cos si si ib It is same as complex umber with imagiary umber i, that is = ( b i ( b e cos si. 7. De Moivre Theorem (for Fidig Power of Complex Number I complex umber if is a positive iteger, the ( θ + i ( θ = ( θ + i ( θ cos si cos si ([], p. 5] Proof of this theorem i complex umber theory without i (CNWOI is as follows Let a + ib is ay complex umber Its equivalet CNWOI is acos ( θ + bsi ( θ ow we have to prove that where c = R cos( θ & d R si ( θ ad θ = ta ( cd. Formula for multiplicatio of two CNWOI is where ( θ + ( θ = ( θ + ( θ acos bsi ccos dsi ( = are the umbers came out after multiplyig same CNWOI times ( + + RR ( + RR cos θ θ cosα si θ θ siα ( RR ( RR ( α = ta cos θ + θ si θ + θ Sice we were multiplyig same CNWOI times So product is R = R = R = = R = R (some modulus/radius associated with give CNWOI θ = θ = θ = = θ = θ (some agle associated with give CNWOI + = ( ( acos θ bsi θ R R Rcos θ θ θ cos α R R Rsi θ θ θ si α + = + acos θ bsi θ R cos θ cosα R si θ siα where α will be equal to θ. Puttig this i Equatio ( we get where Example ( R ( R ( α = θ θ ta cos si ( θ + ( θ = ( θ + ( θ acos bsi ccos dsi cos ( θ ad si ( θ c = R d = R Let 6+ 8i is ay complex umber. Now we fid out its cube usig De Moivre theorem of CNWOI i.e. ( 6+ 8i is equal to 6cos( θ + 8si ( θ ( θ + ( θ = ( θ + ( θ acos bsi ccos dsi i CNWOI θ = = ta OALibJ DOI:0.46/oalib October 04 Volume e856

8 Now usig De Moivre theorem R = 6 cos si 5.0 R = = = 0 R = 0 ( θ + ( θ = ( ( α + ( ( α 6 cos 8si 0 cos 5.0 cos 0 si 5.0 si = + = + = + α = = = ( 000 cos( cos( α ( 000 si ( si ( α ( 000( 0.96 cos( α ( 000( 0.5 si ( α 96 cos( α 5si ( α ta ( θ 6cos ( θ + 8si ( θ = 96 cos( θ + 5si ( θ This is equal to cube of i 8. De Moivre Theorem (for Fidig Root of Complex Number For fidig root of complex umber take complimet of De Moivre theorem (for fidig power of CNWOI. If is a positive iteger, the th root for ay CNWOI acos ( θ + bsi ( θ is equal to ( θ + ( θ = ( θ + ( θ acos bsi ccos dsi where c = R cos ( θ & d = R si ( θ ad θ ( cd Example = ta = θ. Let 96 + i5 is ay complex umber. Now fid out its cubic root usig De Moivre theorem of CNWOI where ad ( θ + ( θ = ( θ + ( θ acos bsi ccos dsi ( = cos ( θ & = si ( θ ad θ c R d R ( θ ( θ 96 cos + 5si = ta cd = θ ( Here θ = ta ( 96 5 = Puttig value θ i CNWOI 96 cos( θ + 5si ( θ we get R 96 cos( si ( = 96 ( ( 0.5 = = 000 Puttig this R ad θ i Equatio ( we get c & d R = 000 c = 000 cos = 0 cos 5.0 = = 6 d = 000 si = 0 si 5.0 = = 8 Puttig these values of c & d i Equatio ( we get 96 cos si = 6 cos si 5.0 This is a cubic root of complex umber 96 + i5. OALibJ DOI:0.46/oalib October 04 Volume e856

9 9. Solutio of Quadratic Equatio Usig CNWOI Quadratic equatios ( ax bx c = for which b 4ac < 0. First fid out θ (i.e. agle/argumet ad R (i.e. modulus/radius, which are equal to Hece roots for quadratic equatio are 9.. Example Let x + x+ = 0 for which b ( abs b 4ac a = θ = ta b a abs b 4ac a ( θ ( θ R = b a cos + abs b 4ac asi ( cos + si ad R( cos( θ si ( θ R θ θ 4ac < 0, the ( b a = = 0.5 ad θ i.e. agle/argumet ad R i.e. modulus/radius are equal to ( θ θ = = = ta b a abs b 4ac a ta ( θ R = b a cos + abs b 4ac asi = 0.5cos si 0 = = = Hece roots for quadratic equatio are Puttig first root i quadratic equatio we get ( cos( 0 + si ( 0 & ( cos( 0 si ( 0 cos 0 + si 0 + cos 0 + si 0 + = cos 40 + si 40 + cos 0 + si 0 + = = = 0 Puttig secod root i quadratic equatio we get cos 0 si 0 + cos 0 si 0 + = cos 40 si 40 + cos 0 si 0 + = = = Example Let x 0x+ 40 = 0 for which b 4ac 0, b a 5 θ i.e. agle/argumet ad R i.e. modulus/radius are equal to < the = ad θ = = = abs b 4ac a = ta b a abs b 4ac a ta ( θ ( θ R = b a cos + abs b 4ac asi = 5cos si = = = Hece roots for quadratic equatio are ( + & ( cos( si ( cos si Puttig first root i quadratic equatio we get OALibJ DOI:0.46/oalib October 04 Volume e856

10 6.4555( cos( si ( ( ( cos( si ( ( 50 ( = 40 cos si cos si = = + = Puttig secod root i quadratic equatio we get ( cos( si ( ( ( cos( si ( ( 50 ( = 40 cos si cos si = + + = + = 9.. Example Let x 4x 0 + = for which ( abs b 4ac a = b 4ac < 0, the ( b a = ad θ i.e. agle/argumet ad R i.e. modulus/radius are equal to b a ( abs b ac a ( θ θ = = = ta 4 ta ( θ R = b a cos + abs b 4ac a si = cos si = = = Hece roots for quadratic equatio are ( + & ( cos( si ( cos si Puttig first root i quadratic equatio we get ( cos( si ( cos si = = + = 0 Puttig secod root i quadratic equatio we get 9.4. Example 4 Let 4x + x 7 = 0 for which ( ( cos( si ( cos si = = + = 0 abs b 4ac a = b 4ac < 0, the ( b a = 0.5 ad θ i.e. agle/argumet ad R i.e. modulus/radius are equal to θ = = = = ta b a abs b 4ac a ta OALibJ DOI:0.46/oalib October 04 Volume e856

11 ( θ ( θ R = b a cos + abs b 4ac asi = 0.5cos si = = =.8757 Hece roots for quadratic equatio are + &.8757 cos ( si ( cos si Puttig first root i quadratic equatio we get cos si cos si = = 7 7 = 0 Puttig secod root i quadratic equatio we get 9.5. Example cos si cos si = = 7 7 = 0 Let x + = 0 for which b 4ac 0, b a 0 abs b 4ac a =. First fid out θ i.e. agle/argumet ad R i.e. modulus/radius which are equal to < the = ad ( θ θ = = = ta b a abs b 4ac a ta 0 90 ( θ R = b a cos + abs b 4ac a si = 0 cos 90 + si 90 = Hece roots for quadratic equatio are & cos ( 90 si ( 90 cos 90 + si 90 Puttig first root i quadratic equatio we get cos 90 + si 90 + Puttig secod root i quadratic equatio we get cos 80 + si 80 + = = 0 cos 90 si 90 + cos 80 si 80 + = 0 + = 0 0. About Square Root of Negative Real Number Imagiary umber i = = where is a egative real umber. Its modulus or radius is equal to ad agle is equal to 80 degrees, so usig De Moivre theorem of CNWOI ( θ + ( θ = ( θ + ( θ acos bsi ccos dsi where c = R cos( θ ad d = R si ( θ ad θ 80 deg calculatig c & d usig above formula = ta cd = θ sice ( / has R = ad θ = OALibJ DOI:0.46/oalib October 04 Volume e856

12 ad c = cos( 80 cos( 90 0 c = = d = si ( 80 si ( 90 d = = So puttig values of c & d i above formula we get ( = 0 cos( 90 + si ( 90 ( = 0 + si ( 90 ( = si ( 90 It is equivalet to the graphical positio ad value of imagiary umber i. So the square root of egative umber is equal to the square root of same positive real umber but with a agle of a 90 deg to real lie. Why the square root of egative umber is at 90 deg to real lie? It is i the roots of rules of multiplicatio where Positive Positive = Positive Positive Negative = Negative Negative Negative = Positive These rules build a limitatio whe we come across a situatio where we eed square root of egative umber. There is o egative umber multiplied by itself will give a egative umber. Complex umber is a aswer to this limitatio. Complex umber is othig but a iteractio of two mutually opposite umber systems, particularly i multiplicatio rules. Their idividual ad combie behavior uder multiplicatio operatio are give below. 0.. Whe Oly X Axis Is Cosidered It Gives ad Positive Negative = Negative i.e. acos(0 bcos(80 usig multiplicatio rule of CNWOI is equal to [abcos(80] Negative Negative = Positive i.e. acos(80 bcos(80 usig multiplicatio rule of CNWOI is equal to [abcos(0]. 0.. Whe Oly Y Axis Is Cosidered It Gives ad Positive Negative = Positive i.e. asi(90 bsi(70 usig multiplicatio rule of CNWOI is equal to [abcos(0] Negative Negative = Negative i.e. a si(70 bsi(70 usig multiplicatio rule of CNWOI is equal to [abcos(80]. 0.. But Joitly They Give Back Traditioal Results Such As Positive Negative = Negative i.e. acos(0 bsi(70 usig product rule of CNWOI is equal to absi(70 or OALibJ DOI:0.46/oalib October 04 Volume e856

13 ad acos (80 bsi(90 usig product rule of CNWOI is equal to absi(70 Negative Negative = Positive i.e. acos(80 bsi(70 usig product rule of CNWOI is equal to absi(90.. Fial Reflectio As said earlier i graphical represetatio of complex umber, the legth of i show o vertical axis is equal to ; it is a cotradictio sice value of imagiary umber i is ot kow. Also divisio process is ot smooth, for dividig oe complex umber with other eeds to multiply it by cojugate of divisor. But i CNWOI theory it looks smooth ad exactly opposite to the process of multiplicatio. If additio, subtractio, multiplicatio, divisio, power ad root ca be foud/calculated without imagiary umber i, preseted i this paper, so there is othig mystical or imagiary about imagiary umber i. Complex umbers exist without imagiary umber i. Refereces [] Naraya, S. ad Mittal, P.K. (005 Theory of Fuctios of a Complex Variable. S. Chad & Compay Ltd., New Delhi. [] Scheider, C. (0 De Moivre s Theorem A Literature ad Curriculum Project o Roots, Powers, ad Complex Numbers. Departmet of Mathematics ad Statistics, Portlad State Uiversity. [] Bogomoly, A. ( Cut the Kot! A Iteractive Colum Usig Java Applets. [4] Cai, G. ( Complex Aalysis. OALibJ DOI:0.46/oalib October 04 Volume e856

Complex Numbers. Brief Notes. z = a + bi

Complex Numbers. Brief Notes. z = a + bi Defiitios Complex Numbers Brief Notes A complex umber z is a expressio of the form: z = a + bi where a ad b are real umbers ad i is thought of as 1. We call a the real part of z, writte Re(z), ad b the