Elementary mathematics
Based on "A Calculus of Number Based on Spatial Forms" by Jeffrey M. James 1993.f1(x) = x f2(x) = -x f3(x) = ex f4(x) = ln x f3(f4(x)) = x eln x = x f4(f3(x)) = x ln ex = x <> - Inversion -0 = 0 () = o - Instanziation e0 = 1 [] = ∎ - Abstraction ln 0 = -∞ + [-∞...∞]i [(a)] = ([a]) = a Involution Axiom 1 ln ea = eln a = a (a[bc]) = (a[b])(a[c]) Distribution of a Axiom 2 ea + ln (b + c) = ea + ln b + ea + ln c a<a> = Inversion of a Axiom 3 a - a = 0 <<a>> = a Inverse Cancellation (double Inversion) Theorem 1 -(-a) = a <a><b> = <ab> Inverse Collection Theorem 2 (-a) + (-b) = -(a + b) Proof of Inverse Cancellation for a != ∎ <<a>> <<a>><a>a -Inversion of a a Inversion of <a> Proof of Inverse Collection for a,b != ∎ <a><b> <a><b>ab<ab> -Inversion of ab <ab> Inversion of a and b Proof of additive self-inversion of void (ambiguous sign) -0 = 0 <> Inversion of void Cardinality of a (aggregation second degree) a = ([a]) = ([a][o]) -Involution2, -Involution1 aa = ([a][o])([a][o]) = ([a][oo]) 2x a, -Distribution of [a] a...a = ([a][o...o]) induction step to all natural numbers a * b => ([a][b]) a / b => ([a]<[b]>) for b != Proof of Associativity of multiplication (a*b)*c = a*(b*c) ([([a] [b])][c] ) ( [a] [b] [c] ) Involution1 ( [a][([b] [c])]) -Involution1 Proof of multiplicative self-inversion of o 1/1 = 1 (<[o]>) (< >) Involution1 ( ) Inversion of void
Real part of the complex logarithm
Imaginary part of the complex logarithm - Riemann surface Absorption ([a]∎) = Zero cardinality of a Theorem 3 a * 0 = 0 ∎ a = ∎ Absorption of a (black hole) Theorem 4 ∎ + a = ∎ Proof of Zero cardinality ([a]∎) ([a]∎)([a][b])<([a][b])> -Inversion of ([a][b]) ([a][b])<([a][b])> -Distribution of [a] Inversion of ([a][b]) Proof of Absorption ∎ a [(∎ [(a)])] 2x -Involution1 [ ] Zero cardinality of (a) Proof of Zero cardinality ([a]∎) ( ∎) Absorption of [a] Involution2 Indefiniteness of cardinality of ∎ ∎ = ∎∎ = ∎∎∎ = ∎∎∎∎ = ... -Absorption of ∎ = <> = [o] = (∎) = [(<>)] = ([<>]) = <[o]> = <(∎)> 5x Inversion of void, 3x Involution1, 3x Involution2 Indefiniteness of Inversion of ∎ (<∎>) = ([o]<∎>) <= 1/0 is undefined (singularity) 0 * ? = 1 zero don't have multiplicative inverse 0/0 ambiguous 0/a = 0 a/a = 1 a/0 = ∞ inverse Absorption <∎> a <∎><<a>> -Inverse Cancellation <∎ <a>> Inverse Collection <∎ > Absorption of <a> ∎<∎> -Inversion of ∎ => Absorption of <∎> or inverse Absorption of ∎ => Possibility of arbitrary absorption of everything everywhere => <∎> undefined (Inversion of a black hole is not allowed) => Division by 0 is undefined (singularity) Cardinality of [a] (aggregation third degree) ([a]...[a]) = (([[a]][o...o])) Since multiplication is as commutative as addition, factors can be aggregated just like summands. Addition 2 + 4 = 6 Multiplikation 2 * 4 = 2 + 2 + 2 + 2 = 8 Exponentiation 24 = 2 * 2 * 2 * 2 = 16 Tetration 42 = 2222 = 224 = 216 = 65.536 Hyper-5 24 = 2222 = 422 = 65.5362 = 2...2 65.536 times - therefore, our universe is far too small ... Since exponentiation is not commutative, exponents cannot be represented aggregated like summands or factors. Thus aggregations lose their beauty and thus their usefulness from here on. oo ooo 2 + 3 = 5 ( [oo] [ooo] ) = oo oo oo 2 * 3 = 2 + 2 + 2 = 6 (([[oo]] [ooo])) = ([oo][oo][oo]) 23 = 2 * 2 * 2 = 8 (([[oo]][ (([[oo]][oo])) ])) = (([[oo]][oooo])) 32 = 222 = 24 = 16 (([[oo]][ (([[oo]][ (([[oo]][oo])) ])) ])) 23 = 222 = 42 = 2222 = 65.536 ... ... = 22 = 22 = 22 = 2*2 = 2+2 = 4 ab => (([[a]][b])) a0 = 1 => (([[a]]∎)) = ((∎)) = o Absorption of [[a]], Involution2 a1 = a => (([[a]][o])) = (([[a]])) = ([a]) = a Involution1, 2x Involution2 Inversion ([<a>]b) = <([a]b)> Inverse Promotion1 Theorem 5 -a * eb = -(a * eb) (<[<a>]>) = <(<[a]>)> Inverse Promotion2 Axiom 4 1/-a = -1/a (<[<a>]>b) = <(<[a]>b)> Inverse Promotion3 Theorem 6 1/-a * eb = -(1/a * eb) Proof of Inverse Promotion1 for a != ∎ ([<a>]b) <([a]b)>([a]b)([<a>]b) -Inversion of ([a]b) <([a]b)>([a <a>]b) -Distribution von b <([a]b)>([ ]b) Inversion of a <([a]b)>([ ] ) Absorption von b <([a]b)> Involution2 Proof of Inverse Promotion3 for a != ∎ ( <[<a>]> b) <(<[a]>b)>( <[a]> b)( <[<a>]> b) -Inversion of (<[a]>b) <(<[a]>b)>([(<[a]>)]b)([ (<[<a>]>) ]b) 2x -Involution1 <(<[a]>b)>([(<[a]>) (<[<a>]>) ]b) -Distribution von b <(<[a]>b)>([(<[a]>) <(<[ a ]>)>]b) Inverse Promotion2 <(<[a]>b)>([ ]b) Inversion of (<[a]>) <(<[a]>b)>([ ] ) Absorption von b <(<[a]>b)> Involution2 Proof of irrationality of √2 √2 = p/q and p, q ∈ ℤ shall be divisor extraneous 2 = (p/q)2 2q2 = p2 -> p2 is even -> p is even sei p = 2r für r ∈ ℤ -> 2q2 = p2 = 4r2 -> q2 = 2r2 -> q is even -> p und q are not divisor extraneous => √2 is irrational (([[oo]]<[oo]>)) = ([p]<[q]>) ([[oo]]<[oo]>) = [p]<[q]> q = o -> p = (([[oo]]<[ oo ]>)) p = o -> q = (([[oo]]<[<oo>]>)) q = oo -> p = (([[oo]]<[ oo ]>)[oo]) p = oo -> q = (([[oo]]<[<oo>]>)[oo]) q = ooo -> p = (([[oo]]<[ oo ]>)[ooo]) p = ooo -> q = (([[oo]]<[<oo>]>)[ooo]) ... -> p and q are not naturally resolvable together => (([[oo]]<[oo]>)) is irrational Operators -a => <a> for a != ∎ a+b => ab a+b+c => abc a-b => a<b> for b != ∎ a*b => ([a][b]) a*b*c => ([a][b][c]) a*-b => ([a][<b>]) = <([a][b])> = ([<a>][b]) +-Inverse Promotion1 for a,b != ∎ a/b => ([a]<[b]>) for b != 1/b => ([o]<[b]>) = (<[b]>) Involution1 for b != a/a => ([a]<[a]>) = o Inversion of [a] for a != ab => (([[a]][b])) a1 => (([[a]][o])) = (([[a]])) = a Involution1, 2x Involution2 a-b => (([[a]][<b>])) for b != ∎ a1/b => (([[a]]<[b]>)) = (([[a]][(<[b]>)])) -Involution1 for b != ac/b => (([[a]]<[b]>[c])) = (([[a]][(<[b]>[c])])) -Involution1 for b != abc (([[a]][(([[b]][c]))])) (([[a]] ([[b]][c]) )) Involution1 ((ab)c)1/d = ab*c/d (([[(([[(([[a]][b]))]][c]))]]<[d]>)) (( [[a]][b] [c] <[d]>)) 4x Involution1 ab * ac = ab+c ([(([[a]][b]))][(([[a]][c]))]) ( ([[a]][b]) ([[a]][c]) ) 2x Involution1 ( ([[a]][b c]) ) Distribution of [[a]] ac * bc = (a*b)c ([(([[a]][c]))][(([[b]][c]))]) ( ([[a]][c]) ([[b]][c]) ) 2x Involution1 ( ([[a] [b]][c]) ) Distribution of [c] 1 / e = e-1 => ([o]<[(o)]>) = (<[(o)]>) = (<o>) 2x Involution1 1 / e2 = e-2 => ([o]<[(oo)]>) = (<[(oo)]>) = (<oo>) 2x Involution1 ln (a*b) = ln a + ln b => [([a][b])] = [a][b] Involution1 ln (a/b) = ln a - ln b => [([a]<[b]>)] = [a]<[b]> Involution1 ln (ab) = ln a * b => [(([[a]][b]))] = ([[a]][b]) Involution1 logb a = ln a/ln b => ([[a]]<[[b]]>) 1/(1/a) = a (<[(<[a]>)]>) (< <[a]> >) Involution1 ( [a] ) Inverse Cancellation a Involution2 (1/a)*(1/b) = 1/(a*b) ([(< [a]>)][(<[b] >)]) ( < [a]> <[b] > ) 2x Involution1 ( < [a] [b] > ) Inverse Collection ( <[([a] [b])]> ) -Involution1 (1/a)b = a-b ( ([[(<[a]>)]][ b ]) ) ( ([ <[a]> ][ b ]) ) Involution1 (<([ [a] ][ b ])>) Inverse Promotion1 ( ([ [a] ][<b>]) ) -Inverse Promotion1 a/c + b/d = (a*d + b*c) / c*d ( [a] <[c]>)( [b] <[d]>) ( [a][d] <[d]><[c]>)( [b][c] <[c]><[d]>) -Inversion of [d] and [c] ( [a][d] <[d] [c]>)( [b][c] <[c] [d]>) 2x Inverse Collection ([([a][d])]<[d] [c]>)([([b][c])]<[c] [d]>) 2x -Involution1 ([([a][d]) ([b][c])]<[c] [d]>) -Distribution of <[c][d]> 1/a + 1/b = (a + b) / a*b ( <[a]>)( <[b]>) ([b]<[b]><[a]>)([a]<[a]><[b]>) -Inversion of [b] and [a] ([b]<[b] [a]>)([a]<[a] [b]>) 2x Inverse Collection ([ab]<[a][b]>) -Distribution of <[a][b]> (a + b) * (a - b) = a² - b² ([a b][a <b>]) ([a b][a]) ([a b][<b>]) Distribution of [ab] ([a][a])([b][a]) ([a][<b>]) ([b][<b>]) Distribution of [a] and [<b>] ([a][a])([b][a])<([a][ b ])><([b][ b ])> 2x -Inverse Promotion1 ([a][a]) <([b][ b ])> Inversion of ([a][b]) (([[a]][oo])) <(([[b]][oo]))> Cardinality of [a] and [b] Numbers 0 => 1 => o 2 => oo -1 => <o> -2 => <oo> 1/2 => (<[oo]>) 2/3 => ([oo]<[ooo]>) 43 => ([b][oooo])ooo for b = oooooooooo 243 => ([b][([b][oo])oooo])ooo for b = oooooooooo 1243 => ([b][([b][boo])oooo])ooo for b = oooooooooo Arity {a} = ([oooooooooo][a]) Definition of {} oooooooooo{a} = {o a} Carry+ Cardinality of oooooooooo {a}{b} = {ab} Collection+ ([{a}]b) = {([a]b)} Promotion+ {} = Zero cardinality+ 23 * 114 = 2622 ([ {oo}ooo ] [ {{o}o}oooo ]) ([{oo}][{{o}o}oooo]) ([ooo][{{o}o}oooo]) Distribution of [{{o}o}oooo] {([ oo ][{{o}o}oooo])} ([ooo][{{o}o}oooo]) Promotion+ {{{o}o}oooo{{o}o}oooo} {{o}o}oooo{{o}o}oooo{{o}o}oooo 2x -Cardinality of {{o}o}oooo {{{o}o}oooo{{o}o}oooo} {{o}o}{{o}o}{{o}o}oooooooooooo Collection+ of o {{{o}o}oooo{{o}o}oooo} {{o}o}{{o}o}{{o}oo} oo Carry+ {{{o}o} {{o}o} {o}{o}{o} oooooooooooo} oo Collection+ of {o} {{{o}o} {{o}o} {o}{o}{oo} oo} oo Carry+ {{{o} {o} oooooo} oo} oo Collection+ of {{o}} {{{ oo} oooooo} oo} oo Collection+ of {{{o}}} Inverse Arity /a\ = (<[oooooooooo]>[a]) Definition of /\ /oooooooooo a\ = o/a\ Carry- Cardinality of oooooooooo /a\/b\ = /ab\ Collection- ([/a\]b) = /([a]b)\ Promotion- /\ = Zero cardinality- {/a\} = /{a}\ = a Arity Cancellation 12,1 * 1,012 = 12,2452 ([ {o}oo/o\ ] [ o//o/oo\\\ ]) ([{o}][o//o/oo\\\]) ([oo][o//o/oo\\\]) ([/o\][o//o/oo\\\]) 2x Distribution of [o//o/oo\\\] {([ o ][o//o/oo\\\])}([oo][o//o/oo\\\])/([ o ][o//o/oo\\\])\ 2x Promotion ± { o//o/oo\\\ } o//o/oo\\\o//o/oo\\\ /o//o/oo\\\ \ 3x -Cardinality of o//o/oo\\\ {o}{ //o/oo\\\ } o//o/oo\\\o//o/oo\\\ /o//o/oo\\\ \ -Collection+ {o} /o/oo\\ o//o/oo\\\o//o/oo\\\ /o//o/oo\\\ \ Arity Cancellation1 {o}oo /o/oo\\ //o/oo\\\ //o/oo\\\ /o//o/oo\\\ \ Collection+ of o {o}oo /oo/oo\ /o/oo\\ /o/oo\\ //o/oo\\\ \ Collection- of /o\ {o}oo /oo/oooo /oo\ /oo\ /o/oo\\\ \ Collection- of //o\\ {o}oo /oo/oooo /ooooo /oo\\\ \ Collection- of ///o\\\ 100 / 3 = 33,3... ([ {{o}} ]<[ooo]>) {([ {o} ]<[ooo]>)} Promotion+ {([oooooooooo]<[ooo]>)} -Carry+ {([ooo]<[ooo]>)([ooo]<[ooo]>)([ooo]<[ooo]>) ([o]<[ooo]>)} 3x Distribution of <[ooo]> {( )( )( ) ([o]<[ooo]>)} 3x Inversion of [ooo] {ooo} {([o]<[ooo]>)} -Collection+ {ooo} ([{o}]<[ooo]>) -Promotion+ {ooo} ([oooooooooo]<[ooo]>) -Carry+ {ooo}([ooo]<[ooo]>)([ooo]<[ooo]>)([ooo]<[ooo]>) ([o]<[ooo]>) 3x Distribution of <[ooo]> {ooo}( )( )( ) ([o]<[ooo]>) 3x Inversion of [ooo] {ooo}ooo ([/oooooooooo\]<[ooo]>) -Carry- {ooo}ooo /([ oooooooooo ]<[ooo]>) \ Promotion- {ooo}ooo/([ooo]<[ooo]>)([ooo]<[ooo]>)([ooo]<[ooo]>)([o]<[ooo]>) \ 3x Distribution of <[ooo]> {ooo}ooo/( )( )( )([o]<[ooo]>) \ 3x Inversion of [ooo] {ooo}ooo/ooo ([/oooooooooo\]<[ooo]>) \ -Carry- {ooo}ooo/ooo /([ oooooooooo ]<[ooo]>)\\ Promotion- ... 100 / 3,3 = 30,3030... ([ {{o}} ]<[ooo/ooo\]>) {([ {o} ]<[ooo/ooo\]>)} Promotion+ {([oooooooooo]<[ooo/ooo\]>)} -Carry+ {([ooooooooo/ooo\/ooo\/ooo\/o\]<[ooo/ooo\]>)} -Carry- and 3x -Collection- {([ooo/ooo\]<[ooo/ooo\]>)(dito)(dito) ([ /o\ ]<[ooo/ooo\]>)} 3x Distribution of <[ooo/ooo\]> {( )( )( ) ([ /o\ ]<[ooo/ooo\]>)} 3x Inversion of [ooo/ooo\] {ooo} {([ /o\ ]<[ooo/ooo\]>)} -Collection+ {ooo} ([{/o\}]<[ooo/ooo\]>) -Promotion+ {ooo} ([ o ]<[ooo/ooo\]>) Arity Cancellation1 {ooo} ([/oooooooooo\]<[ooo/ooo\]>) -Carry- {ooo} /([ oooooooooo ]<[ooo/ooo\]>) \ Promotion- {ooo} /([ooooooooo/ooo\/ooo\/ooo\/o\]<[ooo/ooo\]>) \ -Carry- and 3x -Collection- {ooo}/([ooo/ooo\]<[ooo/ooo\]>)(dito)(dito) ([/o\]<[ooo/ooo\]>) \ 3x Distribution of <[ooo/ooo\]> {ooo}/( )( )( ) ([/o\]<[ooo/ooo\]>) \ 3x Inversion of [ooo/ooo\] {ooo}/ooo ([//oooooooooo\\]<[ooo/ooo\]>) \ -Carry- {ooo}/ooo //([ oooooooooo ]<[ooo/ooo\]>) \\\ 2x Promotion- {ooo}/ooo //([ooooooooo/ooo\/ooo\/ooo\/o\]<[ooo/ooo\]>) \\\ -Carry- and 3x -Collection- {ooo}/ooo//([ooo/ooo\]<[ooo/ooo\]>)(dito)(dito)([/o\]<[ooo/ooo\]>) \\\ 3x Distribution of <[ooo/ooo\]> {ooo}/ooo//( )( )( )([/o\]<[ooo/ooo\]>) \\\ 3x Inversion of [ooo/ooo\] {ooo}/ooo//ooo ([//oooooooooo\\]<[ooo/ooo\]>) \\\ -Carry- {ooo}/ooo//ooo //([ oooooooooo ]<[ooo/ooo\]>)\\\\\ 2x Promotion- ...
Real part of the complex exponential function
Imaginary part of the complex exponential function Transcendental J = [<o>] Phase Element Definition ln -1 = πi (~3.1415926535i) (J) = ([<o>]) = <o> Involution2 eJ = eπi = -1 (o) Euler number e1 = e (~2.71828182845) ((o)) ee (~15.1542622414) [<(a)>] = a[<o>] Phase Independence of a Theorem 7 ln -ea = a + ln -1 (a JJ ) = (a) J-cancellation1 Theorem 8 ea + 2*ln -1 = ea + ln (-1)² = ea + ln 1 = ea = ea + 2πi (a<bJJ>) = (a<b>) J-cancellation2 Theorem 9 ea - b - 2*ln -1 = ea - b - ln (-1)² = ea - b - ln 1 = ea - b = ea - b - 2πi (J) = (<J>) Self-Inversion of J Theorem 10 eJ = e-J (([J]<[oo]>)) i and <i> eJ/2 = √-1 = ±i i = (<[<i>]>) = <(<[i]>)> imaginary unit Theorem 11 i = 1/-i = -e-ln i ([<J>][i]) = ([J][<i>]) Circle number Pi -J*i = J/i = J*-i = π (~3.1415926535) Like ln in the imaginary, by an arbitrary multiple of 2π is ambiguous, so at e all values are repeated every 2πi. To represent this, the J-cancellation and self-inversion of J is permissible only within an instantiation, or they must be contained in at least one more instantiation than in abstractions. Proof of Phase Independence of a for a != [∎] [<(a)> ] [<(a)>( [ ])] -Involution2 [<(a)>(a[ ])] -Absorption of a [<(a)>(a[o <o>])] -Inversion of o [<(a)>(a[o])(a[<o>])] Distribution of a [<(a)>(a )(a[<o>])] Involution1 [ (a[<o>])] Inversion of (a) a[<o>] Involution1 Proof of J-cancellation within () [<o>][<o>] [<([<o>])>] -Phase Independence von [<o>] [< <o> >] Involution2 [ o ] Inverse Cancellation Involution1 Proof of J-cancellation1 within () (a[<o>][<o>]) -Involution1 <<(a[ o ][ o ])>> 2x Inverse Promotion1 (a[ o ][ o ]) Inverse Cancellation (a ) 2x Involution1 Proof of Self-Inversion of J within () (J ) (J< >) -Inversion of void (J<J J>) -J-cancellation2 (J<J><J>) -Inverse Collection ( <J>) Inversion of J
f1(x) = x2 f2(x) = √x f1(f2(x)) = x (√x)2 = x f2(f1(x)) = ±x √(x2) = ±x
Real part and imaginary part of the complex square root
Real part and imaginary part of the complex cube root Proof of ambiguity of (([J]<[oo]>)) or √-1 Whenever the exponent is a non-integer rational number, there is ambiguity. And if the exponent is less than or equal to 0, the base equals 0 gives a degenerate point. i * i = -i * -i = (±i)2 = -1 ([(([J]<[oo]>))][(([J]<[oo]>))]) (([[(([J]<[oo]>))]][oo])) Cardinality of [(([J]<[oo]>))] (( [J]<[oo]> [oo])) 2x Involution1 (( [J] )) Inversion of [oo] <o> 2x Involution2 and ([<(([J]<[oo]>))>][<(([J]<[oo]>))>]) <<([ (([J]<[oo]>)) ][ (([J]<[oo]>)) ])>> 2x Inverse Promotion1 ([ (([J]<[oo]>)) ][ (([J]<[oo]>)) ]) Inverse Cancellation (([[(([J]<[oo]>))]][oo]) ) Cardinality of [(([J]<[oo]>))] (( [J]<[oo]> [oo]) ) 2x Involution1 (( [J] ) ) Inversion of [oo] <o> 2x Involution2 Proof of imaginary unit1 (<[<<(([ J ]<[oo]>))>>]>) (< ([ J ]<[oo]>) >) Inverse Cancellation, Involution1 ( ([<J>]<[oo]>) ) -Inverse Promotion1 ( ([ J ]<[oo]>) ) -Self-Inversion of J Proof of imaginary unit2 <(<[(([ J ]<[oo]>))]>)> <(< ([ J ]<[oo]>) >)> Involution1 <( ([<J>]<[oo]>) )> -Inverse Promotion1 <( ([ J ]<[oo]>) )> -Self-Inversion of J -1 * -1 = 1 ([<o>][<o>]) ( ) J-cancellation1 ee = ee (([[(o)]][(o)])) (( o )) 3x Involution1 < a > = ([<([a])>]) = ([a][<o>]) = ([a]J) 2x-Involution2, Phase Independence of [a] [< a >] = [<([a])>] = [a][<o>] = [a]J -Involution2, Phase Independence of [a] <( a )> = ([<( a )>]) = (a[<o>]) = (aJ) -Involution2, Phase Independence von a Indefiniteness of Cardinality of J within () ... = (<JJJJ>) = (<JJ>) = o = (JJ) = (JJJJ) = ... J-cancellation1, J-cancellation2, Inversion of void ... = (<JJJ>) = (<J>) = (J) = (JJJ) = (JJJJJ) = ... J-cancellation1, J-cancellation2, Self-Inversion of J Indefiniteness of [∎] within () ( [∎] ) ( [∎] JJ) -J-cancellation1 ([<([∎])>]J) -Phase Independence of [∎] ([< ∎ >]J) Involution2 - Indefiniteness der Inversion of ∎ Indefiniteness of 00 (([∎]∎)) (( ∎)) Absorption of [∎] ( ) Involution2 => 00 = 1 but from the Indefiniteness of [∎] follows in principle the Indefiniteness of 00 a0 = 1 (([[a]]∎)) (( ∎)) Absorption of [[a]] ( ) Involution2 0n = 0 n ∈ ℕ (([∎][o...o])) (∎...∎) Cardinality of ∎ (∎ ) Absorption of ∎ Involution2 ln 0 + ln -1 = ln 0 ∎J [<(∎)>] -Phase Independence of ∎ [< >] Involution2 [ ] Inversion of void Proof of multiplicative Self-Inversion of (([J]<[oo]>)) (like o) 1/±i = ∓i (<[(([ J ]<[oo]>))]>) (< ([ J ]<[oo]>) >) Involution1 ( ([<J>]<[oo]>) ) -Inverse Promotion1 ( ([ J ]<[oo]>) ) -Self-Inversion of J and (<[<(([ J ]<[oo]>))>]>) <(<[ (([ J ]<[oo]>)) ]>)> -Inverse Promotion2 <(< ([ J ]<[oo]>) >)> Involution1 <( ([<J>]<[oo]>) )> -Inverse Promotion1 <( ([ J ]<[oo]>) )> -Self-Inversion of J ∓i = ±i <(([ J ]<[oo]>))> (<[<<(([ J ]<[oo]>))>>]>) imaginary unit11 (< ([ J ]<[oo]>) >) Inverse Cancellation, Involution1 ( ([<J>]<[oo]>) ) -Inverse Promotion1 ( ([ J ]<[oo]>) ) -Self-Inversion of J ±i = ∓i (([ J ]<[oo]>)) <(<[(([ J ]<[oo]>))]>)> imaginary unit12 <(< ([ J ]<[oo]>) >)> Involution1 <( ([<J>]<[oo]>) )> -Inverse Promotion1 <( ([ J ]<[oo]>) )> -Self-Inversion of J J/i = i*π / i = π ([J]<[(([ J ]<[oo]>))]>) ([J]< ([ J ]<[oo]>) >) Involution1 ([J] ([<J>]<[oo]>) ) -Inverse Promotion1 ([J] ([ J ]<[oo]>) ) -Self-Inversion of J and ([ J ]<[<(([ J ]<[oo]>))>]>) <([ J ]<[ (([ J ]<[oo]>)) ]>)> Inverse Promotion3 ([<J>]<[ (([ J ]<[oo]>)) ]>) -Inverse Promotion1 ([<J>]< ([ J ]<[oo]>) >) Involution1 ([<J>] ([<J>]<[oo]>) ) -Inverse Promotion1 ([<J>] ([ J ]<[oo]>) ) -Self-Inversion of J J * -i = π ([ J ][<(([ J ]<[oo]>))>]) <([ J ][ (([ J ]<[oo]>)) ])> Inverse Promotion1 ([<J>][ (([ J ]<[oo]>)) ]) -Inverse Promotion1 ([<J>] ([ J ]<[oo]>) ) Involution1 and ([J][<<(([ J ]<[oo]>))>>]) ([J][ (([ J ]<[oo]>)) ]) Inverse Cancellation ([J] ([ J ]<[oo]>) ) Involution1 π * i = J ([([<J>][i])][i]) ( [<J>][i] [i]) Involution1 ( [<J>][<o> ]) i * i = -1 <<( [ J ][ o ])>> 2x Inverse Promotion1 J Inverse Cancellation, Involution1, 2x Involution2 ii = eJ*i/2 = e-π/2 = e-π/2-2πk k ∈ ℤ (([[(([J]<[oo]>))]][(([J]<[oo]>))])) (( [J]<[oo]> ([J]<[oo]>) )) 3x Involution1 and ( ([[(([ J ]<[oo]>))]][<(([J]<[oo]>))>]) ) (<( [ J ]<[oo]> [ (([J]<[oo]>)) ])>) 2x Involution1, Inverse Promotion1 ( ( [<J>]<[oo]> [ (([J]<[oo]>)) ]) ) -Inverse Promotion1 ( ( [ J ]<[oo]> ([J]<[oo]>) ) ) Involution1, -Self-Inversion of J i-2i = eπ (([[(([J]<[oo]>))]][(([J]<[oo]>))][oo])) (( [J]<[oo]> ([J]<[oo]>) [oo])) 3x Involution1 (( [J] ([J]<[oo]>) )) Inversion of [oo] and ( ([[(([ J ]<[oo]>))]][<(([J]<[oo]>))>][oo]) ) (<( [ J ]<[oo]> [ (([J]<[oo]>)) ][oo])>) 2x Involution1, Inverse Promotion1 ( ( [<J>]<[oo]> [ (([J]<[oo]>)) ][oo]) ) -Inverse Promotion1 ( ( [ J ] ([J]<[oo]>) ) ) Involution1, -Self-Inversion of J 1/√a = √(1/a) (<[(([ [a] ]<[oo]>))]>) (< ([ [a] ]<[oo]>) >) Involution1 ( ([<[a]>]<[oo]>) ) -Inverse Promotion1 √9 = √±32 = ±3 (([[ ooooooooo ]]<[oo]>)) (([[( [ooo][ooo] )]]<[oo]>)) Cardinality of ooo (([[(([[ooo]][oo]))]]<[oo]>)) Cardinality of [ooo] (( [[ooo]][oo] <[oo]>)) 2x Involution1 (( [[ooo]] )) Inversion of [oo] ooo 2x Involution2 and (([[ ooooooooo ]]<[oo]>)) (([[ ( [ ooo ][ ooo ]) ]]<[oo]>)) Cardinality of ooo (([[<<( [ ooo ][ ooo ])>>]]<[oo]>)) -Inverse Cancellation (([[ ( [<ooo>][<ooo>]) ]]<[oo]>)) 2x -Inverse Promotion1 (([[ (([[<ooo>]][oo]) ) ]]<[oo]>)) Cardinality of [<ooo>] (( [[<ooo>]][oo] <[oo]>)) 2x Involution1 (( [[<ooo>]] )) Inversion of [oo] <ooo> 2x Involution2 -22 = 22 (([[<oo>]][oo])) ([<oo>][<oo>]) -Cardinality of [<oo>] <<([ oo ][ oo ])>> 2x Inverse Promotion1 ([ oo ][ oo ]) Inverse Cancellation (([[oo]][oo])) Cardinality of [oo] ab = -ab (( [[ a ]] [b])) (( [[ a ]][oo] <[oo]>[b])) -Inversion of [oo] (([[ (([[ a ]][oo])) ] ]<[oo]>[b])) 2x -Involution1 (([[ ( [ a ][ a ] ) ] ]<[oo]>[b])) -Cardinality of [a] (([[<<( [ a ][ a ] )>>] ]<[oo]>[b])) -Inverse Cancellation (([[ ( [<a>][<a>] ) ] ]<[oo]>[b])) 2x -Inverse Promotion (([[ (([[<a>]][oo])) ] ]<[oo]>[b])) Cardinality of [<a>] (( [[<a>]][oo] <[oo]>[b])) 2x Involution1 !!! ambiguity (( [[<a>]] [b])) Inversion of [oo] Self-Inversion problem -0 = 0 1/1 = 1 <∎> ∎ = ∎∎ = ∎∎∎ = ∎∎∎∎ = ... J = <J> ... = <JJJJ> = <JJ> = = JJ = JJJJ = ... , ambiguity of (([J]<[oo]>)) 1/±i = ∓i ab = -ab
Three-dimensional representation of the Eulerian formula ez*i = cos z + i sin z => (([z]([J]<[oo]>)))
Real part of the complex sine function
Imaginary part of the complex sine function an+1 = ([an]J) o -> (J) -> o -> (J) -> o -> ... <= (1 -> -1 -> 1 -> -1 -> 1 -> ...) an+1 = ([an][i]) o -> i -> (J) -> <i> -> o -> ... <= (1 -> i -> -1 -> -i -> 1 -> ...) an+1 = <([an][i])> o -> <i> -> (J) -> i -> o -> ... <= (1 -> -i -> -1 -> i -> 1 -> ...) ea*i = cos a + i sin a => (([a][i])) e-a*i = cos a - i sin a => (([<a>][i])) sin a = (ea*i - e-a*i)/2i => ([ (([a][i]))<(([<a>][i]))>]<[oo][i]>) cos a = (ea*i + e-a*i)/2 => ([ (([a][i])) (([<a>][i])) ]<[oo] >) ea = cosh a + sinh a => (a) e-a = cosh a - sinh a => (<a>) sinh a = (ea - e-a)/2 => ([(a)<(<a>)>]<[oo]>) cosh a = (ea + e-a)/2 => ([(a) (<a>) ]<[oo]>) sin(x + i y) = sin(x) cosh(y) + i cos(x) sinh(y) cos(x + i y) = cos(x) cosh(y) - i sin(x) sinh(y) sin(z) = -i sinh(i z) sinh(z) = -i sin(i z) cos(z) = cosh(i z) cosh(z) = cos(i z) sin'(z) = cos(z) sinh'(z) = cosh(z) cos'(z) = -sin(z) cosh'(z) = sinh(z) unit circle x2 + y2 = 1 (([[x]][oo])) (([[y]][oo])) = o unit hyperbole x2 - y2 = 1 (([[x]][oo]))<(([[y]][oo]))> = o
