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 (second degree) (Count)
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] (third degree)
([a]) = (([[a]])) = (([[a]][o])) -Involution2, -Involution1
([a][a]) = (([[a]][o])([[a]][o])) = (([[a]][oo])) 2x [a], -Distribution of [[a]]
([a]...[a]) = (([[a]][o...o])) induction step to all natural numbers
a b Addition ooo ooo 3 + 3 = 6
( [a] [b]) Multiplication ( [ooo] [ooo]) 3 * 3 = 3 + 3 + 3 = 9
(( [[a]] [b])) Exponentiation (( [[ooo]] [ooo])) 33 = 3 * 3 * 3 = 27
((( [[[a]]] [b]))) Tetration ((( [[[ooo]]] [ooo]))) 333 = 327 = 7.625.597.484.987
(((([[[[a]]]][b])))) Hyper-5 (((([[[[ooo]]]][ooo])))) 3...3 7.625.597.484.987 times - our universe is already too small for that
...
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 = a/b
2 = (a/b)2
2 * b2 = (a/b)2 * b2
2 * b * b = a * a
a mod 1 = 0 & b mod 1 = 0 ->
-> (a*a) mod 2 = 0 -> a mod 2 = 0 -> (2*b*b) mod 4 = 0 -> (b*b) mod 2 = 0 -> b mod 2 = 0
-> (a*a) mod 8 = 0 -> a mod 4 = 0 -> (2*b*b) mod 16 = 0 -> (b*b) mod 8 = 0 -> b mod 4 = 0
-> (a*a) mod 32 = 0 -> a mod 8 = 0 -> (2*b*b) mod 64 = 0 -> (b*b) mod 32 = 0 -> b mod 8 = 0
-> (a*a) mod 128 = 0 -> ...
=> √2 is irrational
(([[oo]]<[oo]>)) = ([a]<[b]>)
([[oo]]<[oo]>) = [a]<[b]>
b = o -> a = (([[oo]]<[ oo ]>)) eln 2 / 2
a = o -> b = (([[oo]]<[<oo>]>)) eln 2 / -2
b = oo -> a = (([[oo]]<[ oo ]>)[oo]) eln 2 / 2 + ln 2
a = oo -> b = (([[oo]]<[<oo>]>)[oo]) eln 2 / -2 + ln 2
b = ooo -> a = (([[oo]]<[ oo ]>)[ooo]) eln 2 / 2 + ln 3
a = ooo -> b = (([[oo]]<[<oo>]>)[ooo]) eln 2 / -2 + ln 3
...
=> a and b 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