Musean hypernumber - Citizendia

Musean Hypernumbers are an algebraic concept envisioned by Charles A. Musès (1919–2000) to form a complete, integrated, connected, and natural number system. Charles A Muses (1919&ndash2000 a figure who wrote articles and books under various pseudonyms (including Musès Musaios Kyril Demys Arthur Fontaine Kenneth Demarest and Carl ^[1]^[2]^[3]^[4]^[5] Musès sketched certain fundamental types of hypernumbers and arranged them in ten "levels", each with its own associated arithmetic and geometry. Arithmetic or arithmetics (from the Greek word αριθμός = number is the oldest and most elementary branch of mathematics used by almost everyone Geometry ( Greek γεωμετρία; geo = earth metria = measure is a part of Mathematics concerned with questions of size shape and relative position

Mostly criticized for lack of mathematical rigor and unclear defining relations, Musean hypernumbers are often perceived as an unfounded mathematical speculation. This impression was not helped by Musès' outspoken confidence in applicability for fields far beyond what one might expect from a number system, including consciousness, religion, and metaphysics.

The term "M-algebra" was used by Musès for investigation into a subset of his hypernumber concept (the 16 dimensional conic sedenions and certain subalgebras thereof), which is at times confused with the Musean hypernumber level concept itself. In Abstract algebra, sedenions form a 16- dimensional algebra over the reals. The current article separates this well-understood "M-algebra" after Musès from the remaining controversial hypernumbers, and lists certain applications envisioned by the inventor.

"M-algebra" and "hypernumber levels"

Musès was convinced that the basic laws of arithmetic on the reals are in direct correspondence to a concept where numbers could be arranged in "levels", where fewer arithmetical laws would be applicable with increasing level number. Arithmetic or arithmetics (from the Greek word αριθμός = number is the oldest and most elementary branch of mathematics used by almost everyone ^[3] However, this concept was not developed much further beyond the initial idea, and providing defining relations to most of these levels is outstanding.

Higher dimensional numbers built on the first three levels were called "M-algebra"^[6] by Musès if they yielded a distributive multiplication, unit element, and multiplicative norm. In Mathematics, and in particular in Abstract algebra, distributivity is a property of Binary operations that generalises the distributive law In Linear algebra, Functional analysis and related areas of Mathematics, a norm is a function that assigns a strictly positive length It contains kinds of octonions and historical quaternions (except A. In Mathematics, the octonions are a nonassociative extension of the Quaternions Their 8-dimensional Normed division algebra over the Real Quaternions, in Mathematics, are a non-commutative extension of Complex numbers They were first described by the Irish Mathematician MacFarlane's hyperbolic quaternion) as subalgebras. In Mathematics, a hyperbolic quaternion is a mathematical concept first suggested by Alexander MacFarlane in 1891 in a speech to the American Association A proof of completeness of M-algebra has not been provided.

Conic sedenions / "16 dimensional M-algebra"

"M-algebra" after C. Musès^[6] refers to number systems with dimensionality that are vector spaces over the reals, to bases that are roots of -1 or +1, and which possess a multiplicative modulus. In Mathematics, a vector space (or linear space) is a collection of objects (called vectors) that informally speaking may be scaled and added In Mathematics, the real numbers may be described informally in several different ways While the idea of such numbers was far from new and contains many known isomorphic number systems (like e. g. split-complex numbers or tessarines), certain results from 16 dimensional (conic) sedenions were a novelty: Musès demonstrated the logarithm and real powers in number systems built to non-real roots of +1. In Linear algebra, a split-complex number is of the form z = x + y j where j2 = +1, and x and y are Real The tessarines are a mathematical idea introduced by James Cockle in 1848

Multiplication table

Conic sedenions^[7]^[8] form an algebra with non-commutative, non-associative, but alternative multiplication and a multiplicative modulus. In Mathematics, commutativity is the ability to change the order of something without changing the end result In Mathematics, associativity is a property that a Binary operation can have In Abstract algebra, an alternative algebra is an algebra in which multiplication need not be Associative, only alternative. It consists of one real axis (to basis $1$ ), eight imaginary axes (to bases $i n$ with $i_n^2=-1$ ), and seven counterimaginary^[9] axes (to bases $\varepsilon$ with $\varepsilon{}_n^2=+1$ ).

The multiplication table is:

Image:ConicSedenionsMultTable.png

Similar to unity (1), the imaginary basis $i 0$ is always commutative and associative under multiplication. Musès at times used the symbol $\varepsilon_0 := 1$ to highlight this similarity. ^[6] In fact, conic sedenions are isomorphic to complex octonions, i. In Mathematics, the octonions are a nonassociative extension of the Quaternions Their 8-dimensional Normed division algebra over the Real e. octonions with complex number coefficients. Complex plane In Mathematics, the complex numbers are an extension of the Real numbers obtained by adjoining an Imaginary unit, denoted By examining $\varepsilon_n$ as bases to real number coefficients, however, Musès was able to show certain algebraic relations, including power and logarithm of $\varepsilon_n$ .

Select findings

Musès showed that a countercomplex basis $\varepsilon{}_n$ ( $n = 1. . . 7$ ) not only has an exponential function^[10]

$e ^ { \varepsilon{}_n \alpha } = \cosh ~\alpha + \varepsilon{}_n ( \sinh ~\alpha )$

( $α$ real) but also possesses real powers:^[7]^[11]

$\varepsilon{}_n ^ \alpha = \frac{1}{2} [ (1 - \varepsilon{}_n ) + (1 + \varepsilon{}_n ) e^{- \pi i_n \alpha } ]$

This is referred to as "power orbit" of $\varepsilon{}_n$ by Musès. The exponential function is a function in Mathematics. The application of this function to a value x is written as exp( x) Also, a logarithm

$\ln \varepsilon{}_n = \frac{\pi }{2} ( i_0 - i_n )$

is possible in this arithmetic. ^[7] Their multiplicative modulus $| z |$ is^[8]

$|z| = |a + \sum{b_n i_n} + \sum{c_n \varepsilon_n } + d| := \sqrt[4]{ (a^2 + b_n^2 - c_n^2 - d^2)^2 + 4(ad - b_n c_n)^2 }$

List of number types^[7] and their isomorphisms

Circular quaternions and octonions

Circular quaternions and octonions from the Musean hypernumbers are identical to quaternions and octonions from Cayley-Dickson construction. Quaternions, in Mathematics, are a non-commutative extension of Complex numbers They were first described by the Irish Mathematician In Mathematics, the octonions are a nonassociative extension of the Quaternions Their 8-dimensional Normed division algebra over the Real They are built on imaginary bases $i n$ only.

Hyperbolic quaternions

Hyperbolic quaternions after Musès, to bases { $1, \varepsilon{}_1 , \varepsilon{}_2 , i_3$ } are isomorphic to coquaternions (split-quaternions). In Abstract algebra, the split-quaternions or coquaternions are elements of an Associative algebra introduced by James Cockle in 1849 under the They are different from A. MacFarlane's hyperbolic quaternions (first mention in 1891), which are not associative. Alexander Macfarlane ( April 21 1851 – August 28, 1913) was a Scottish - Canadian Logician Physicist In Mathematics, a hyperbolic quaternion is a mathematical concept first suggested by Alexander MacFarlane in 1891 in a speech to the American Association In Mathematics, associativity is a property that a Binary operation can have

Conic quaternions

Conic quaternions are built on bases { $1, i, \varepsilon, i_0$ } and form a commutative, associative, and distributive arithmetic. In Mathematics, commutativity is the ability to change the order of something without changing the end result In Mathematics, associativity is a property that a Binary operation can have In Mathematics, and in particular in Abstract algebra, distributivity is a property of Binary operations that generalises the distributive law Arithmetic or arithmetics (from the Greek word αριθμός = number is the oldest and most elementary branch of mathematics used by almost everyone They contain non-trivial idempotents and zero divisors, but no nilpotents. Idempotence ˌaɪdɨmˈpoʊtəns describes the property of operations in Mathematics and Computer science which means that multiple applications of the operation In Abstract algebra, a nonzero element a of a ring is a left zero divisor if there exists a nonzero b such that ab = 0 In Mathematics, an element x of a ring R is called nilpotent if there exists some positive Integer n such that Conic quaternions are isomorphic to tessarines, and also to bicomplex numbers (from the multicomplex numbers). The tessarines are a mathematical idea introduced by James Cockle in 1848 In Mathematics, a bicomplex number (from the Multicomplex numbers see e In Mathematics, the multicomplex numbers, {\Bbb{MC}}_n form an n dimensional algebra generated by one element e which satisfies ~e^n

In contrast, circular and hyperbolic quaternions are not commutative, hyperbolic quaternions also contain nilpotents. In Mathematics, an element x of a ring R is called nilpotent if there exists some positive Integer n such that

Hyperbolic octonions

Hyperbolic octonions are isomorphic to split-octonion algebra. In Mathematics, the octonions are a nonassociative extension of the Quaternions Their 8-dimensional Normed division algebra over the Real In Mathematics, the split-octonions are a Nonassociative extension of the Quaternions (or the Split-quaternions. They consist of one real, three imaginary ( $\sqrt{-1}$ ), and four counterimaginary ( $\varepsilon$ ) bases, e. In Mathematics, the real numbers may be described informally in several different ways Geometric interpretation Geometrically imaginary numbers are found on the vertical axis of the complex number plane g. { $1, i_1, i_2, i_3, \varepsilon{}_4 , \varepsilon{}_5, \varepsilon{}_6 , \varepsilon{}_7$ }.

Conic octonions

Conic octonions to bases $\{ 1, i_1, i_2, i_3,~i_0, \varepsilon{}_1, \varepsilon{}_2, \varepsilon{}_3 \}$ form an associative, non-commutative octonionic number system. They are isomorphic to biquaternions. The biquaternions are the numbers w + xi + yj + zk \ \! where w x y and z are complex numbers and the elements of {1 i j k} multiply as in the Quaternion group

External links

Mention in zero-divisor analysis by R. de Marrais on arXiv. The term hypercomplex number has been used in Mathematics for the elements of algebras that extend or go beyond Complex number arithmetic In Mathematics, an element x of a ring R is called nilpotent if there exists some positive Integer n such that In Mathematics, the octonions are a nonassociative extension of the Quaternions Their 8-dimensional Normed division algebra over the Real Quaternions, in Mathematics, are a non-commutative extension of Complex numbers They were first described by the Irish Mathematician In Abstract algebra, sedenions form a 16- dimensional algebra over the reals. In Linear algebra, a split-complex number is of the form z = x + y j where j2 = +1, and x and y are Real In Mathematics, the split-octonions are a Nonassociative extension of the Quaternions (or the Split-quaternions. In Abstract algebra, the split-quaternions or coquaternions are elements of an Associative algebra introduced by James Cockle in 1849 under the In Abstract algebra, the split-quaternions or coquaternions are elements of an Associative algebra introduced by James Cockle in 1849 under the The tessarines are a mathematical idea introduced by James Cockle in 1848 In Abstract algebra, a nonzero element a of a ring is a left zero divisor if there exists a nonzero b such that ab = 0 org
Zero-divisor algebras on Tony Smith's personal home page (as of 12 Jan 2007)

The hypernumber "level" concept

In^[3] Musès paired certain fundamental laws of arithmetic with suggested number levels, where fewer of these laws would be applicable with increasing level number. Musès envisioned ". . . sensitivity to operational distinctions on the part of hypernumbers". In the absence of rigorous mathematical treatment, however, Musès' hypernumber level concept has only been adapted for metaphysical or religious ideas. Rigour or rigor (see spelling differences) has a number of meanings in relation to intellectual life and discourse ^[12]^[13]^[14]

Providing defining relations for hypernumbers remains a fringe interest today,^[15] though it could benefit description of physical law that is based on the lower, well-understood levels. ^[16]^[17]

The following lists an overview of the levels as envisioned by Musès.

Real, complex, and epsilon numbers

The first two levels in hypernumber arithmetic correspond to real and imaginary number arithmetic. In Mathematics, the real numbers may be described informally in several different ways Geometric interpretation Geometrically imaginary numbers are found on the vertical axis of the complex number plane The $\varepsilon$ basis after Musès is identical to j from the split-complex numbers, and is a non-real root of $+ 1$ . Epsilon numbers are assigned the 3rd level in the hypernumbers program.

w arithmetic

Beginning with w arithmetic,^[1]^[4]^[11] Musès envisioned hypernumber types that are increasingly unfamiliar and speculative. While providing certain rules on how to use these numbers, many open questions remain to date. w numbers are assigned the 4th level in the hypernumbers program.

In the two-dimensional (real, w) plane, the power orbit $~w^\alpha$ (with $~\alpha$ real) is periodic with $w 0 = w 6 = 1$ and the following integral powers:

$w^1 = ~w$

$w^2 = ~-1 + w$

$w^3 = ~-1$

$w^4 = ~-w$

$w^5 = ~1 - w$ .

They offer a multiplicative modulus:

$||a + bw|| = \sqrt{a^2 + ab + b^2}$

If a and b are real number coefficients, the arithmetic <(1,w), +, *> is a field (in fact the complex numbers with basis 1 and a primitive sixth root of unity rather than the usual fourth). In Mathematics, the real numbers may be described informally in several different ways In Abstract algebra, a field is an Algebraic structure in which the operations of Addition, Subtraction, Multiplication and division Complex plane In Mathematics, the complex numbers are an extension of the Real numbers obtained by adjoining an Imaginary unit, denoted In Mathematics, the n th roots of unity, or de Moivre numbers are all the Complex numbers that yield 1 when raised to a given power However, the dual base number to (w) is (-w), which is different from the conjugate of (w), which is 1-(w). This is in contrast to e. g. the imaginary base $i := \sqrt{-1}$ , for which both dual and conjugate are the same (-i). The resulting (-w) arithmetic is therefore distinct from -(w) arithmetic, while coexisting on the same number plane.

Image:HypernumbersPowerOrbitW.gif

p and q numbers

So-called p and q numbers^[4] are assigned the 5th level in the hypernumbers program, and form a nearly dual system. Each being nilpotent ( $p 2 = q 2 = 0$ ), the arithmetic is envisioned to offer a multiplicative modulus, an argument, and a polar form. In Mathematics, an element x of a ring R is called nilpotent if there exists some positive Integer n such that In Mathematics, the absolute value (or modulus) of a Real number is its numerical value without regard to its sign. Complex plane In Mathematics, the complex numbers are an extension of the Real numbers obtained by adjoining an Imaginary unit, denoted Complex plane In Mathematics, the complex numbers are an extension of the Real numbers obtained by adjoining an Imaginary unit, denoted

The integral powers are:

$p^0 = q^0 = p^2 = q^2 =~0$

$p^1 =~p$

$q^1 =~q$

$p^3 =~q$

$q^3 =~p$

In the {p, q} plane, both $~p^\alpha$ and $~q^\alpha$ (with $~\alpha$ real) lie on a two-leaved rose, described through $ap +~bq$ with

$(a^2 + b^2)^2 =~(a + b)(a - b)^2$ .

Image:HypernumbersPowerOrbitPQ.gif

Note on (-p), p^(-1), 1/p

From:^[4]

". . . Note that -p is generated via w, thus: $(qw)^3 = (wq)^3 = (w^3)(q^3) = (-1)p =~-p$ . It must be remembered that because p is nilpotent ( $p^2 = 0, p \ne 0$ ), its zeroth power cannot be 1; in fact $p^0 =~0$ . Hence also $p^{-1} \ne 1/p$ , and since $(1/p)(1/p) = 1/p^2 = \infty$ , we see that $~1/p$ is panpotent, i. e. a root of infinity. Compare $1/(1 \pm \varepsilon)$ , which are a pair of divisors of infinity. "

m numbers

The 6th level in the Musean hypernumbers is governed by cassinoids or Cassinian ovals,^[4] which geometrically describe their multiplication.

In the {real, m} plane, they offer the following relations:

$m^2 =~m$

$(\sqrt{2} m )^2 =~0$

$(\sqrt{3} m )^2 =~-1$

It is speculated that a number system like this would use coefficients such as $\sqrt{2}$ in the expression $\sqrt{2} m$ , that are not actually real numbers. Instead, one would need to look at +1, -1, +m, and -m as units, and the coefficients as absolute numbers which are distinct from real numbers and are never negative.

Image:HypernumbersPowerOrbitM.gif

The Cassinian ovals are described by:

$s^4 :=~(a^2 + b^2)^2 + 2(a^2 - b^2) + 1$

The remaining levels

In the 7th level, Musès pictured a number $Ω$ where $Ω n = Ω$ for any finite n, $\Omega^\infty = 0$ , but $\Omega^{\infty - n}$ would be a number of the form $a + b Ω$ (with a, b real). ^[4]

The 8th level, v is envisioned as unifying concept to allow to transition between all the lower hypernumber types. ^[5]

The 9th level, $σ$ is envisioned as the creator of axes, and has somewhat the characteristic of an operator (rather than a number). The product $σ v$ is proposed to be the unit step function. ^[5]

The 10th level consists of 0 and antinumbers. Antinumbers are envisioned to be numbers beyond positive and negative infinity. With use of v one would be able to span entire spaces consisting of axes of zeros, and connect numbers beyond positive and negative infinity. ^[5]

Visions of applicability

The range of applications envisioned by Musès of his hypernumber concept is grandiose: A full and complete understanding of all laws of physics (in particular quantum mechanics^[6]^[18]), description of consciousness in terms of physical formulations,^[1]^[4]^[5] spiritual growth, religious enlightenment, solving mathematical problems (including the Riemann hypothesis), and exploration of para-psychological phenomena (e. The Riemann hypothesis (also called the Riemann zeta-hypothesis) first formulated by Bernhard Riemann in 1859 is one of the most famous and important unsolved g. ^[19]). But none of his visions have been realized. Much of Musès' own writings mix the mathematical content outlined above with one or more of these visions,^[20] and most secondary literature is about this speculative context.

References

^ ^a ^b ^c Musès, Charles A. (1972). "Hypernumbers and their Spaces: a Summary of New Findings". J. Study. Consciousness 5: 251–256.
^ Musès, Charles A. (1977). "Explorations in mathematics". impact of science on society 27: 67–85.
^ ^a ^b ^c Musès, Charles A. (1978). "Hypernumbers—II. further concepts and computational applications". Appl. Math. Comput. 4: 45–66. doi:10.1016/0096-3003(78)90026-7. A digital object identifier ( DOI) is a permanent identifier given to an Electronic document.
^ ^a ^b ^c ^d ^e ^f ^g Musès, Charles A. (1979). "Computing in the bio-sciences with hypernumbers: a survey". Intl. J. Bio-Med. Comput. 10: 519–525. doi:10.1016/0020-7101(79)90032-1. A digital object identifier ( DOI) is a permanent identifier given to an Electronic document.
^ ^a ^b ^c ^d ^e Musès, Charles A. (1983). "Hypernumbers and time operators". Appl. Math. Comput. 12: 139–167. doi:10.1016/0096-3003(83)90004-8. A digital object identifier ( DOI) is a permanent identifier given to an Electronic document.
^ ^a ^b ^c ^d Musès, Charles A. (1980). "Hypernumbers and quantum field theory with a summary of physically applicable hypernumber arithmetics and their geometries". Appl. Math. Comput. 6: 63–94. doi:10.1016/0096-3003(80)90016-8. A digital object identifier ( DOI) is a permanent identifier given to an Electronic document.
^ ^a ^b ^c ^d Carmody, Kevin (1988). "Circular and hyperbolic quaternions, octonions, and sedenions". Appl. Math. Comput. 28: 47–72. doi:10.1016/0096-3003(88)90133-6. A digital object identifier ( DOI) is a permanent identifier given to an Electronic document.
^ ^a ^b Carmody, Kevin (1997). "Circular and hyperbolic quaternions, octonions, and sedenions— further results". Appl. Math. Comput. 84: 27–48. doi:10.1016/S0096-3003(96)00051-3. A digital object identifier ( DOI) is a permanent identifier given to an Electronic document.
^ The terms "counterimaginary" and "countercomplex" used by Musès are synonymous to the more common term split-complex
^ Musès, Charles A. In Linear algebra, a split-complex number is of the form z = x + y j where j2 = +1, and x and y are Real (1977). "Applied hypernumbers: computational concepts". Appl. Math. Comput. 3: 211–226. doi:10.1016/0096-3003(77)90002-9. A digital object identifier ( DOI) is a permanent identifier given to an Electronic document.
^ ^a ^b Musès, Charles A. (1994). "Hypernumbers applied, or how they interface with the physical world". Appl. Math. Comput. 60: 25–36. doi:10.1016/0096-3003(94)90203-8. A digital object identifier ( DOI) is a permanent identifier given to an Electronic document.
^ Musaios (a pseudonym of Musès'), "The Lion Path", House of Horus (1990)
^ House of Horus web site
^ Private Lion Path web site
^ "Hypercomplex" number discussion group on Yahoo (R)
^ Köplinger, Jens (2006). A pseudonym is a fictitious alternative to a person's legal name (see Alias) "Hypernumbers and relativity". Appl. Math. Comput. 188: 954. doi:10.1016/j.amc.2006.10.051. A digital object identifier ( DOI) is a permanent identifier given to an Electronic document.
^ Köplinger, Jens (2006). "Gravity and electromagnetism on conic sedenions". Appl. Math. Comput. 188: 954. doi:10.1016/j.amc.2006.10.050. A digital object identifier ( DOI) is a permanent identifier given to an Electronic document.
^ Musès, Charles A. (1984). "Some current dilemmas in applied physical mathematics with some solutions". Appl. Math. Comput. 14: 207–211. doi:10.1016/0096-3003(84)90038-9. A digital object identifier ( DOI) is a permanent identifier given to an Electronic document.
^ Charles Musès - "Time and destiny", Thinking Allowed Productions (#S460) online)
^ "The nature of hypernumbers can reveal the projection process . . . (and) on the source of the hologram world or ordinary bodily experience . . . to be able to go between the image world and the source world at will (time travel). " (from C. Musès, A. M. Young: "Consciousness and reality: the human pivot point", Outerbridge & Lazard, New York, 1972)

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Contents

"M-algebra" and "hypernumber levels"

Conic sedenions / "16 dimensional M-algebra"

Multiplication table

Select findings

List of number types^[7] and their isomorphisms

Circular quaternions and octonions

Hyperbolic quaternions

Conic quaternions

Hyperbolic octonions

Conic octonions

See also

External links

The hypernumber "level" concept

Real, complex, and epsilon numbers

w arithmetic

p and q numbers

Note on (-p), p^(-1), 1/p

m numbers

The remaining levels

Visions of applicability

References

Contents

"M-algebra" and "hypernumber levels"

Conic sedenions / "16 dimensional M-algebra"

Multiplication table

Select findings

List of number types[7] and their isomorphisms

Circular quaternions and octonions

Hyperbolic quaternions

Conic quaternions

Hyperbolic octonions

Conic octonions

See also

External links

The hypernumber "level" concept

Real, complex, and epsilon numbers

w arithmetic

p and q numbers

Note on (-p), p^(-1), 1/p

m numbers

The remaining levels

Visions of applicability

References

List of number types^[7] and their isomorphisms