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Musean Hypernumbers are a concept envisioned by Charles A. Musès (1919–2000) to form a complete, integrated, connected, and natural number system [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. Charles A. Musès (1919-2000), also known as Musaios, was the founder of the Lion Path, a shamanistic movement. ...
Arithmetic or arithmetics (from the Greek word αÏιθμός = number) is the oldest and most elementary branch of mathematics, used by almost everyone, for tasks ranging from simple daily counting to advanced science and business calculations. ...
Table of Geometry, from the 1728 Cyclopaedia. ...
Mostly criticized for lack of mathematical rigor and unclear defining relations, Musean hypernumbers are often precieved as an unfounded mathematical speculation. This impression is 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. The current article separates this well-understood "M-algebra" after Musès from the remaining controversial hypernumbers, and lists certain applications envisioned by the inventor. The sedenions form a 16-dimensional algebra over the reals obtained by applying the Cayley-Dickson construction to the octonions. ...
"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 [3]. However, this concept was not developed much further beyond the initial idea, and providing defining relations to some of these levels is subject to ongoing investigation. Arithmetic or arithmetics (from the Greek word αÏιθμός = number) is the oldest and most elementary branch of mathematics, used by almost everyone, for tasks ranging from simple daily counting to advanced science and business calculations. ...
The higher dimensional numbers built on the first three levels were called "M-algebra" [6] by Musès if they yielded a distributive algebra with multiplicative norm. It contains kinds of octonions and historical quaternions (except A. MacFarlane's hyperbolic quaternion) as subalgebras. It was proposed (through unproven) to yield all multiplicatively normed linear algebras that are vector spaces over the reals. In mathematics, and in particular in abstract algebra, distributivity is a property of binary operations that generalises the distributive law from elementary algebra. ...
In linear algebra, functional analysis and related areas of mathematics, a norm is a function which assigns a positive length or size to all vectors in a vector space, other than the zero vector. ...
In mathematics, the octonions are a nonassociative extension of the quaternions. ...
In mathematics, the quaternions are a non-commutative extension of the complex numbers. ...
To meet Wikipedias quality standards, this article or section may require cleanup. ...
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. 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), the 16 dimensional (conic) sedenions were a novelty. In addition, Musès demonstrated the logarithm and real powers in number systems built to non-real roots of +1. 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. ...
In mathematics, the split-complex numbers are an extension of the real numbers defined analogously to the complex numbers. ...
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. It consists of one real axis (to basis 1), eight imaginary axes (to bases in with ), and seven counterimaginary axes (to bases with ; the terms "counterimaginary" and "countercomplex" used by Musès are synonymous to the more common term split-complex). In mathematics, especially abstract algebra, a binary operation * on a set S is commutative if x * y = y * x for all x and y in S. Otherwise * is noncommutative. ...
In mathematics, associativity is a property that a binary operation can have. ...
Look up alternative in Wiktionary, the free dictionary. ...
Mathematical meanings Especially in British/European usage, the modulus of a number is its absolute value. ...
In mathematics, the split-complex numbers are an extension of the real numbers defined analogously to the complex numbers. ...
The multiplication table is:
Similar to unity (1), the imaginary basis i0 is always commutative and associative under multiplication. Musès at times used the symbol to highlight this similarity [6].
Select findings Musès showed that a countercomplex basis (n = 1...7) not only has an exponential function [9] The exponential function is one of the most important functions in mathematics. ...
(α real) but also possesses real powers [7][10]:
This is refered to as "power orbit" of by Musès. Also, a logarithm
is possible in this arithmetic [7].
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. They are built on imaginary bases in only. In mathematics, the quaternions are a non-commutative extension of the complex numbers. ...
In mathematics, the octonions are a nonassociative extension of the quaternions. ...
In mathematics, the Cayley-Dickson construction produces a sequence of algebras over the field of real numbers, each with twice the dimension of the previous one. ...
Hyperbolic quaternions Hyperbolic quaternions after Musès, to bases {} are isomorphic to coquaternions (split-quaternions). They are different from A. MacFarlane's hyperbolic quaternions (first mention in 1891), which are not associative. In abstract algebra, a coquaternion is an idea put forward by James Cockle in 1849. ...
Alexander MacFarlane (1851 - 1913) was a Scottish-Canadian logician, physicist, and mathematician. ...
To meet Wikipedias quality standards, this article or section may require cleanup. ...
In mathematics, associativity is a property that a binary operation can have. ...
Conic quaternions Conic quaternions are built on bases {} and form a commutative, associative, and distributive arithmetic. They contain idempotents and zero divisors, but no nilpotents. Conic quaternions are isomorphic to tessarines, and also to bicomplex numbers (from the multicomplex numbers). In mathematics, especially abstract algebra, a binary operation * on a set S is commutative if x * y = y * x for all x and y in S. Otherwise * is noncommutative. ...
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 from elementary algebra. ...
Arithmetic or arithmetics (from the Greek word αÏιθμός = number) is the oldest and most elementary branch of mathematics, used by almost everyone, for tasks ranging from simple daily counting to advanced science and business calculations. ...
In mathematics, an idempotent element is an element which, intuitively, leaves something unchanged. ...
In abstract algebra, a non-zero element a of a ring R is a left zero divisor if there exists a non-zero 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 xn = 0. ...
The tessarines are a mathematical idea introduced by James Cockle in 1848. ...
A bicomplex number is a number written in the form, a + bi1 + ci2 + dj, where i1, i2 and j are imaginary units. ...
In mathematics, the multicomplex numbers form a commutative n dimensional algebra generated by one element e which satisfies . ...
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 xn = 0. ...
Hyperbolic octonions Hyperbolic octonions are isomorphic to split-octonion algebra. They consist of one real, three imaginary (), and four counterimaginary () bases, e.g. {}. In mathematics, the octonions are a nonassociative extension of the quaternions. ...
In mathematics, the split-octonions are a nonassociative extension of the quaternions (or the split-quaternions). ...
In mathematics, the real numbers may be described informally in several different ways. ...
Conic octonions Conic octonions to bases form an associative, non-commutative octonionic number system. They are isomorphic to biquaternions. In mathematics, a biquaternion is a numeric and geometric concept developed by William Kingdon Clifford, William Rowan Hamilton, and Alexander MacAuley in the nineteenth century. ...
See also In mathematics, a biquaternion is a numeric and geometric concept developed by William Kingdon Clifford, William Rowan Hamilton, and Alexander MacAuley in the nineteenth century. ...
To meet Wikipedias quality standards, this article or section may require cleanup. ...
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 xn = 0. ...
In mathematics, the octonions are a nonassociative extension of the quaternions. ...
In mathematics, the quaternions are a non-commutative extension of the complex numbers. ...
The sedenions form a 16-dimensional algebra over the reals obtained by applying the Cayley-Dickson construction to the octonions. ...
In mathematics, the split-complex numbers are an extension of the real numbers defined analogously to the complex numbers. ...
In mathematics, the split-octonions are a nonassociative extension of the quaternions (or the split-quaternions). ...
In abstract algebra, a coquaternion is an idea put forward by James Cockle in 1849. ...
In abstract algebra, a coquaternion is an idea put forward by James Cockle in 1849. ...
The tessarines are a mathematical idea introduced by James Cockle in 1848. ...
In abstract algebra, a non-zero element a of a ring R is a left zero divisor if there exists a non-zero b such that ab = 0. ...
External links - Mention in zero-divisor analysis by R. de Marrais on arXiv.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 meta-physical or religious ideas [11][12][13]. Look up Rigour in Wiktionary, the free dictionary. ...
Providing defining relations for hypernumbers remains a fringe interest today [14], though it could benefit description of physical law that is based on the lower, well-understood levels [15][16].
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. The 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. In mathematics, the real numbers may be described informally in several different ways. ...
w arithmetic Beginning with w arithmetic [1][4][10], 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 (with real) is elliptic, with w0 = w6 = 1 and the following integral powers:
They offer a multiplicative modulus:
If a and b are real number coefficients, the arithmetic <(1,w), +, *> is a field. 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 , 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. 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 (except division by zero) may be performed, and the same rules hold which are familiar from the arithmetic of ordinary numbers. ...
 Image File history File links HypernumbersPowerOrbitW.gif‎ Summary Power orbit of w arithmetic from the hypernumbers program. ...
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 (p2 = q2 = 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 xn = 0. ...
In mathematics, the absolute value (or modulus1) of a real number is its numerical value without regard to its sign. ...
In mathematics, a complex number is a number of the form where a and b are real numbers, and i is the imaginary unit, with the property i 2 = −1. ...
In mathematics, a complex number is a number of the form where a and b are real numbers, and i is the imaginary unit, with the property i 2 = −1. ...
The integral powers are:
In the {p, q} plane, both and (with real) lie on a two-leaved rose, described through with
.
 Image File history File links HypernumbersPowerOrbitPQ.gif‎ Summary Power Orbit of Rose Numbers (p and q Numbers) from the Hypernumbers Program. ...
Note on (-p), p^(-1), 1/p From [4]:
"...Note that -p is generated via w, thus: . It must be remembered that because p is nilpotent (), its zeroth power cannot be 1; in fact . Hence also , and since , we see that is panpotent, i.e. a root of infinity. Compare , 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:
It is speculated that a number system like this would use coefficients such as in the expression , 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 File history File links HypernumbersPowerOrbitM.gif‎ Summary Isonorms of Cassinoid Numbers (with characteristic s, norm t, and handle k). ...
The Cassinian ovals are described by:
The remaining levels In the 7th level, Musès pictured a number Ω where Ωn = Ω for any finite n, , but would be a number of the form a + bΩ (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 σv 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 zero's, and connect numbers beyond positive and negative infinity [5].
Visions of applicability The range of applications envisioned by Musès of his hypernumber concept is grandious: A full and complete understanding of all laws of physics (in particular quantum mechanics [6][17]), 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.g. [18]). But none of his visions has been realized. Much of Musès' own writings mix the mathematical content outlined above with one or more of these visions [19], and most secondary literature is about this speculative context. This feeds to common criticism which discards Musean hypernumbers as serious field of research [20] Unsolved problems in mathematics: Is the real part of a non-trivial zero of the Riemann zeta function always ½? In mathematics, the Riemann hypothesis (also called the Riemann zeta-hypothesis), first formulated by Bernhard Riemann in 1859, is one of the most famous unsolved problems. ...
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 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.
- ^ 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 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 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.
- ^ 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.
- ^ Musès, Charles A. (1977). "Applied hypernumbers: computational concepts". Appl. Math. Comput. 3: 211-226. DOI:10.1016/0096-3003(77)90002-9.
- ^ 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.
- ^ 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). "Hypernumbers and relativity". Appl. Math. Comput.. DOI:10.1016/j.amc.2006.10.051.
- ^ Köplinger, Jens (2006). "Gravity and electromagnetism on conic sedenions". Appl. Math. Comput.. DOI:10.1016/j.amc.2006.10.050.
- ^ 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.
- ^ 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)
- ^ Initial perception of this article, prior to the Jan 2007 rewrite
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