Musean hypernumber

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Musean Hypernumbers are an algebraic 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.

Mostly criticized for lack of mathematical rigor and unclear defining relations, Musean hypernumbers are often perceived 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.

[edit] "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.

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.

[edit] 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.

[edit] 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 $i n$ with $i_n^2=-1$ ), and seven counterimaginary axes (to bases $\varepsilon$ with $\varepsilon{}_n^2=+1$ ; the terms "counterimaginary" and "countercomplex" used by Musès are synonymous to the more common term split-complex).

The multiplication table is:

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]

[edit] Select findings

Musès showed that a countercomplex basis $\varepsilon{}_n$ ( $n = 1...7$ ) not only has an exponential function^[9]

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

( $α$ real) but also possesses real powers:^[7]^[10]

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

This is referred to as "power orbit" of $\varepsilon{}_n$ by Musès. Also, a logarithm

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

is possible in this arithmetic.^[7]

[edit] List of number types^[7] and their isomorphisms

[edit] 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 $i n$ only.

[edit] Hyperbolic quaternions

Hyperbolic quaternions after Musès, to bases { $1, \varepsilon{}_1 , \varepsilon{}_2 , i_3$ } are isomorphic to coquaternions (split-quaternions). They are different from A. MacFarlane's hyperbolic quaternions (first mention in 1891), which are not associative.

[edit] Conic quaternions

Conic quaternions are built on bases { $1, i, \varepsilon, i_0$ } 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 contrast, circular and hyperbolic quaternions are not commutative, hyperbolic quaternions also contain nilpotents.

[edit] Hyperbolic octonions

Hyperbolic octonions are isomorphic to split-octonion algebra. They consist of one real, three imaginary ( $\sqrt{-1}$ ), and four counterimaginary ( $\varepsilon$ ) bases, e.g. { $1, i_1, i_2, i_3, \varepsilon{}_4 , \varepsilon{}_5, \varepsilon{}_6 , \varepsilon{}_7$ }.

[edit] 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.

[edit] See also

Biquaternion
Hyperbolic quaternion (per A. MacFarlane)
Hypercomplex number
Nilpotents
Octonion
Quaternion
Sedenion
Split-complex number
Split-octonions
Split-quaternion / Coquaternion
Tessarines
Zero divisor

[edit] 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)

[edit] 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]

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.

[edit] Real, complex, and epsilon numbers

The first two levels in hypernumber arithmetic correspond to real and imaginary number arithmetic. 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.

[edit] 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 $~w^\alpha$ (with $~\alpha$ real) is elliptic, 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. 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.

[edit] 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.

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

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

[edit] 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."

[edit] 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.

The Cassinian ovals are described by:

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

[edit] 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 zero's, and connect numbers beyond positive and negative infinity.^[5]

[edit] 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 have 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]

[edit] 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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Categories: Hypercomplex numbers | Nonassociative algebra

Musean hypernumber

From Wikipedia, the free encyclopedia

Contents

[edit] "M-algebra" and "hypernumber levels"

[edit] Conic sedenions / "16 dimensional M-algebra"

[edit] Multiplication table

[edit] Select findings

[edit] List of number types^[7] and their isomorphisms

[edit] Circular quaternions and octonions

[edit] Hyperbolic quaternions

[edit] Conic quaternions

[edit] Hyperbolic octonions

[edit] Conic octonions

[edit] See also

[edit] External links

[edit] The hypernumber "level" concept

[edit] Real, complex, and epsilon numbers

[edit] w arithmetic

[edit] p and q numbers

[edit] Note on (-p), p^(-1), 1/p

[edit] m numbers

[edit] The remaining levels

[edit] Visions of applicability

[edit] References

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Musean hypernumber

From Wikipedia, the free encyclopedia

Contents

[edit] "M-algebra" and "hypernumber levels"

[edit] Conic sedenions / "16 dimensional M-algebra"

[edit] Multiplication table

[edit] Select findings

[edit] List of number types[7] and their isomorphisms

[edit] Circular quaternions and octonions

[edit] Hyperbolic quaternions

[edit] Conic quaternions

[edit] Hyperbolic octonions

[edit] Conic octonions

[edit] See also

[edit] External links

[edit] The hypernumber "level" concept

[edit] Real, complex, and epsilon numbers

[edit] w arithmetic

[edit] p and q numbers

[edit] Note on (-p), p^(-1), 1/p

[edit] m numbers

[edit] The remaining levels

[edit] Visions of applicability

[edit] References

Views

Navigation

interaction

Search

In other languages

[edit] List of number types^[7] and their isomorphisms