Hypernombre

The hypernombres are numbers associated with dimensions, discovered by Dr. Charles A. Musès (1919 - 2000). The hypernombres form a system complete, coherent, connected and natural. There exist ten levels of hypernombres, each one has its clean Arithmétique and Géométrie.

Selection of the types of hypernombres

Real numbers and complexes

The first two levels of arithmetic of the hypernombres correspond to arithmetic of the real and Imaginary number.

Epsilon numbers

With the numbers $\backslash \; varepsilon$, the program of the hypernombres is able to define a broad interval of mathematical operations on the systems of numbers containing of the bases with the square roots not-real of 1, i.e $\backslash \; varepsilon^2\; =\; 1\; \backslash ,$ but $\backslash \; varepsilon\; \backslash \; \backslash \; pm\; 1\; \backslash ,$ (see also Complex number split). The concept of orbit of power of the hypernombres allows the powers, the roots and the Logarithme S on the systems of Nombre S which contain the bases of $\backslash \; varepsilon$.

The epsilon numbers are placed in the third level of the program of the hypernombres.

Exponential against-orbit of power

If one considers that for the complex numbers, the orbit of power $i^\; \backslash \; alpha$ as well as the exponential orbit $\backslash \; exp\; ~i\; \backslash \; alpha$ of the imaginary basic number is not valid any more, it is not the case for the numbers $~\; \backslash \; varepsilon$. In the place, we have for the exponential orbit:

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

The orbit of power is written:

$\backslash \; varepsilon\; ^\; \backslash \; alpha\; =\; \backslash \; frac\; \{1\}\; \{2\}\; (1\; -\; \backslash \; varepsilon)\; +\; (1\; +\; \backslash \; varepsilon)\; e^\; \{-\; \backslash \; pi\; I\; \backslash \; alpha\}$

Note : the orbit of power contains a product of $~\; \backslash \; varepsilon$ and $~i$, which requires arithmetic conical quaternions (below).

Examples and isomorphisms

Circular quaternions and octonions

The quaternions and octonions them circular program of the hypernombres are identical to the quaternions and with the octonions of the Construction of Cayley-Dickson.

Hyperbolic quaternions

The hyperbolic quaternions resulting from the program of the hypernombres are isomorphous with the Coquaternion S (split quaternions). They are different from the hyperbolic quaternions of A. MacFarlane (first mention in 1891), they are not associative but they offer like them a multiplicative module.

In the hypernombres, the hyperbolic quaternions consist of a real base , an imaginary base and two against-imaginary bases (e.g. {$1,\; \backslash \; varepsilon\; \{\}\; \_1,\; \backslash \; varepsilon\; \{\}\; \_2,\; i\_3$}, or {$1,\; i\_1,\; \backslash \; varepsilon\; \{\}\; \_2,\; \backslash \; varepsilon\; \{\}\; \_3$}; to also see Complex number split). Like the Quaternion S (circulars), their multiplication is associative but not commutative, the three bases not-real is mutually anti-commutative.

Conical quaternions

The conical quaternions are built on the basis {$1,\; I,\; \backslash \; varepsilon,\; i\_0$}, with $i\_0\; =\; \backslash \; varepsilon\; i$ and form a closed Arithmétique commutative, associative, distributive (containing roots and logarithms), with a multiplicative module. They contain elements Idempotent S and dividing of zero, but not of elements Nilpotent S. the conical quaternions are isomorphous with the Tessarine S and also with the numbers bicomplexes (resulting from the numbers multicomplexes).

By contrast, the quaternions and coquaternions them (circular) (split quaternions) are not commutative (coquaternions them contain also elements Nilpotent S).

The conical quaternions are necessary to describe the orbit of power of $\backslash \; varepsilon\; \backslash ,$ (above) and also the logarithm of $\backslash \; varepsilon\; \backslash ,$:

$\backslash \; ln\; \backslash \; varepsilon\; =\; \backslash \; frac\; \{\backslash \; pi\}\; \{2\}\; (i\_0\; -\; I)$

Octonions hyperbolic

The octonions hyperbolic are isomorphous with the algebra of the octonions split. They consist of a real base , three imaginary bases ($\backslash \; sqrt\; \{-\; 1\}\; \backslash ,$) and four bases against-imaginary ($\backslash \; varepsilon\; \backslash ,$, $\backslash \; sqrt\; \{1\}\; \backslash \; 1\; \backslash ,$).

This algebra was used in physics in the Théorie of the cords. It can be also used to describe the equation of Dirac in physics on a natural system of numbers (in the place of the algebra of the matrices on the complex numbers; to see the references below).

Octonions conical

Octonions conical bases $\backslash \; \{1,\; i\_1,\; i\_2,\; i\_3,\; ~i\_0,\; \backslash \; varepsilon\; \{\}\; \_1,\; \backslash \; varepsilon\; \{\}\; \_2,\; \backslash \; varepsilon\; \{\}\; \_3\; \backslash \}$ form a system of associative but not-commutative number octononic.

The subalgebra $\backslash \; \{1,\; ~i\_0\; \backslash \}$ is isomorphous with the complex numbers, with $i\_0\; =\; i\_n\; \backslash \; varepsilon\_n\; \backslash ,$ (for any N = 1,2,3), is commutative and associative at all the bases of subalgebra of quaternion $\backslash \; \{1,\; ~i\_1,\; i\_2,\; i\_3\; \backslash \}$. The bases of octonions conical can consequently be written in the form $\backslash \; \{1,\; i\_1,\; i\_2,\; i\_3,\; ~i\_0,\; -\; i\_0\; i\_1,\; -\; i\_0\; i\_2,\; -\; i\_0\; i\_3\; \backslash \}$ showing their isomorphism with the quaternions with the coefficients of complex numbers to form the Biquaternion S.

Sédénions conical

A particular case of arithmetic of the hypernombres are the sédénions conical, which form a modular algebra (i.e with a multiplicative module), alternate, flexible device, not commutative; by the Construction of Cayley-Dickson, one obtains octonions them (circular) (built on a real axis and 7 secondary axes), this forms the normalized and modular algebra broadest.

Elliptic complex numbers (arithmetic W )

In the plan (real, W ), the orbit of power $~w^\; \backslash \; alpha\; \backslash ,$ (with $~\; \backslash \; real\; alpha\; \backslash ,$) is elliptic, and the arithmetic one, is consequently also called complex elliptic or numbers W - complex. They are placed in the 4th level of the program of the hypernombres.

The powers of W are cyclic, with $w^0\; =\; w^6\; =\; 1\; \backslash ,$ and the following whole powers:

$w^1\; =\; ~w$

$w^2\; =\; ~-1\; +\; w$

$w^3\; =\; ~-1$

$w^4\; =\; ~-w$

$w^5\; =\; ~1\; -\; w$

They offer a multiplicative module:

$||+\; bw\; has||\; =\; \backslash \; sqrt\; \{a^2\; +\; ab\; +\; b^2\}\; \backslash ,$

If has and B is real coefficients , arithmetic the are a body. Nevertheless, the dual base of number of (W) is (- W) , which different from is combined (W) , which is 1 (W) . This contrasts for example with the imaginary base $i\; =\; \backslash \; sqrt\; \{-\; 1\}\; \backslash ,$, for which the dual one and combined are the same one (- I ). The arithmetic resultant (- W ) is consequently distinct from the arithmetic one - ( W ), while they coexist on the same plan of numbers. Additivement, - ( W ) and (- W ) are identical, but multiplicativement, they are distinct.

Orbits of power

The orbit of power of W is:

$w\; ^\; \backslash \; alpha\; =\; \backslash \; frac\; \{2\}\; \{\backslash \; sqrt\; \{3\}\}\; (\backslash \; cos\; ~\; (\backslash \; frac\; \{\backslash \; alpha\; \backslash \; pi\}\; \{3\}\; +\; \backslash \; frac\; \{\backslash \; pi\}\; \{6\})\; +\; W\; ~\; \backslash \; sin\; ~\; \backslash \; frac\; \{\backslash \; alpha\; \backslash \; pi\}\; \{3\})$

Exponential orbit

The exponential orbit of W is ( has , B real):

$e\; ^\; \{has\; +\; bw\}\; =\; E\; ^\; \{has\; +\; \backslash \; frac\; \{B\}\; \{2\}\}\; W\; ^\; \{\backslash \; frac\; \{3\; \backslash \; sqrt\; \{3\}\; B\}\; \{2\; \backslash \; pi\}\}$

For the particular case of $b\; =\; ~-2a\; \backslash ,$, the orbit of power and the exponential orbit fail both. This gives:

$e\; ^\; \{has\; (-\; 1\; +\; 2w)\}\; =\; W\; ^\; \{\backslash \; frac\; \{3\; \backslash \; sqrt\; \{3\}\; has\}\; \{\backslash \; pi\}\}$

and

$w\; ^\; \backslash \; alpha\; =\; E\; ^\; \{-\; \backslash \; frac\; \{\backslash \; alpha\; \backslash \; pi\}\; \{3\; \backslash \; sqrt\; \{3\}\}\; (1\; -\; 2w)\}$

Numbers of pink (numbers p and Q )

The numbers of pink are placed in the 5th level in the program of the hypernombres and form quasi a dual system. Each one being Nilpotent, the arithmetic offer still a multiplicative module, a argument and a polar form. Geometrically, the powers $~p^\; \backslash \; alpha\; \backslash ,$ and $~q^\; \backslash \; alpha\; \backslash ,$ are pinks with two petals.

The whole powers are:

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

$p^1\; =~p$

$q^1\; =~q$

$p^3\; =~q$

$q^3\; =~p$

They offer a multiplicative module:

$||ap\; +\; bq||\; =\; \backslash \; frac\; \{(a^2\; +\; b^2)\; ^2\}\; \{(has\; +\; b)\; (has\; -\; b)^2\}$

Orbits of power

In the plan { p , Q }, $~p^\; \backslash \; alpha\; \backslash ,$ and $~q^\; \backslash \; alpha\; \backslash ,$ (with $~\; \backslash \; alpha$ real) are connected on a pink with two petals, described by $ap\; +~bq\; \backslash ,$ with

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

While the product $(ap\; +~bq)\; (CP\; +~dq)\; =\; 0$ for any reality { has , B , C , D }, the orbit of power and its geomery connected makes it possible to carry out a not-commonplace multiplication. The factors can be represented in the same orbit of power, e.g.

$p^\; \backslash \; alpha:\; =~\; \backslash \; frac\; \{ap\; +\; bq\}$

$p^\; \backslash \; beta:\; =~\; \backslash \; frac\; \{CP\; +\; dq\}$

and multiplied later with $||N||~p^\; \backslash \; alpha\; p^\; \backslash \; beta\; =||N||~p^\; \{(\backslash \; alpha\; +\; \backslash \; beta)\}$ (with $||N||\; :\; =~||ap\; +\; bq||*||CP\; +\; dq||$ absolute; $~\; \backslash \; alpha$, $~\; \backslash \; beta$ real). Consequently, the multiplication is sensitive to the representation of the point in the plan { p , Q }.

In general, there exists an infinite quantity of representation possible of $||m||~\; (p^\; \backslash \; gamma\; +\; q^\; \backslash \; delta)$ (with $~||m||$ absolute; $~\; \backslash \; gamma$, $~\; \backslash \; delta$ real) for all $ap\; +~bq$ given. But only the multiplication along is orbit of power of p or of Q allows a multiplicative module.

Exponential orbit

The respective orbits exponential are:

$e^\; \{\backslash \; alpha\; p\}\; =\; 1\; +\; \backslash \; frac\; \{p\}\; \{2\}\; (\backslash \; sinh\; \backslash \; alpha\; -\; \backslash \; sin\; \backslash \; alpha)\; +\; \backslash \; frac\; \{Q\}\; \{2\}\; (\backslash \; sinh\; \backslash \; alpha\; +\; \backslash \; sin\; \backslash \; alpha)$

$e^\; \{\backslash \; alpha\; Q\}\; =\; 1\; +\; \backslash \; frac\; \{p\}\; \{2\}\; (\backslash \; sinh\; \backslash \; alpha\; +\; \backslash \; sin\; \backslash \; alpha)\; +\; \backslash \; frac\; \{Q\}\; \{2\}\; (\backslash \; sinh\; \backslash \; alpha\; -\; \backslash \; sin\; \backslash \; alpha)$

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

From C. Muses, Computing in the bio-sciences with hypernumbers: With survey (see the reference supplements below):

" … Notez that - p is generated via W , as follows: $(qw)\; ^3\; =\; (wq)\; ^3\; =\; (w^3)\; (q^3)\; =\; (-\; 1)\; p\; =~-p$. It must be recalled that because p east nilpotent ($p^2\; =\; 0,\; p\; \backslash \; 0$), its power zéro-ième cannot be 1; in fact $p^0\; =~0$. Consequently $p^\; \{-\; 1\}\; \backslash \; 1/p$, and since $(1/p)\; (1/p)\; =\; 1/p^2\; =\; \backslash \; infty$, we see that $~1/p$ east panpotent, i.e a root of the infinite one. To compare $1/(1\; \backslash \; pm\; \backslash \; varepsilon)$, which is a pair of dividers of the infini."

Cassinioïdes numbers (numbers m )

The arithmetic one of the numbers cassinioïdes, the 6th level of the program of the hypernombres, is controlled by the cassinoïdes or the ovals of Cassini. Their relationship to the geometry illustrates the multiplicative multiplication and their modules. The coefficients of the base of number m are absolute numbers, which are similar to the positive real numbers; nevertheless, arithmeric the m is sensitive to the size of its coefficients.

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

$m^2\; =~m$

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

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

Characteristic, module and handle

For a number $a\; +~bm$, the " caractéristique" S is defined as follows:

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

A multiplicative module T and a handle K are then defined through:

$t\; =\; |+\; bm\; has|\; =\; \backslash \; sqrt\; \{\; |\; s^2\; -\; 1\; |\; \}$

$k\; =\; \backslash \; sqrt\; \{s^2\; +\; 1\}$

Distinction between coefficients and real numbers

By quoting K. Carmody, " Cassinoid Numbers: The Musèan Hypernumber m " (April 27th, 2006, on http://www.kevincarmody.com/math/hypernumbers.html):

“The coefficients such as $\backslash \; sqrt\; \{2\}$ in the expression $\backslash \; sqrt\; \{2\}\; m\; \backslash ,$ are not truly real numbers. For example, if we multiply -1 like a real number by $+m$, we can obtain $+m\; \backslash ,$, but we cannot obtain $-m\; \backslash ,$.

In a correct way, +1, -1, + m and - m is units, and the coefficients of their multiples along their respective axes are absolute numbers , which are distinct from the real numbers and which are never negative. ”

See too

• hyperbolic Biquaternion

• Quaternion (by A. MacFarlane)
• Number hypercomplexe
• Elements Nilpotent S
• Octonion
• Quaternion
• Sédénion
• Complex number split
• Octonion split
• Quaternion split/Coquaternion
• Tessarine S
• Dividing of zero

References on the hypernombres and publications

General

For a bibliography supplements on the hypernombre and of information on the hypernombres in general , to see http://www.kevincarmody.com/math/hypernumbers.html. Majority of the contents above at summer obtained starting from this page (in English).

Selections of articles:

• C. Muses, “Applied hypernumbers: Computational concepts”, in Appl. Maths. Comput. , 3 (1977) 211-226.

• C. Muses, “Hypernumbers II-Further concepts and computational applications”, in Appl. Maths. Comput. , 4 (1978) 45-66.
• C. Muses, “Computing in the bio-sciences with hypernumbers: survey has”, in Intl. J. Bio-Med. Comput. , 10 (1979) 519-525.
• K. Carmody, “Circular and hyperbolic quaternions, octonions, and sedenions”, in Appl. Maths. Comput. 28 (1988) 47-72.
• C. Muses, “Hypernumbers applied, but how they interface with the physical world”, in Appl. Maths. Comput. 60 (1994) 25-36.
• K. Carmody, “Circular and hyperbolic quaternions, octonions, and sedenions- further results”, in Appl. Maths. Comput. 84 (1997) 27-48.

Applications

For a true use of the hypernombres and isomorphous systems in physics , to see for example.

• C. Muses, “Hypernumbers and quantum field theory with has summary off physically applicable hypernumbers and to their geometries”, in Appl. Maths. Comput. 6 (1980) 63-94.

• C. Muses, “Hypernumber Applied gold How They Interfaces with the Physical World”, in Appl. Maths. Computation 60 (1994) 25-36.
• Mr. Gogberashvili, “Octonionic Electrodynamics”, in J. Phys. a: Maths. Gen. 39 (2006) 7099-7104.
• J. Köplinger, “Dirac equation one hyperbolic octonions”, in Appl. Maths. Computation (2006).

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