Hypernumber

Hypernumbers are numbers with associated dimensionalities, discovered by Dr. Charles A. Musès (1919–2000). Hypernumbers form a complete, integrated, connected, and natural system. There are ten levels of hypernumbers, each with its own arithmetic and geometry.

For detailed information on hypernumbers see e.g. http://www.kevincarmody.com/math/hypernumbers.html, from which most of the content of this page was obtained.

Select hypernumber types

Traditional real and complex numbers

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

Epsilon numbers

With $\varepsilon$ numbers, the hypernumbers program is able to define a wide range of mathematical operations on number systems that contain bases with are non-real square roots of 1, i.e. $\varepsilon ^{2}=1$ but $\varepsilon \neq 1$ (see also Split-complex number). The power orbit concept in hypernumbers allows for powers, roots, and logarithms on number systems that contain $\varepsilon$ bases.

Epsilon numbers are assigned the 3rd level in the hypernumbers program.

Exponential vs. power orbit

Whereas for complex numbers the power orbit $~i^{\alpha }$ falls together with the exponential orbit $\exp ~i\alpha$ of the imaginary base number, this is not the case anymore for $~\varepsilon$ numbers. Instead we have for the exponential orbit:

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

The power orbit is:

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

Please note that the power orbit contains a product of $~\varepsilon$ and $~i$, which requires conic quaternion arithmetic (below).

Example: Circular quaternions and octonions

Circular quaternions and octonions from the hypernumbers program are identical to traditional quaternions and octonions.

Example: conic quaternions

Conic quaternions are built on bases {$1,i,\varepsilon ,i_{0}$} and form a commutative, associative, distributive, closed arithmetic (containing roots and logarithms), with multiplicative modulus.

In comparison, traditional (circular) quaternions and split-quaternions (coquaternions) are not commutative. Split-quaternions also contain zero divisors and nilpotents.

Conic quaternions are needed to describe the power orbit of $\varepsilon$ (above), and also the logarithm of $\varepsilon$:

$\ln \varepsilon ={\frac {\pi }{2}}(i_{0}-i)$

Example: Hyperbolic quaternions

Hyperbolic quaternions from the hypernumbers program are isomorph to split-quaternions / coquaternions. They are different from A. MacFarlane's hyperbolic quaternions (first mention in 1891), which are not associative.

In hypernumbers, hyperbolic quaternions consist of one real, one imaginary, and two counterimaginary bases (e.g. {$1,\varepsilon {}_{1},\varepsilon {}_{2},i_{3}$}, or {$1,i_{1},\varepsilon {}_{2},\varepsilon {}_{3}$}; see also split-complex number). Like traditional (circular) quaternions, their multiplication is associative but not commutative, the three non-real bases are mutually anti-commutative.

Example: Hyperbolic octonions

Hyperbolic octonions are computationally identical to split-octonion algebra. They consist of one real, three imaginary (${\sqrt {-1}}$), and four counterimaginary ($\varepsilon$, ${\sqrt {1}}\neq 1$) bases.

This algebra has been used in physics e.g. in String theory. Also, it can be used to describe the Dirac equation in physics on a native non-traditional number system (see references below).

Example: Conic sedenions

A special case of hypernumber arithmetic are conic sedenions, which form a 16 dimensional modular (i.e. with multiplicative modulus), alternative, flexible, non-commutative, non-associative algebra; whereas traditional Cayley-Dickson construction yields 8 dimensional (circular) octonions (built on one real and 7 imaginary axes) as widest modular and normed algebra.

Elliptic complex numbers (w arithmetic)

In the (real, w) plane, the power orbit $~w^{\alpha }$ (with $~\alpha$ real) is elliptic, and the arithmetic is therefore also called elliptic complex or w-complex numbers. They are assigned the 4th level in the hypernumbers program.

The powers of w are cyclic, 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. Additively, -(w) and (-w) are identical, but multiplicatively they are distinct.

Power orbit

The power orbit of w is:

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

File:HypernumbersPowerOrbitW.gif

Exponential orbit

The exponential orbit of w is (a, b real):

$e^{a+bw}=e^{a+{\frac {b}{2}}}w^{\frac {3{\sqrt {3}}b}{2\pi }}$

For the special case of $b=~-2a$ the power orbit and exponential orbit fall together. This yields:

$e^{a(-1+2w)}=w^{\frac {3{\sqrt {3}}a}{\pi }}$

and

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

Rose numbers (p and q numbers)

Rose numbers are assigned the 5th level in the hypernumbers program, and form a nearly dual system. Each being nilpotent, the arithmetic yet offers a multiplicative modulus, an argument, and a polar form. Geometrically, the powers $~p^{\alpha }$ and $~q^{\alpha }$ are two-leaved roses.

Integral 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 modulus:

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

Power orbit

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

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

File:HypernumbersPowerOrbitPQ.gif

While the product $(ap+~bq)(cp+~dq)=0$ for any real {a, b, c, d}, the power orbit and its related geometry allow to carry out non-trivial multiplication. The factors can be represented to the same power orbit, e.g.

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

$p^{\beta }:=~{\frac {cp+dq}{||cp+dq||}}$

and multiplied subsequently to $||n||~p^{\alpha }p^{\beta }=||n||~p^{(\alpha +\beta )}$ (with $||n||:=~||ap+bq||*||cp+dq||$ absolute; $~\alpha$, $~\beta$ real). Therefore, multiplication is sensitive to the representation of the point in the {p, q} plane. In general, there is an infinite amount of possible representations $||m||~(p^{\gamma }+q^{\delta })$ (with $~||m||$ absolute; $~\gamma$, $~\delta$ real) for any given $ap+~bq$.

Exponential orbit

The respective exponential orbits are:

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

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

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

From C. Musès, Computing in the bio-sciences with hypernumbers: A survey (see full reference below):

"...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 ($p^{2}=0,p\neq 0$), its zeroth power cannot be 1; in fact $p^{0}=~0$. Hence also $p^{-1}\neq 1/p$, 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."

Cassinoid numbers (m numbers)

Cassiniod number arithmetic, the 6th level in the hypernumbers program, is governed by cassinoids or Cassinian ovals. Their relation to geometry illustrates multiplication and their multiplicative modulus. Coefficients to the m number base are absolute numbers, which are similar to positive real numbers; however, m arithmetic is sensitive to the magnitude of its coefficients, therefore departing from traditional concepts.

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

$m^{2}=~m$

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

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

Characteristic, modulus, and handle

For a number $a+~bm$ the "characteristic" s is defined as:

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

A multiplicative modulus t and a handle k are then defined through:

$t=|a+bm|={\sqrt {|s^{2}-1|}}$

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

File:HypernumbersPowerOrbitM.gif

Distinction between coefficients and real numbers

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

"Coefficients such as ${\sqrt {2}}$ in the expression ${\sqrt {2}}m$ are not actually real numbers. For example, if we multiply -1 as a real number by $+m$, we can get $+m$, but we cannot get $-m$. [...]

Properly, +1, -1, +m, and -m are units, and the coefficients of their multiples along their respective axes are absolute numbers, which are distinct from real numbers and are never negative."

See also

• A. MacFarlane's Hyperbolic quaternion
• Hypercomplex number
• Nilpotents
• Octonion
• Quaternion
• Sedenion
• Split-complex number
• Split-quaternion / Coquaternion
• Split-octonions
• Zero divisor

Hypernumber publications

General

For a complete reference to hypernumber publications, and information on hypernumbers in general, see http://www.kevincarmody.com/math/hypernumbers.html . Most of the content above was obtained from this page.

Selection of journal articles:

• C. Musès, Applied hypernumbers: Computational concepts, Appl. Math. Comput. 3 (1977) 211–226. doi:10.1016/0096-3003(77)90002-9
• C. Musès, Hypernumbers II—Further concepts and computational applications, Appl. Math. Comput. 4 (1978) 45–66. doi:10.1016/0096-3003(78)90026-7
• C. Musès, Computing in the bio-sciences with hypernumbers: a survey, Intl. J. Bio-Med. Comput. 10 (1979) 519–525.
• K. Carmody, Circular and hyperbolic quaternions, octonions, and sedenions, Appl. Math. Comput. 28 (1988) 47–72. doi:10.1016/0096-3003(88)90133-6
• C. Musès, Hypernumbers applied, or how they interface with the physical world, Appl. Math. Comput. 60 (1994) 25–36. doi:10.1016/0096-3003(94)90203-8
• K. Carmody, Circular and hyperbolic quaternions, octonions, and sedenions— further results, Appl. Math. Comput. 84 (1997) 27–48. doi:10.1016/S0096-3003(96)00051-3

Applied

For genuine use of hypernumbers (and equivalent non-traditional numbers) in physics see e.g.

• C. Musès, Hypernumbers and quantum field theory with a summary of physically applicable hypernumbers and their geometries, Appl. Math. Comput. 6 (1980) 63–94. doi:10.1016/0096-3003(80)90016-8
• C. Musès, Hypernumber Applied or How They Interface with the Physical World. Appl. Math. Computation 60 (1994) 25-36. doi:10.1016/0096-3003(94)90203-8
• M. Gogberashvili, Octonionic Electrodynamics, J. Phys. A: Math. Gen. 39 (2006) 7099-7104. doi:10.1088/0305-4470/39/22/020
• J. Köplinger, Dirac equation on hyperbolic octonions. Appl. Math. Computation (2006), doi:10.1016/j.amc.2006.04.005