# Musean hypernumber

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 was not helped by Musès' outspoken confidence in applicability to 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" 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 with 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 defining relations for most of these levels have not been constructed.

Higher-dimensional numbers built on the first three levels were called "M-algebra"[6][7] by Musès if they yielded a distributive multiplication, unit element, and multiplicative norm. It contains kinds of octonions and historical quaternions (except A. MacFarlane's hyperbolic quaternions) as subalgebras. A proof of completeness of M-algebra has not been provided.

## Conic sedenions / "16 dimensional M-algebra"

The term "M-algebra" (after C. Musès[6]) refers to number systems that are vector spaces over the reals, whose bases consist in 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), certain results from 16 dimensional (conic) sedenions were a novelty. Musès demonstrated the existence of a logarithm and real powers in number systems built to non-real roots of +1.

### Multiplication table

The conic sedenions[8][9] form an algebra with a 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[10] axes (to bases $\varepsilon$ with $\varepsilon{}_n^2=+1$).

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] In fact, conic sedenions are isomorphic to complex octonions, i.e. octonions with complex number coefficients. 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 $\varepsilon{}_n$ ($n = 1, \ldots, 7$) not only has an exponential function[11]

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

($\alpha$ real) but also possesses real powers:[8][12]

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

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.[8] Their multiplicative modulus $|z|$ is[9]

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

### List of number types[8] 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 $i_n$ only.

#### 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 Alexander Macfarlane's hyperbolic quaternions (first mention in 1891), which are not associative.

#### Conic quaternions

Conic quaternions are built on bases {$1, i, \varepsilon, i_0$} and form a commutative, associative, and distributive arithmetic. They contain non-trivial 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.

#### 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. {$1, i_1, i_2, i_3, \varepsilon{}_4 , \varepsilon{}_5, \varepsilon{}_6 , \varepsilon{}_7$}.

#### 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.