Tessarine

The tessarines are a mathematical idea introduced by James Cockle in 1848. The concept includes both ordinary complex numbers and split-complex numbers. A tessarine "t" may be described as a 2 × 2 matrix

:

where "w" and "z" can be any complex number.

Isomorphisms to other number systems

In general the tessarines form an algebra of dimension two over the complex numbers, isomorphic to the direct sum $mathbf\{C\}\; oplus\; mathbf\{C\}$.

Complex number

When "z" = 0, then "t" amounts to an ordinary complex number, which is "w" itself.

plit-complex number

When "w" and "z" are both real numbers, then we have an algebra of dimension two over the real numbers, isomorphic to the direct sum $mathbf\{R\}\; oplus\; mathbf\{R\}$: that is, "t" amounts to a split-complex number, "w" + j "z". The particular tessarine

:

has the property that its matrix product square is the identity matrix. This property led Cockle to call the tessarine j a "new imaginary in algebra". The commutative and associative ring of all tessarines also appears in the following forms:

Conic quaternion / octonion / sedenion, bicomplex number

When "w" and "z" are both complex numbers

: $w\; :=~a\; +\; ib$

: $z\; :=~c\; +\; id$

("a", "b", "c", "d" real) then "t" algebra is isomorphic to conic quaternions $a\; +\; bi\; +\; c\; varepsilon\; +\; d\; i\_0$, to bases $\{\; 1,~i,~varepsilon\; ,~i\_0\; \}$, in the following identification:

:

They are also isomorphic to bicomplex numbers (from multicomplex numbers) to bases $\{\; 1,~i\_1,\; i\_2,\; j\; \}$ if one identifies:

:

Note that "j" in bicomplex numbers is identified with the opposite sign as "j" from above.

When "w" and "z" are both quaternions (to bases $\{\; 1,~i\_1,~i\_2,~i\_3\; \}$), then "t" algebra is isomorphic to conic octonions; allowing octonions for "w" and "z" (to bases $\{\; 1,~i\_1,\; dots,\; ~i\_7\; \}$) the resulting algebra is identical to conic sedenions.

elect algebraic properties

Tessarines with "w" and "z" complex numbers form a commutative and associative quaternionic ring (whereas quaternions are not commutative). They allow for powers, roots, and logarithms of $j\; equiv\; varepsilon$, which is a non-real root of 1 (see conic quaternions for examples and references). They do not form a field because the idempotents

:

have determinant / modulus 0 and therefore cannot be inverted multiplicatively. In addition, the arithmetic contains zero divisors

:

In contrast, the quaternions form a skew field without zero-divisors, and can also be represented in 2×2 matrix form.

References

* James Cockle in London-Dublin-Edinburgh Philosophical Magazine, series 3
** 1848 On Certain Functions Resembling Quaternions and on a New Imaginary in Algebra, 33:435–9.
** 1849 On a New Imaginary in Algebra 34:37–47.
** 1849 On the Symbols of Algebra and on the Theory of Tessarines 34:406–10.
** 1850 On Impossible Equations, on Impossible Quantities and on Tessarines 37:281–3.
** 1850 On the True Amplitude of a Tessarine 38:290–2.