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 .
When z = 0, then t amounts to an ordinary complex number, which is w itself.
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 : 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
(a, b, c, d real) then t algebra is isomorphic to conic quaternions , to bases , in the following identification:
They are also isomorphic to bicomplex numbers (from multicomplex numbers) to bases 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 ), then t algebra is isomorphic to conic octonions; allowing octonions for w and z (to bases ) the resulting algebra is identical to conic sedenions.
Select 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 , 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
equiv z^2 (1 + j )(1 - j)
equiv z^2 (1 + varepsilon )(1 - varepsilon) = 0.
In contrast, the quaternions form a skew field without zero-divisors, and can also be represented in 2×2 matrix form.
- 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.