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.

Split-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.

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 $j\; equiv\; varepsilon$, 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

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.

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.