Tessarine

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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

$\begin{pmatrix} w & z \\ z & w\end{pmatrix},$

where w and z can be any complex number.

1 Isomorphisms to other number systems
2 References

[edit] Isomorphisms to other number systems

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

[edit] Complex number

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

[edit] 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

$j = \begin{pmatrix} 0 & 1 \\ 1 & 0\end{pmatrix}$

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:

[edit] 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:

$1 \equiv \begin{pmatrix} 1 & 0 \\ 0 & 1\end{pmatrix} \qquad i \equiv \begin{pmatrix} i & 0 \\ 0 & i\end{pmatrix} \qquad \varepsilon \equiv \begin{pmatrix} 0 & 1 \\ 1 & 0\end{pmatrix} \qquad i_0 \equiv \begin{pmatrix} 0 & i \\ i & 0\end{pmatrix}$

They are also isomorphic to bicomplex numbers (from multicomplex numbers) to bases $\{ 1,~i_1, i_2, j \}$ if one identifies:

$1 \equiv \begin{pmatrix} 1 & 0 \\ 0 & 1\end{pmatrix} \qquad i_1 \equiv \begin{pmatrix} i & 0 \\ 0 & i\end{pmatrix} \qquad i_2 \equiv \begin{pmatrix} 0 & i \\ i & 0\end{pmatrix} \qquad j \equiv \begin{pmatrix} 0 & -1 \\ -1 & 0\end{pmatrix}$

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.

[edit] 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

$\begin{pmatrix} z & \pm z \\ \pm z & z \end{pmatrix} \equiv z (1 \pm j) \equiv z (1 \pm \varepsilon)$

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

$\begin{pmatrix} z & z \\ z & z \end{pmatrix} \begin{pmatrix} z & -z \\ -z & z \end{pmatrix} \equiv z^2 (1 + j )(1 - j) \equiv z^2 (1 + \varepsilon )(1 - \varepsilon) = 0.$

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

[edit] 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.

v • d • e Number systems

Basic	Natural numbers $\mathbb{N}$ · Negative numbers · Integers $\mathbb{Z}$ · Rational numbers $\mathbb{Q}$ · Irrational numbers · Real numbers $\mathbb{R}$ · Imaginary numbers $\mathbb{I}$ · Complex numbers $\mathbb{C}$ · Algebraic numbers $\overline{\mathbb{Q}}$ · Transcendental numbers · Quaternions $\mathbb{H}$ · Octonions $\mathbb{O}$ · Sedenions $\mathbb{S}$ · Cayley–Dickson construction · Split-complex numbers $\mathbb{R}^{1,1}$

Complex extensions	Bicomplex numbers · Biquaternions · Split-quaternions · Tessarines · Hypercomplex numbers · Musean hypernumbers · Superreal numbers · Hyperreal numbers · Supernatural numbers · Surreal numbers

Other extensions	Dual numbers · Transfinite numbers · Extended real numbers · Cardinal numbers · Ordinal numbers · p-adic numbers