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.

Contents

[edit] Isomorphisms to other number systems

[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 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, ..., ~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 2x2 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.
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