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# 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

$begin\left\{pmatrix\right\} w & z z & wend\left\{pmatrix\right\},$

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 $mathbf\left\{C\right\} oplus mathbf\left\{C\right\}$.

### 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 $mathbf\left\{R\right\} oplus mathbf\left\{R\right\}$: that is, t amounts to a split-complex number, w + j z. The particular tessarine

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

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 $\left\{ 1,~i,~varepsilon ,~i_0 \right\}$, in the following identification:

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

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

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

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 $\left\{ 1,~i_1,~i_2,~i_3 \right\}$), then t algebra is isomorphic to conic octonions; allowing octonions for w and z (to bases $\left\{ 1,~i_1, dots, ~i_7 \right\}$) 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

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

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

$begin\left\{pmatrix\right\} z & z z & z end\left\{pmatrix\right\} begin\left\{pmatrix\right\} z & -z -z & z end\left\{pmatrix\right\}$
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.

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