Tessarine - Citizendia

The tessarines are a mathematical idea introduced by James Cockle in 1848. Sir James Cockle FRS FRAS FCPS FMS ( 14 January 1819 &ndash 27 January 1895) was an English The concept includes both ordinary complex numbers and split-complex numbers. Complex plane In Mathematics, the complex numbers are an extension of the Real numbers obtained by adjoining an Imaginary unit, denoted In Linear algebra, a split-complex number is of the form z = x + y j where j2 = +1, and x and y are Real 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. In Mathematics, a matrix (plural matrices) is a rectangular table of elements (or entries) which may be Numbers or more generally Complex plane In Mathematics, the complex numbers are an extension of the Real numbers obtained by adjoining an Imaginary unit, denoted

1 Isomorphisms to other number systems
2 References

Isomorphisms to other number systems

Complex number

When z = 0, then t amounts to an ordinary complex number, which is w itself. Complex plane In Mathematics, the complex numbers are an extension of the Real numbers obtained by adjoining an Imaginary unit, denoted

Split-complex number

When w and z are both real numbers, then t amounts to a split-complex number, w + j z. In Mathematics, the real numbers may be described informally in several different ways In Linear algebra, a split-complex number is of the form z = x + y j where j2 = +1, and x and y are Real 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". Geometric interpretation Geometrically imaginary numbers are found on the vertical axis of the complex number plane 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:

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

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. In Mathematics, commutativity is the ability to change the order of something without changing the end result In Mathematics, associativity is a property that a Binary operation can have In Mathematics, a ring is an Algebraic structure which generalizes the algebraic properties of the Integers though the rational, real Complex plane In Mathematics, the complex numbers are an extension of the Real numbers obtained by adjoining an Imaginary unit, denoted Musean hypernumbers are an algebraic concept envisioned by Charles A In Mathematics, a bicomplex number (from the Multicomplex numbers see e In Mathematics, the multicomplex numbers, {\Bbb{MC}}_n form an n dimensional algebra generated by one element e which satisfies ~e^n

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. Quaternions, in Mathematics, are a non-commutative extension of Complex numbers They were first described by the Irish Mathematician Musean hypernumbers are an algebraic concept envisioned by Charles A In Mathematics, the octonions are a nonassociative extension of the Quaternions Their 8-dimensional Normed division algebra over the Real Musean hypernumbers are an algebraic concept envisioned by Charles A

Select algebraic properties

Tessarines with w and z complex numbers form a commutative and associative quaternionic ring (whereas quaternions are not commutative). In Mathematics, commutativity is the ability to change the order of something without changing the end result In Mathematics, associativity is a property that a Binary operation can have In Mathematics, a ring is an Algebraic structure which generalizes the algebraic properties of the Integers though the rational, real Quaternions, in Mathematics, are a non-commutative extension of Complex numbers They were first described by the Irish Mathematician 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). Musean hypernumbers are an algebraic concept envisioned by Charles A 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 Abstract algebra, a field is an Algebraic structure in which the operations of Addition, Subtraction, Multiplication and division Idempotence ˌaɪdɨmˈpoʊtəns describes the property of operations in Mathematics and Computer science which means that multiple applications of the operation In Algebra, a determinant is a function depending on n that associates a scalar, det( A) to every n × n In Mathematics, the absolute value (or modulus) of a Real number is its numerical value without regard to its sign. 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 Abstract algebra, a nonzero element a of a ring is a left zero divisor if there exists a nonzero b such that ab = 0

In contrast, the quaternions form a skew field without zero-divisors, and can also be represented in 2x2 matrix form. Quaternions, in Mathematics, are a non-commutative extension of Complex numbers They were first described by the Irish Mathematician In Abstract algebra, a division ring, also called a skew field, is a ring in which division is possible Quaternions, in Mathematics, are a non-commutative extension of Complex numbers They were first described by the Irish Mathematician

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. The Philosophical Magazine is arguably the world’s oldest commercially published Scientific journal.
- 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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