abstract algebra In mathematics, more specifically algebra, abstract algebra or modern algebra is the study of algebraic structures, which are set (mathematics), sets with specific operation (mathematics), operations acting on their elements. Algebraic structur ...

, a bicomplex number is a pair of

complex number In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the for ...

s constructed by the Cayley–Dickson process that defines the bicomplex conjugate

(w,z)^* = (w, -z)

, and the product of two bicomplex numbers as :

(u,v)(w,z) = (u w - v z, u z + v w).

Then the bicomplex norm is given by :

(w,z)^* (w,z) = (w, -z)(w,z) = (w^2 + z^2, 0),

quadratic form In mathematics, a quadratic form is a polynomial with terms all of degree two (" form" is another name for a homogeneous polynomial). For example, 4x^2 + 2xy - 3y^2 is a quadratic form in the variables and . The coefficients usually belong t ...

in the first component. The bicomplex numbers form a commutative algebra over C of dimension two that is

isomorphic In mathematics, an isomorphism is a structure-preserving mapping or morphism between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between the ...

to the direct sum of algebras . The product of two bicomplex numbers yields a quadratic form value that is the product of the individual quadratic forms of the numbers: a verification of this property of the quadratic form of a product refers to the Brahmagupta–Fibonacci identity. This property of the quadratic form of a bicomplex number indicates that these numbers form a

composition algebra In mathematics, a composition algebra over a field is a not necessarily associative algebra over together with a nondegenerate quadratic form that satisfies :N(xy) = N(x)N(y) for all and in . A composition algebra includes an involution ...

. In fact, bicomplex numbers arise at the binarion level of the Cayley–Dickson construction based on

\mathbb

with norm z². The general bicomplex number can be represented by the matrix

\beginw & iz \\ iz & w \end

, which has

determinant In mathematics, the determinant is a Scalar (mathematics), scalar-valued function (mathematics), function of the entries of a square matrix. The determinant of a matrix is commonly denoted , , or . Its value characterizes some properties of the ...

w^2 + z^2

. Thus, the composing property of the quadratic form concurs with the composing property of the determinant. Bicomplex numbers feature two distinct

imaginary unit The imaginary unit or unit imaginary number () is a mathematical constant that is a solution to the quadratic equation Although there is no real number with this property, can be used to extend the real numbers to what are called complex num ...

s. Multiplication being associative and commutative, the product of these imaginary units must have positive one for its square. Such an element as this product has been called a

hyperbolic unit In algebra, a split-complex number (or hyperbolic number, also perplex number, double number) is based on a hyperbolic unit satisfying j^2=1, where j \neq \pm 1. A split-complex number has two real number components and , and is written z=x+yj ...

As a real algebra

Bicomplex numbers form an algebra over C of dimension two, and since C is of dimension two over R, the bicomplex numbers are an algebra over R of dimension four. In fact the real algebra is older than the complex one; it was labelled ''tessarines'' in 1848 while the complex algebra was not introduced until 1892. A basis for the tessarine 4-algebra over R uses the following

units Unit may refer to: General measurement * Unit of measurement, a definite magnitude of a physical quantity, defined and adopted by convention or by law **International System of Units (SI), modern form of the metric system **English units, histo ...

(with matrix representations given): the

multiplicative identity In mathematics, an identity element or neutral element of a binary operation is an element that leaves unchanged every element when the operation is applied. For example, 0 is an identity element of the addition of real numbers. This concept is use ...

1 = \begin 1 & 0 \\ 0 & 1 \end

, the same

i = \begin i & 0 \\ 0 & i \end

as in the

s, the same

j = \begin 0 & 1 \\ 1 & 0 \end

as in the

split-complex number In algebra, a split-complex number (or hyperbolic number, also perplex number, double number) is based on a hyperbolic unit satisfying j^2=1, where j \neq \pm 1. A split-complex number has two real number components and , and is written z=x+y ...

s and a second imaginary unit

k = i j = j i = \begin 0 & i \\ i & 0 \end

, which multiply according to the table given.

History

The subject of multiple

s was examined in the 1840s. In a long series "On quaternions, or on a new system of imaginaries in algebra" beginning in 1844 in

Philosophical Magazine The ''Philosophical Magazine'' is one of the oldest scientific journals published in English. It was established by Alexander Tilloch in 1798;John Burnett"Tilloch, Alexander (1759–1825)" Dictionary of National Biography#Oxford Dictionary of ...

William Rowan Hamilton Sir William Rowan Hamilton (4 August 1805 – 2 September 1865) was an Irish astronomer, mathematician, and physicist who made numerous major contributions to abstract algebra, classical mechanics, and optics. His theoretical works and mathema ...

communicated a system multiplying according to the

quaternion group In group theory, the quaternion group Q8 (sometimes just denoted by Q) is a nonabelian group, non-abelian group (mathematics), group of Group order, order eight, isomorphic to the eight-element subset \ of the quaternions under multiplication. ...

. In 1848

Thomas Kirkman Thomas Penyngton Kirkman FRS (31 March 1806 – 3 February 1895) was a British mathematician and ordained minister of the Church of England. Despite being primarily a churchman, he maintained an active interest in research-level mathematics, a ...

reported on his correspondence with

Arthur Cayley Arthur Cayley (; 16 August 1821 – 26 January 1895) was a British mathematician who worked mostly on algebra. He helped found the modern British school of pure mathematics, and was a professor at Trinity College, Cambridge for 35 years. He ...

regarding equations on the units determining a system of hypercomplex numbers.

Tessarines

In 1848 James Cockle introduced the tessarines in a series of articles in ''Philosophical Magazine''. A tessarine is a hypercomplex number of the form :

t = w + x i + y j + z k, \quad w, x, y, z \in \mathbb

where

i j = j i = k, \quad i^2 = -1, \quad j^2 = +1 .

Cockle used tessarines to isolate the hyperbolic cosine series and the hyperbolic sine series in the exponential series. He also showed how

zero divisor In abstract algebra, an element of a ring is called a left zero divisor if there exists a nonzero in such that , or equivalently if the map from to that sends to is not injective. Similarly, an element of a ring is called a right ze ...

s arise in tessarines, inspiring him to use the term "impossibles". The tessarines are now best known for their subalgebra of real tessarines

t = w + y j \

, also called

s, which express the parametrization of the

unit hyperbola In geometry, the unit hyperbola is the set of points (''x'',''y'') in the Cartesian plane that satisfy the implicit equation x^2 - y^2 = 1 . In the study of indefinite orthogonal groups, the unit hyperbola forms the basis for an ''alternative rad ...

Bicomplex numbers

In an 1892 ''

Mathematische Annalen ''Mathematische Annalen'' (abbreviated as ''Math. Ann.'' or, formerly, ''Math. Annal.'') is a German mathematical research journal founded in 1868 by Alfred Clebsch and Carl Neumann. Subsequent managing editors were Felix Klein, David Hilbert, ...

'' paper, Corrado Segre introduced bicomplex numbers, which form an algebra isomorphic to the tessarines. Segre read W. R. Hamilton's ''Lectures on Quaternions'' (1853) and the works of W. K. Clifford. Segre used some of Hamilton's notation to develop his system of bicomplex numbers: Let ''h'' and ''i'' be elements that square to −1 and that commute. Then, presuming

associativity In mathematics, the associative property is a property of some binary operations that rearranging the parentheses in an expression will not change the result. In propositional logic, associativity is a Validity (logic), valid rule of replaceme ...

of multiplication, the product ''hi'' must square to +1. The algebra constructed on the basis is then the same as James Cockle's tessarines, represented using a different basis. Segre noted that elements :

g = (1 - hi)/2, \quad g' = (1 + hi)/2

are

idempotent Idempotence (, ) is the property of certain operations in mathematics and computer science whereby they can be applied multiple times without changing the result beyond the initial application. The concept of idempotence arises in a number of pl ...

s. When bicomplex numbers are expressed in terms of the basis , their equivalence with tessarines is apparent, particularly if the vectors in this basis are reordered as . Looking at the linear representation of these

algebras shows agreement in the fourth dimension when the negative sign is used; consider the sample product given above under linear representation.

Bibinarions

The modern theory of

s positions the algebra as a binarion construction based on another binarion construction, hence the bibinarions. The unarion level in the Cayley-Dickson process must be a field, and starting with the real field, the usual complex numbers arises as division binarions, another field. Thus the process can begin again to form bibinarions. Kevin McCrimmon noted the simplification of nomenclature provided by the term ''binarion'' in his text ''A Taste of Jordan Algebras'' (2004).

Polynomial roots

Write and represent elements of it by ordered pairs (''u'',''v'') of complex numbers. Since the algebra of tessarines T is isomorphic to ²C, the rings of polynomials T and ²C 'X''are also isomorphic, however polynomials in the latter algebra split: :

\sum_^n (a_k, b_k ) (u, v)^k \quad = \quad \left(,\quad  \sum_^n b_k v^k \right).

In consequence, when a polynomial equation

f(u,v) = (0,0)

in this algebra is set, it reduces to two polynomial equations on C. If the degree is ''n'', then there are ''n''

roots A root is the part of a plant, generally underground, that anchors the plant body, and absorbs and stores water and nutrients. Root or roots may also refer to: Art, entertainment, and media * ''The Root'' (magazine), an online magazine focusin ...

for each equation:

u_1, u_2, \dots, u_n,\ v_1, v_2, \dots, v_n .

Any ordered pair

( u_i, v_j ) \!

from this set of roots will satisfy the original equation in ²C 'X'' so it has ''n''² roots. Due to the isomorphism with T 'X'' there is a correspondence of polynomials and a correspondence of their roots. Hence the tessarine polynomials of degree ''n'' also have ''n''² roots, counting multiplicity of roots.

Applications

Bicomplex number appears as the center of CAPS (complexified

algebra of physical space Algebra is a branch of mathematics that deals with abstract systems, known as algebraic structures, and the manipulation of expressions within those systems. It is a generalization of arithmetic that introduces variables and algebraic ope ...

), which is Clifford algebra

Cl(3,\mathbb)

. Since the linear space of CAPS can be viewed as the four dimensional space span over . Tessarines have been applied in

digital signal processing Digital signal processing (DSP) is the use of digital processing, such as by computers or more specialized digital signal processors, to perform a wide variety of signal processing operations. The digital signals processed in this manner are a ...

. Bicomplex numbers are employed in fluid mechanics. The use of bicomplex algebra reconciles two distinct applications of complex numbers: the representation of two-dimensional potential flows in the complex plane and the complex exponential function.