In abstract algebra, the splitcomplex numbers (or hyperbolic numbers) are a twodimensional commutative algebra over the real numbers different from the complex numbers. Every splitcomplex number has the form
where x and y are real numbers. The number j is similar to the imaginary unit i, except that
As an algebra over the reals, the splitcomplex numbers are related to the direct sum of algebras R⊕R (under the isomorphism sending x + y j to (x + y, x − y) ). The name split comes from this characterization: as a real algebra, the splitcomplex numbers split into the direct sum R⊕R.
Geometrically, multiplication of splitcomplex numbers preserves the (square) Minkowski norm (x^{2} − y^{2}) in the same way that multiplication of complex numbers preserves the (square) Euclidean norm (x^{2} + y^{2}). Unlike the complex numbers, the splitcomplex numbers contain nontrivial idempotents (other than 0 and 1), as well as zero divisors, and therefore they do not form a field.
The splitcomplex number is one of the concepts necessary to read a 2 × 2 real matrix.
Splitcomplex numbers have many other names; see the synonyms section below.
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A splitcomplex number is one of the form
where x and y are real numbers and the quantity j satisfies
Choosing results in the complex numbers. It is this sign change which distinguishes the splitcomplex numbers from the ordinary complex ones. The quantity j here is not a real number but an independent quantity; that is, it is not equal to ±1.
The collection of all such z is called the splitcomplex plane. Addition and multiplication of splitcomplex numbers are defined by
This multiplication is commutative, associative and distributes over addition.
Just as for complex numbers, one can define the notion of a splitcomplex conjugate. If
the conjugate of z is defined as
The conjugate satisfies similar properties to usual complex conjugate. Namely,
These three properties imply that the splitcomplex conjugate is an automorphism of order 2.
The modulus of a splitcomplex number z = x + j y is given by the quadratic form
It has an important property that it is preserved by splitcomplex multiplication:
However, this quadratic form is not positivedefinite but rather has signature (1,−1), so the modulus is not a norm.
The associated bilinear form is given by
where z = x + j y and w = u + j v. Another expression for the modulus is then
Since it is not positivedefinite, this bilinear form is not an inner product; nevertheless the bilinear form is frequently referred to as an indefinite inner product. A similar abuse of language refers to the modulus as a norm.
A splitcomplex number is invertible if and only if its modulus is nonzero (). The inverse of such an element is given by
Splitcomplex numbers which are not invertible are called null elements. These are all of the form (a ± j a) for some real number a.
There are two nontrivial idempotents given by e = (1 − j)/2 and e* = (1 + j)/2. Recall that idempotent means that ee = e and e*e* = e*. Both of these elements are null:
It is often convenient to use e and e* as an alternate basis for the splitcomplex plane. This basis is called the diagonal basis or null basis. The splitcomplex number z can be written in the null basis as
If we denote the number z = ae + be* for real numbers a and b by (a,b), then splitcomplex multiplication is given by
In this basis, it becomes clear that the splitcomplex numbers are isomorphic to the direct sum RR with addition and multiplication defined pairwise.
The splitcomplex conjugate in the diagonal basis is given by
and the modulus by
A twodimensional real vector space with the Minkowski inner product is called 1+1 dimensional Minkowski space, often denoted R^{1,1}. Just as much of the geometry of the Euclidean plane R^{2} can be described with complex numbers, the geometry of the Minkowski plane R^{1,1} can be described with splitcomplex numbers.
The set of points
is a hyperbola for every nonzero a in R. The hyperbola consists of a right and left branch passing through (a, 0) and (−a, 0). The case a = 1 is called the unit hyperbola. The conjugate hyperbola is given by
with an upper and lower branch passing through (0, a) and (0, −a). The hyperbola and conjugate hyperbola are separated by two diagonal asymptotes which form the set of null elements:
These two lines (sometimes called the null cone) are perpendicular in R^{2} and have slopes ±1.
Splitcomplex numbers z and w are said to be hyperbolicorthogonal if <z, w> = 0. While analogous to ordinary orthogonality, particularly as it is known with ordinary complex number arithmetic, this condition is more subtle. It forms the basis for the simultaneous hyperplane concept in spacetime.
The analogue of Euler's formula for the splitcomplex numbers is
This can be derived from a power series expansion using the fact that cosh has only even powers while that for sinh has odd powers. For all real values of the hyperbolic angle θ the splitcomplex number λ = exp(jθ) has norm 1 and lies on the right branch of the unit hyperbola. Numbers such as λ have been called hyperbolic versors.
Since λ has modulus 1, multiplying any splitcomplex number z by λ preserves the modulus of z and represents a hyperbolic rotation (also called a Lorentz boost or a squeeze mapping). Multiplying by λ preserves the geometric structure, taking hyperbolas to themselves and the null cone to itself.
The set of all transformations of the splitcomplex plane which preserve the modulus(or equivalently, the inner product) forms a group called the generalized orthogonal group O(1,1). This group consists of the hyperbolic rotations — which form a subgroup denoted SO^{+}(1,1) — combined with four discrete reflections given by
The exponential map
sending θ to rotation by exp(jθ) is a group isomorphism since the usual exponential formula applies:
If a splitcomplex number z does not lie on one of the diagonals, then z has a polar decomposition.
In abstract algebra terms, the splitcomplex numbers can be described as the quotient of the polynomial ring R[x] by the ideal generated by the polynomial x^{2} − 1,
The image of x in the quotient is the "imaginary" unit j. With this description, it is clear that the splitcomplex numbers form a commutative ring with characteristic 0. Moreover if we define scalar multiplication in the obvious manner, the splitcomplex numbers actually form a commutative and associative algebra over the reals of dimension two. The algebra is not a division algebra or field since the null elements are not invertible. In fact, all of the nonzero null elements are zero divisors. Since addition and multiplication are continuous operations with respect to the usual topology of the plane, the splitcomplex numbers form a topological ring.
The splitcomplex numbers do not form a normed algebra in the usual sense of the word since the "norm" is not positivedefinite. However, if one extends the definition to include norms of general signature, they do form such an algebra. This follows from the fact that
For an exposition of normed algebras in general signature, see the reference by Harvey.
From the definition it is apparent that the ring of splitcomplex numbers is isomorphic to the group ring R[C_{2}] of the cyclic group C_{2} over the real numbers R.
The splitcomplex numbers are a special case of a Clifford algebra. Namely, they form a Clifford algebra over a onedimensional vector space with a positivedefinite quadratic form. Contrast this with the complex numbers which form a Clifford algebra over a onedimensional vector space with a negativedefinite quadratic form. (NB: some authors switch the signs in the definition of a Clifford algebra which will interchange the meaning of positivedefinite and negativedefinite). In mathematics, the splitcomplex numbers are members of the Clifford algebra Cℓ_{1,0}(R) = Cℓ^{0}_{1,1}(R). This is an extension of the real numbers defined analogously to the complex numbers C = Cℓ_{0,1}(R) = Cℓ^{0}_{2,0}(R).
One can easily represent splitcomplex numbers by matrices. The splitcomplex number
can be represented by the matrix
Addition and multiplication of splitcomplex numbers are then given by matrix addition and multiplication. The modulus of z is given by the determinant of the corresponding matrix. In this representation, splitcomplex conjugation corresponds to multiplying on both sides by the matrix
For any real number a, a hyperbolic rotation by through a hyperbolic angle a corresponds to multiplication by the matrix
The diagonal basis for the splitcomplex number plane can be invoked by using an ordered pair (x,y) for and making the mapping
Now the quadratic form is uv = (x + y)(x − y) = x^{2} − y^{2}. Furthermore,
so the two parametrized hyperbolas are brought into correspondence. The action of hyperbolic versor then corresponds under this linear transformation to a squeeze mapping
The commutative diagram interpretation of this correspondence has A = B = {splitcomplex number plane}, C = D = R^{2}, f is the action of a hyperbolic versor, g & h are the linear transformation by the matrix of ones, and k is the squeeze mapping.
Note that in the context of 2 × 2 real matrices there are in fact a great number of different representations of splitcomplex numbers.
The use of splitcomplex numbers dates back to 1848 when James Cockle revealed his Tessarines. William Kingdon Clifford used splitcomplex numbers to represent sums of spins. Clifford introduced the use of splitcomplex numbers as coefficients in a quaternion algebra now called splitbiquaternions. He called its elements "motors", a term in parallel with the "rotor" action of an ordinary complex number taken from the circle group. Extending the analogy, functions of a motor variable contrast to functions of an ordinary complex variable.
In the twentiethcentury the splitcomplex numbers became a common platform to describe the Lorentz boosts of special relativity, in a spacetime plane, because a velocity change between frames of reference is essentially the action of a hyperbolic versor in a space of mixed signature.
In 1935 J.C. Vignaux and A. Durañona y Vedia developed the splitcomplex geometric algebra and function theory in four articles in Contribución a las Ciencias Físicas y Matemáticas, National University of La Plata, República Argentina (in Spanish). These expository and pedagogical essays presented the subject for broad appreciation.
In 1941 E.F. Allen used the splitcomplex geometric arithmetic to establish the ninepoint hyperbola of a triangle inscribed in zz* = 1.
Different authors have used a great variety of names for the splitcomplex numbers. Some of these include:
Splitcomplex numbers and their higherdimensional relatives (splitquaternions / coquaternions and splitoctonions) were at times referred to as "Musean numbers", since they are a subset of the hypernumber program developed by Charles Musès.
Higherorder derivatives of splitcomplex numbers, obtained through a modified CayleyDickson construction:
Enveloping algebras and number programs:

