# Split-complex number
A **split-complex number** is a number of the form \( z = x + yj \), where \( x \) and \( y \) are real numbers and the hyperbolic unit \( j \) satisfies \( j^2 = 1 \) with \( j \neq \pm 1 \)[1](https://books.physics.oregonstate.edu/GELG/csplit.html). The system is also called the hyperbolic numbers, tessarines, double numbers, perplex numbers, bireal numbers, motors, or Lorentz numbers[2](https://www.commsp.ee.ic.ac.uk/~mandic/complexbook/files/c01.pdf) • [3](https://www.mdpi.com/2227-7390/13/3/504). It is one of the three two-dimensional commutative algebras over the real numbers obtained by changing the square of the new unit: \( i^2 = -1 \) gives the complex numbers, \( j^2 = +1 \) gives the split-complex numbers, and \( \varepsilon^2 = 0 \) gives the [dual numbers](https://www.edgepedia.org/dual-number)[3](https://www.mdpi.com/2227-7390/13/3/504).
| Key fact | Detail |
|---|---|
| Definition | \( z = x + yj \) with \( x, y \in \mathbb{R} \) and \( j^2 = 1 \), \( j \notin \mathbb{R} \)[1](https://books.physics.oregonstate.edu/GELG/csplit.html) |
| Construction | Cayley–Dickson process with \( \epsilon = -1 \); equivalently \( \mathbb{R}[j]/(j^2 - 1) \)[1](https://books.physics.oregonstate.edu/GELG/csplit.html) • [4](https://knowlpedia.com/catalog/algebras/split-complex-numbers/) |
| Algebraic type | Commutative and associative ring with zero divisors, not a field; isomorphic to \( \mathbb{R} \oplus \mathbb{R} \) with pointwise multiplication[1](https://books.physics.oregonstate.edu/GELG/csplit.html) • [5](https://ncatlab.org/nlab/show/split-complex%2Bnumbers) |
| Norm | \( \lvert x + yj \rvert^2 = x^2 - y^2 \), not positive-definite[1](https://books.physics.oregonstate.edu/GELG/csplit.html) |
| Idempotents | \( \tfrac{1}{2}(1 \pm j) \) are orthogonal projections splitting the algebra into two null subspaces[1](https://books.physics.oregonstate.edu/GELG/csplit.html) |
| Geometry | \( e^{j\beta} = \cosh\beta + j\sinh\beta \); multiplication by such an element is a Lorentz boost in 2-dimensional Minkowski space[1](https://books.physics.oregonstate.edu/GELG/csplit.html) |
| History | Introduced by James Cockle in 1848 as the tessarines[2](https://www.commsp.ee.ic.ac.uk/~mandic/complexbook/files/c01.pdf) |
## Arithmetic
Addition is componentwise. Multiplication follows from the distributive law and the rule \( j^2 = 1 \): for \( a = a' + ja'' \) and \( b = b' + jb'' \),
\[ ab = (a'b' + a''b'') + j(a'b'' + a''b'). \]
Like the complex numbers, the split-complex numbers are commutative and associative under this multiplication[1](https://books.physics.oregonstate.edu/GELG/csplit.html). Conjugation sends \( a' + ja'' \) to \( a' - ja'' \), and the squared norm is
\[ \lvert a' + ja'' \rvert^2 = (a' + ja'')(a' - ja'') = a'^2 - a''^2, \]
which is not positive-definite[1](https://books.physics.oregonstate.edu/GELG/csplit.html). The multiplicative inverse is \( a^{-1} = \bar{a}/N(a) \), where \( N(a) = a'^2 - a''^2 \); it is not always defined, so the split-complex numbers form a commutative ring rather than a field[6](https://www.researchgate.net/publication/228565818_On_hyperbolic_complex_LTI_digital_systems).
**Zero divisors.** Because \( j^2 = 1 \), the algebra contains nonzero elements whose product is zero:
\[ (1 + j)(1 - j) = 1 - j^2 = 0. \]
Such elements are zero divisors, and their existence is what separates the split-complex numbers from the complex numbers, which form a field[1](https://books.physics.oregonstate.edu/GELG/csplit.html). The two idempotents \( \tfrac{1}{2}(1 + j) \) and \( \tfrac{1}{2}(1 - j) \), which satisfy \( e^2 = e \), act as orthogonal projection operators dividing the algebra into two null subspaces[1](https://books.physics.oregonstate.edu/GELG/csplit.html). Through these idempotents the map \( x + yj \mapsto (x+y,\, x-y) \) identifies the split-complex numbers with the direct sum \( \mathbb{R} \oplus \mathbb{R} \) with pointwise multiplication[5](https://ncatlab.org/nlab/show/split-complex%2Bnumbers).
**Square roots.** The algebra has four square roots of unity: \( \pm 1 \) and \( \pm j \). More generally, since \( (a + bj)^2 = (a^2 + b^2) + 2abj \), a split-complex number is a square only if its real part is at least as large as its imaginary part; in particular \( j \) itself is not the square of any split-complex number[1](https://books.physics.oregonstate.edu/GELG/csplit.html).
## Representations
The algebra can be written as a quotient of polynomial rings, \( \mathbb{C}_s = \mathbb{R}[e]/(e^2 - 1) \), with elements \( a + be \) and the quadratic form \( N(a + be) = a^2 - b^2 \)[4](https://knowlpedia.com/catalog/algebras/split-complex-numbers/). It is also a Clifford algebra: the split-complex numbers are isomorphic to \( \mathrm{Cl}_{1,0}(\mathbb{R}) \), the [Clifford algebra](https://www.edgepedia.org/clifford-algebra) generated by one element squaring to \( +1 \)[7](https://www.researchgate.net/publication/308514051_INTRODUCTION_TO_CLIFFORD_ANALYSIS). Changing the sign rule in the Clifford product instead yields the dual numbers or the complex numbers[8](https://konrad.burnik.org/wordpress/split-complex-numbers-as-clifford-algebras/).
A split-complex number \( a' + ja'' \) can also be represented by the real \( 2 \times 2 \) matrix
\[ M_a = \begin{pmatrix} a' & a'' \\ a'' & a' \end{pmatrix}, \]
a representation in which multiplication is commutative and the determinant equals the norm \( a'^2 - a''^2 \)[6](https://www.researchgate.net/publication/228565818_On_hyperbolic_complex_LTI_digital_systems). The algebra is obtained from the real numbers by the Cayley–Dickson process with parameter \( \epsilon = -1 \), which is also the route to the other split algebras such as the [split-quaternions](https://www.edgepedia.org/split-quaternion)[1](https://books.physics.oregonstate.edu/GELG/csplit.html) • [9](https://www.imperial.ac.uk/media/imperial-college/research-centres-and-groups/theoretical-physics/msc/dissertations/2013/Amel-Durakovic-Dissertation.pdf).
## Geometry and special relativity
The squared norm \( x^2 - y^2 \) is the inner product of two-dimensional [Minkowski space](https://www.edgepedia.org/minkowski-space), with the spacetime vector \( (x, t) \) corresponding to the split-complex number \( x + tj \)[1](https://books.physics.oregonstate.edu/GELG/csplit.html). The exponential formula
\[ e^{j\beta} = \cosh(\beta) + j\sinh(\beta), \]
checkable by expanding the power series, makes multiplication by \( e^{j\beta} \) a hyperbolic rotation. For this reason the split-complex numbers are called hyperbolic numbers, and they can be used to describe Lorentz transformations between reference frames, which are hyperbolic rotations of the spacetime plane[1](https://books.physics.oregonstate.edu/GELG/csplit.html). This contrasts with the complex numbers, where multiplication by \( e^{i\theta} = \cos\theta + i\sin\theta \) gives ordinary circular rotations[10](https://www.math4wisdom.com/wiki/Research/SplitComplexNumbers).
## History
James Cockle introduced the system in 1848 under the name tessarines[2](https://www.commsp.ee.ic.ac.uk/~mandic/complexbook/files/c01.pdf). In 1876 William Kingdon Clifford, working to model spins, introduced his system of hypercomplex numbers by combining the quaternion algebra with the split-complex numbers[2](https://www.commsp.ee.ic.ac.uk/~mandic/complexbook/files/c01.pdf). In 1883 Weierstrass showed that every finite-dimensional commutative and associative algebra over the real numbers of dimension greater than 2 contains zero divisors; however, not every commutative associative real algebra with \( n > 2 \) contains a hyperbolic subalgebra, since, for example, ℝ[t]/(t³) has no nontrivial idempotents and therefore cannot contain one[6](https://www.researchgate.net/publication/228565818_On_hyperbolic_complex_LTI_digital_systems).
## Applications
**Analysis.** Matvei Libine, a mathematician working on quaternionic analysis and representation theory, proved in 2007 a hyperbolic analog of the Cauchy integral formula: an integral formula, formally identical to the classical Cauchy formula, for solutions of the wave equation \( \square_{1,1} F = 0 \) on \( \mathbb{R}^{1,1} \), with singularities of the kernel concentrated on the light cone \( N(Z) = 0 \) rather than at a single point[11](https://arxiv.org/abs/0712.0375).
**Signal processing.** Alfsmann and Göckler of the Digital Signal Processing Group at Ruhr-Universität Bochum investigated digital systems based on hyperbolic numbers as the simplest instance of a non-division algebra, analyzing a general first-order hyperbolic system to determine the impact of zero divisors on digital signal processing applications[6](https://www.researchgate.net/publication/228565818_On_hyperbolic_complex_LTI_digital_systems).
**Machine learning and other uses.** Split-complex arithmetic has been tried in neural networks: a Stanford CS229 project training real, complex, and split-complex networks on CIFAR-10 found little performance difference between the arithmetics, and found the complex and split-complex models harder to regularize than real-valued ones[12](https://cs229.stanford.edu/proj2017/final-reports/5231880.pdf). Steven Deckelman and Barry Robson used the split-complex numbers to define a Hyperbolic Dirac Network, a hyperbolic analog of a classical probabilistic graphical model, in biomedical informatics[13](https://www.researchgate.net/publication/272489181_Split-complex_numbers_and_Dirac_bra-kets). More broadly, commutative analogs of Clifford algebras, studied by Heerak Sharma and Dmitry Shirokov, turn out to be either multicomplex spaces or "multi split-complex" spaces built from split-complex rather than complex numbers[14](https://arxiv.org/abs/2504.19763).
## References
1. [The Split Complex Numbers, Section 2.8, Oregon State University open textbook](https://books.physics.oregonstate.edu/GELG/csplit.html)
2. [The Magic of Complex Numbers, chapter 1 of a book by D. Mandic and colleagues, Imperial College London](https://www.commsp.ee.ic.ac.uk/~mandic/complexbook/files/c01.pdf)
3. [Hypercomplex Numbers—A Tool for Enhanced Efficiency and Intelligence in Digital Signal Processing, Mathematics (MDPI), 2025](https://www.mdpi.com/2227-7390/13/3/504)
4. [Split complex numbers Cs, Knowlpedia](https://knowlpedia.com/catalog/algebras/split-complex-numbers/)
5. [Split-complex numbers, nLab](https://ncatlab.org/nlab/show/split-complex%2Bnumbers)
6. [D. Alfsmann and H. G. Göckler, On hyperbolic complex LTI digital systems](https://www.researchgate.net/publication/228565818_On_hyperbolic_complex_LTI_digital_systems)
7. [Introduction to Clifford Analysis (ResearchGate preprint)](https://www.researchgate.net/publication/308514051_INTRODUCTION_TO_CLIFFORD_ANALYSIS)
8. [Split-Complex Numbers as a Clifford Algebra, K. Burnik, The Hitchhiker's Guide to Computable Mathematics](https://konrad.burnik.org/wordpress/split-complex-numbers-as-clifford-algebras/)
9. [A. Durakovic, Split algebras, MSc dissertation, Imperial College London, 2013](https://www.imperial.ac.uk/media/imperial-college/research-centres-and-groups/theoretical-physics/msc/dissertations/2013/Amel-Durakovic-Dissertation.pdf)
10. [SplitComplexNumbers, Math 4 Wisdom research wiki](https://www.math4wisdom.com/wiki/Research/SplitComplexNumbers)
11. [M. Libine, Hyperbolic Cauchy Integral Formula for the Split Complex Numbers, arXiv:0712.0375 (2007)](https://arxiv.org/abs/0712.0375)
12. [Split-Complex Convolutional Neural Networks, Stanford CS229 project report (2017)](https://cs229.stanford.edu/proj2017/final-reports/5231880.pdf)
13. [S. Deckelman and B. Robson, Split-complex numbers and Dirac bra-kets, Communications in Information and Systems 14 (2014)](https://www.researchgate.net/publication/272489181_Split-complex_numbers_and_Dirac_bra-kets)
14. [H. Sharma and D. Shirokov, On Commutative Analogues of Clifford Algebras and Their Decompositions, arXiv:2504.19763](https://arxiv.org/abs/2504.19763)
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*Topic: Encyclopedia › Physical world and mathematics › Mathematics and statistics › Numbers and algebra › Arithmetic and number systems › Number systems › Real and complex number constructions › Constructions and models of the complex numbers*
*Initially written Sep 17, 2026 · Reviewed: Oct 15, 2026 · Edited: Oct 15, 2026 · Last review: Oct 15, 2026*
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