Split-complex number

In algebra, a split-complex number (or hyperbolic number, also perplex number, double number) is based on a hyperbolic unit $j$ satisfying $j^{2}=1$ , where $j\neq \pm 1$ . A split-complex number has two real number components $x$ and $y$ , and is written $z=x+yj.$ The conjugate of $z$ is $z^{*}=x-yj.$ Since $j^{2}=1,$ the product of a number $z$ with its conjugate is $N(z):=zz^{*}=x^{2}-y^{2},$ an isotropic quadratic form.

The collection $D$ of all split-complex numbers $z=x+yj$ for ⁠ $x,y\in \mathbb {R}$ ⁠ forms an algebra over the field of real numbers. Two split-complex numbers $w$ and $z$ have a product $wz$ that satisfies $N(wz)=N(w)N(z).$ This composition of $N$ over the algebra product makes $(D, +, \times, *)$ a composition algebra.

A similar algebra based on ⁠ $\mathbb {R} ^{2}$ ⁠ and component-wise operations of addition and multiplication, ⁠ $(\mathbb {R} ^{2},+,\times ,xy),$ ⁠ where $xy$ is the quadratic form on ⁠ $\mathbb {R} ^{2},$ ⁠ also forms a quadratic space. The ring isomorphism

${\begin{aligned}D&\to \mathbb {R} ^{2}\\x+yj&\mapsto (x-y,x+y)\end{aligned}}$

relates proportional quadratic forms, but the mapping is not an isometry since the multiplicative identity $(1, 1)$ of ⁠ $\mathbb {R} ^{2}$ ⁠ is at a distance ⁠ ${\sqrt {2}}$ ⁠ from 0, which is normalized in $D$ .

Split-complex numbers have many other names; see § Synonyms below. See the article Motor variable for functions of a split-complex number.

Definition

A split-complex number is an ordered pair of real numbers, written in the form

$z=x+jy$

where $x$ and $y$ are real numbers and the hyperbolic unit^[1] $j$ satisfies

$j^{2}=+1$

In the field of complex numbers the imaginary unit i satisfies $i^{2}=-1.$ The change of sign distinguishes the split-complex numbers from the ordinary complex ones. The hyperbolic unit $j$ is not a real number but an independent quantity.

The collection of all such $z$ is called the split-complex plane. Addition and multiplication of split-complex numbers are defined by

${\begin{aligned}(x+jy)+(u+jv)&=(x+u)+j(y+v)\\(x+jy)(u+jv)&=(xu+yv)+j(xv+yu).\end{aligned}}$

This multiplication is commutative, associative and distributes over addition.

Conjugate, modulus, and bilinear form

Just as for complex numbers, one can define the notion of a split-complex conjugate. If

$z=x+jy~,$

then the conjugate of $z$ is defined as

$z^{*}=x-jy~.$

The conjugate is an involution which satisfies similar properties to the complex conjugate. Namely,

${\begin{aligned}(z+w)^{*}&=z^{*}+w^{*}\\(zw)^{*}&=z^{*}w^{*}\\\left(z^{*}\right)^{*}&=z.\end{aligned}}$

The squared modulus of a split-complex number $z=x+jy$ is given by the isotropic quadratic form

$\lVert z\rVert ^{2}=zz^{*}=z^{*}z=x^{2}-y^{2}~.$

It has the composition algebra property:

$\lVert zw\rVert =\lVert z\rVert \lVert w\rVert ~.$

However, this quadratic form is not positive-definite but rather has signature $(1, -1)$ , so the modulus is not a norm.

The associated bilinear form is given by

$\langle z,w\rangle =\operatorname {\mathrm {Re} } \left(zw^{*}\right)=\operatorname {\mathrm {Re} } \left(z^{*}w\right)=xu-yv~,$

where $z=x+jy$ and $w=u+jv.$ Here, the real part is defined by $\operatorname {\mathrm {Re} } (z)={\tfrac {1}{2}}(z+z^{*})=x$ . Another expression for the squared modulus is then

$\lVert z\rVert ^{2}=\langle z,z\rangle ~.$

Since it is not positive-definite, 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 split-complex number is invertible if and only if its modulus is nonzero ( $\lVert z\rVert \neq 0$ ), thus numbers of the form $x \pm j x$ have no inverse. The multiplicative inverse of an invertible element is given by

$z^{-1}={\frac {z^{*}}{{\lVert z\rVert }^{2}}}~.$

Split-complex numbers which are not invertible are called null vectors. These are all of the form $(a \pm j a)$ for some real number $a$ .

The diagonal basis

There are two nontrivial idempotent elements given by $e={\tfrac {1}{2}}(1-j)$ and $e^{*}={\tfrac {1}{2}}(1+j).$ Recall that idempotent means that $ee=e$ and $e^{*}e^{*}=e^{*}.$ Both of these elements are null:

$\lVert e\rVert =\lVert e^{*}\rVert =e^{*}e=0~.$

It is often convenient to use $e$ and $e$ ^∗ as an alternate basis for the split-complex plane. This basis is called the diagonal basis or null basis. The split-complex number $z$ can be written in the null basis as

$z=x+jy=(x-y)e+(x+y)e^{*}~.$

If we denote the number $z=ae+be^{*}$ for real numbers $a$ and $b$ by $(a, b)$ , then split-complex multiplication is given by

$\left(a_{1},b_{1}\right)\left(a_{2},b_{2}\right)=\left(a_{1}a_{2},b_{1}b_{2}\right)~.$

The split-complex conjugate in the diagonal basis is given by $(a,b)^{*}=(b,a)$ and the squared modulus by

$\lVert (a,b)\rVert ^{2}=ab.$

Isomorphism

On the basis {e, e*} it becomes clear that the split-complex numbers are ring-isomorphic to the direct sum ⁠ $\mathbb {R} \oplus \mathbb {R}$ ⁠ with addition and multiplication defined pairwise.

The diagonal basis for the split-complex number plane can be invoked by using an ordered pair $(x, y)$ for $z=x+jy$ and making the mapping

$(u,v)=(x,y){\begin{pmatrix}1&1\\1&-1\end{pmatrix}}=(x,y)S~.$

Now the quadratic form is $uv=(x+y)(x-y)=x^{2}-y^{2}~.$ Furthermore,

$(\cosh a,\sinh a){\begin{pmatrix}1&1\\1&-1\end{pmatrix}}=\left(e^{a},e^{-a}\right)$

so the two parametrized hyperbolas are brought into correspondence with $S$ .

The action of hyperbolic versor $e^{bj}\!$ then corresponds under this linear transformation to a squeeze mapping

$\sigma :(u,v)\mapsto \left(ru,{\frac {v}{r}}\right),\quad r=e^{b}~.$

Though lying in the same isomorphism class in the category of rings, the split-complex plane and the direct sum of two real lines differ in their layout in the Cartesian plane. The isomorphism, as a planar mapping, consists of a counter-clockwise rotation by 45° and a dilation by √2. The dilation in particular has sometimes caused confusion in connection with areas of a hyperbolic sector. Indeed, hyperbolic angle corresponds to area of a sector in the ⁠ $\mathbb {R} \oplus \mathbb {R}$ ⁠ plane with its "unit circle" given by $\{(a,b)\in \mathbb {R} \oplus \mathbb {R} :ab=1\}.$ The contracted unit hyperbola $\{\cosh a+j\sinh a:a\in \mathbb {R} \}$ of the split-complex plane has only half the area in the span of a corresponding hyperbolic sector. Such confusion may be perpetuated when the geometry of the split-complex plane is not distinguished from that of ⁠ $\mathbb {R} \oplus \mathbb {R}$ ⁠.

Geometry

Unit hyperbola: $‖ z ‖ = 1$

  Conjugate hyperbola: $‖ z ‖ = -1$

  Asymptotes: $‖ z ‖ = 0$

A two-dimensional real vector space with the Minkowski inner product is called $(1 + 1)$ -dimensional Minkowski space, often denoted ⁠ $\mathbb {R} ^{1,1}.$ ⁠ Just as much of the geometry of the Euclidean plane ⁠ $\mathbb {R} ^{2}$ ⁠ can be described with complex numbers, the geometry of the Minkowski plane ⁠ $\mathbb {R} ^{1,1}$ ⁠ can be described with split-complex numbers.

The set of points

$\left\{z:\lVert z\rVert ^{2}=a^{2}\right\}$

is a hyperbola for every nonzero $a$ in ⁠ $\mathbb {R} .$ ⁠ 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

$\left\{z:\lVert z\rVert ^{2}=-a^{2}\right\}$

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:

$\left\{z:\lVert z\rVert =0\right\}.$

These two lines (sometimes called the null cone) are perpendicular in ⁠ $\mathbb {R} ^{2}$ ⁠ and have slopes ±1.

Split-complex numbers $z$ and $w$ are said to be hyperbolic-orthogonal 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 split-complex numbers is

$\exp(j\theta )=\cosh(\theta )+j\sinh(\theta ).$

This formula can be derived from a power series expansion using the fact that cosh has only even powers while that for sinh has odd powers.^[2] For all real values of the hyperbolic angle $θ$ the split-complex 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 split-complex 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 split-complex 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

$z\mapsto \pm z$ and $z\mapsto \pm z^{*}.$

The exponential map

$\exp \colon (\mathbb {R} ,+)\to \mathrm {SO} ^{+}(1,1)$

sending $θ$ to rotation by $exp(jθ)$ is a group isomorphism since the usual exponential formula applies:

$e^{j(\theta +\phi )}=e^{j\theta }e^{j\phi }.$

If a split-complex number $z$ does not lie on one of the diagonals, then $z$ has a polar decomposition.

Algebraic properties

In abstract algebra terms, the split-complex numbers can be described as the quotient of the polynomial ring ⁠ $\mathbb {R} [x]$ ⁠ by the ideal generated by the polynomial $x^{2}-1,$

$\mathbb {R} [x]/(x^{2}-1).$

The image of $x$ in the quotient is the "imaginary" unit $j$ . With this description, it is clear that the split-complex numbers form a commutative algebra over the real numbers. The algebra is not a field since the null elements are not invertible. 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 split-complex numbers form a topological ring.

The algebra of split-complex numbers forms a composition algebra since

$\lVert zw\rVert =\lVert z\rVert \lVert w\rVert ~$

for any numbers $z$ and $w$ .

From the definition it is apparent that the ring of split-complex numbers is isomorphic to the group ring ⁠ $\mathbb {R} [C_{2}]$ ⁠ of the cyclic group $C 2$ over the real numbers ⁠ $\mathbb {R} .$ ⁠

Matrix representations

One can easily represent split-complex numbers by matrices. The split-complex number $z=x+jy$ can be represented by the matrix $z\mapsto {\begin{pmatrix}x&y\\y&x\end{pmatrix}}.$

Addition and multiplication of split-complex numbers are then given by matrix addition and multiplication. The squared modulus of $z$ is given by the determinant of the corresponding matrix.

In fact there are many representations of the split-complex plane in the four-dimensional ring of 2x2 real matrices. The real multiples of the identity matrix form a real line in the matrix ring M(2,R). Any hyperbolic unit m provides a basis element with which to extend the real line to the split-complex plane. The matrices

$m={\begin{pmatrix}a&c\\b&-a\end{pmatrix}}$

which square to the identity matrix satisfy $a^{2}+bc=1.$ For example, when a = 0, then (b,c) is a point on the standard hyperbola. More generally, there is a hypersurface in M(2,R) of hyperbolic units, any one of which serves in a basis to represent the split-complex numbers as a subring of M(2,R).^[3]^{[better source needed]}

The number $z=x+jy$ can be represented by the matrix $x\ I+y\ m.$

History

The use of split-complex numbers dates back to 1848 when James Cockle revealed his tessarines.^[4] William Kingdon Clifford used split-complex numbers to represent sums of spins. Clifford introduced the use of split-complex numbers as coefficients in a quaternion algebra now called split-biquaternions. 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.

Since the late twentieth century, the split-complex multiplication has commonly been seen as a Lorentz boost of a spacetime plane.^[5]^[6]^[7]^[8]^[9]^[10] In that model, the number $z = x + y j$ represents an event in a spatio-temporal plane, where x is measured in seconds and $y$ in light-seconds. The future corresponds to the quadrant of events ${z : | y | < x}$ , which has the split-complex polar decomposition $z=\rho e^{aj}\!$ . The model says that $z$ can be reached from the origin by entering a frame of reference of rapidity $a$ and waiting $ρ$ nanoseconds. The split-complex equation

$e^{aj}\ e^{bj}=e^{(a+b)j}$

expressing products on the unit hyperbola illustrates the additivity of rapidities for collinear velocities. Simultaneity of events depends on rapidity $a$ ;

$\{z=\sigma je^{aj}:\sigma \in \mathbb {R} \}$

is the line of events simultaneous with the origin in the frame of reference with rapidity a.

Two events $z$ and $w$ are hyperbolic-orthogonal when $z^{*}w+zw^{*}=0.$ Canonical events $exp(aj)$ and $j exp(aj)$ are hyperbolic orthogonal and lie on the axes of a frame of reference in which the events simultaneous with the origin are proportional to $j exp(aj)$ .

In 1933 Max Zorn was using the split-octonions and noted the composition algebra property. He realized that the Cayley–Dickson construction, used to generate division algebras, could be modified (with a factor gamma, $γ$ ) to construct other composition algebras including the split-octonions. His innovation was perpetuated by Adrian Albert, Richard D. Schafer, and others.^[11] The gamma factor, with $R$ as base field, builds split-complex numbers as a composition algebra. Reviewing Albert for Mathematical Reviews, N. H. McCoy wrote that there was an "introduction of some new algebras of order 2^e over F generalizing Cayley–Dickson algebras."^[12] Taking $F = R$ and $e = 1$ corresponds to the algebra of this article.

In 1935 J.C. Vignaux and A. Durañona y Vedia developed the split-complex 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.^[13]

In 1941 E.F. Allen used the split-complex geometric arithmetic to establish the nine-point hyperbola of a triangle inscribed in $zz * = 1$ .^[14]

In 1956 Mieczyslaw Warmus published "Calculus of Approximations" in Bulletin de l’Académie polonaise des sciences (see link in References). He developed two algebraic systems, each of which he called "approximate numbers", the second of which forms a real algebra.^[15] D. H. Lehmer reviewed the article in Mathematical Reviews and observed that this second system was isomorphic to the "hyperbolic complex" numbers, the subject of this article.

In 1961 Warmus continued his exposition, referring to the components of an approximate number as midpoint and radius of the interval denoted.

Synonyms

Different authors have used a great variety of names for the split-complex numbers. Some of these include:

(real) tessarines, James Cockle (1848)
(algebraic) motors, W.K. Clifford (1882)
hyperbolic complex numbers, J.C. Vignaux (1935), G. Cree (1949)^[16]
bireal numbers, U. Bencivenga (1946)
real hyperbolic numbers, N. Smith (1949)^[17]
approximate numbers, Warmus (1956), for use in interval analysis
double numbers, I.M. Yaglom (1968), Kantor and Solodovnikov (1989), Hazewinkel (1990), Rooney (2014)
hyperbolic numbers, W. Miller & R. Boehning (1968),^[18] G. Sobczyk (1995)
anormal-complex numbers, W. Benz (1973)
perplex numbers, P. Fjelstad (1986) and Poodiack & LeClair (2009)
countercomplex or hyperbolic, Carmody (1988)
Lorentz numbers, F.R. Harvey (1990)
semi-complex numbers, F. Antonuccio (1994)
paracomplex numbers, Cruceanu, Fortuny & Gadea (1996)
split-complex numbers, B. Rosenfeld (1997)^[19]
spacetime numbers, N. Borota (2000)
Study numbers, P. Lounesto (2001)
twocomplex numbers, S. Olariu (2002)
split binarions, K. McCrimmon (2004)

References

^ Vladimir V. Kisil (2012) Geometry of Mobius Transformations: Elliptic, Parabolic, and Hyperbolic actions of SL(2,R), pages 2, 161, Imperial College Press ISBN 978-1-84816-858-9
^ James Cockle (1848) On a New Imaginary in Algebra, Philosophical Magazine 33:438
^ Abstract Algebra/2x2 real matrices at Wikibooks
^ James Cockle (1849) On a New Imaginary in Algebra 34:37–47, London-Edinburgh-Dublin Philosophical Magazine (3) 33:435–9, link from Biodiversity Heritage Library.
^ Francesco Antonuccio (1994) Semi-complex analysis and mathematical physics
^ F. Catoni, D. Boccaletti, R. Cannata, V. Catoni, E. Nichelatti, P. Zampetti. (2008) The Mathematics of Minkowski Space-Time, Birkhäuser Verlag, Basel. Chapter 4: Trigonometry in the Minkowski plane. ISBN 978-3-7643-8613-9.
^ Francesco Catoni; Dino Boccaletti; Roberto Cannata; Vincenzo Catoni; Paolo Zampetti (2011). "Chapter 2: Hyperbolic Numbers". Geometry of Minkowski Space-Time. Springer Science & Business Media. ISBN 978-3-642-17977-8.
^ Fjelstad, Paul (1986), "Extending special relativity via the perplex numbers", American Journal of Physics, 54 (5): 416–422, Bibcode:1986AmJPh..54..416F, doi:10.1119/1.14605
^ Louis Kauffman (1985) "Transformations in Special Relativity", International Journal of Theoretical Physics 24:223–36.
^ Sobczyk, G.(1995) Hyperbolic Number Plane, also published in College Mathematics Journal 26:268–80.
^ Robert B. Brown (1967)On Generalized Cayley-Dickson Algebras, Pacific Journal of Mathematics 20(3):415–22, link from Project Euclid.
^ N.H. McCoy (1942) Review of "Quadratic forms permitting composition" by A.A. Albert, Mathematical Reviews #0006140
^ Vignaux, J.(1935) "Sobre el numero complejo hiperbolico y su relacion con la geometria de Borel", Contribucion al Estudio de las Ciencias Fisicas y Matematicas, Universidad Nacional de la Plata, Republica Argentina
^ Allen, E.F. (1941) "On a Triangle Inscribed in a Rectangular Hyperbola", American Mathematical Monthly 48(10): 675–681
^ M. Warmus (1956) "Calculus of Approximations" Archived 2012-03-09 at the Wayback Machine, Bulletin de l'Académie polonaise des sciences, Vol. 4, No. 5, pp. 253–257, MR0081372
^ Cree, George C. (1949). The Number Theory of a System of Hyperbolic Complex Numbers (MA thesis). McGill University.
^ Smith, Norman E. (1949). Introduction to Hyperbolic Number Theory (MA thesis). McGill University.
^ Miller, William; Boehning, Rochelle (1968). "Gaussian, parabolic, and hyperbolic numbers". The Mathematics Teacher. 61 (4): 377–382. doi:10.5951/MT.61.4.0377. JSTOR 27957849.
^ Rosenfeld, B. (1997) Geometry of Lie Groups, page 30, Kluwer Academic Publishers ISBN 0-7923-4390-5

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