Exponential forms and path integrals for complex numbers in n dimensions

Two distinct systems of commutative complex numbers in n dimensions are described, of polar and planar types. Exponential forms of n-complex numbers are given in each case, which depend on geometric variables. Azimuthal angles, which are cyclic variables, appear in these forms at the exponent, and this leads to the concept of residue for path integrals of n-complex functions. The exponential function of an n-complex number is expanded in terms of functions called in this paper cosexponential functions, which are generalizations to n dimensions of the circular and hyperbolic sine and cosine functions. The factorization of n-complex polynomials is discussed.


Introduction
Hypercomplex numbers are a generalization to several dimensions of the regular complex numbers in 2 dimensions. A well-known example of hypercomplex numbers are the quaternions of Hamilton, which are a system of hypercomplex numbers in four dimensions, the multiplication being a non-commutative operation. [1] Many other hypercomplex systems are possible, [2]- [4] but these systems do not have all the required properties of regular, two-dimensional complex numbers which rendered possible the development of the theory of functions of a 2-dimensional complex variable.
Two distinct systems of complex numbers in n dimensions are described in this paper, for which the multiplication is associative and commutative, and which are rich enough in properties such that exponential forms exist and the concepts of analytic n-complex function, contour integration and residue can be defined. The first type of n-complex numbers described in this article is characterized by the presence, in an odd number of dimensions, of one polar axis, and by the presence, in an even number of dimensions, of two polar axes. Therefore, these numbers will be called polar n-complex numbers. The other type of ncomplex numbers described in this paper exists as a distinct entity only when the number of dimensions n of the space is even. These numbers will be called planar n-complex numbers.
The planar hypercomplex numbers become for n = 2 the usual complex numbers x + iy.
The central result of this paper is the existence of an exponential form of n-complex numbers, which is expressed in terms of geometric variables. The exponential form provides the link between the algebraic side of the operations and the analytic properties of the functions of n-complex variables. The azimuthal angles φ k , which are cyclic variables, appear in these forms at the exponent, and this leads to the concept of n-complex residue for path integrals of n-complex functions. Expressions are given for the elementary functions of n-complex variable. The exponential function of an n-complex number is expanded in terms of functions called in this paper n-dimensional cosexponential functions of the polar and respectively planar type. The polar cosexponential functions are a generalization to n dimensions of the hyperbolic functions cosh y, sinh y, and the planar cosexponential functions are a generalization to n dimensions of the trigonometric functions cos y, sin y. Addition theorems and other relations are obtained for the n-dimensional cosexponential functions.
In the case of polar n-complex numbers, a polynomial can be written as a product of linear or quadratic factors, although several factorizations are in general possible. In the case of planar n-complex numbers, a polynomial can always be written as a product of linear factors, although, again, several factorizations are in general possible.

.1 Operations with polar n-complex numbers
A hypercomplex number in n dimensions is determined by its n components (x 0 , x 1 , ..., x n−1 ).
The polar n-complex numbers and their operations discussed in this paper can be represented by writing the n-complex number (x 0 , x 1 , ..., x n−1 ) as u = x 0 + h 1 x 1 + h 2 x 2 + · · · + h n−1 x n−1 , where h 1 , h 2 , · · · , h n−1 are bases for which the multiplication rules are for j, k, l = 0, 1, ..., n−1, where h 0 = 1. In this relation, [(j +k)/n] denotes the integer part of (j + k)/n, defined as [a] ≤ a < [a] + 1, so that 0 ≤ j + k − n[(j + k)/n] ≤ n − 1. In this paper, brackets larger than the regular brackets [ ] do not have the meaning of integer part. The significance of the composition laws in Eq. (1) can be understood by representing the bases h j , h k by points on a circle at the angles α j = 2πj/n, α k = 2πk/n, as shown in Fig. 1, and the product h j h k by the point of the circle at the angle 2π(j + k)/n. If 2π ≤ 2π(j + k)/n < 4π, the point represents the basis h l of angle α l = 2π(j + k)/n − 2π.
Two n-complex numbers The sum of the n-complex numbers u and u ′ is The product of the n-complex numbers u, u ′ is The product uu ′ can be written as If u, u ′ , u ′′ are n-complex numbers, the multiplication is associative (uu ′ )u ′′ = u(u ′ u ′′ ) and commutative uu ′ = u ′ u, because the product of the bases, defined in Eq. (1), is associative and commutative.
The inverse of polar the n-complex number u is the n-complex number u ′ having the property that uu ′ = 1. This equation has a solution provided that the corresponding determinant ν is not equal to zero, ν = 0. If n is an even number, it can be shown that and if n is an odd number, where Thus, in an even number of dimensions n, an n-complex number has an inverse unless it lies on one of the nodal hypersurfaces v + = 0, or v − = 0, or ρ 1 = 0, or ... or ρ n/2−1 = 0. In an odd number of dimensions n, an n-complex number has an inverse unless it lies on one of the nodal hypersurfaces v+ = 0, or ρ 1 = 0, or ... or ρ (n−1)/2 = 0.
For even n, and for odd n, From these relations it results that if the product of two n-complex numbers is zero, uu ′ = 0, .., n/2, which means that either u = 0, or u ′ = 0, or u, u ′ belong to orthogonal hypersurfaces in such a way that the afore-mentioned products of components should be equal to zero.

Geometric representation of polar n-complex numbers
The polar n-complex number x 0 + h 1 x 1 + h 2 x 2 + · · · + h n−1 x n−1 can be represented by the point A of coordinates (x 0 , x 1 , ..., x n−1 ). If O is the origin of the n-dimensional space, the distance from the origin O to the point A of coordinates (x 0 , x 1 , ..., x n−1 ) has the expression The quantity d will be called modulus of the polar n-complex number The modulus of an n-complex number u will be designated by d = |u|. If ν > 0, the quantity ρ = ν 1/n will be called amplitude of the n-complex number u.
The exponential and trigonometric forms of the n-complex number u can be obtained conveniently in a rotated system of axes defined by the transformation for k = 1, ..., [(n − 1)/2] . This transformation from the coordinates x 0 , ..., x n−1 to the variables ξ + , ξ − , ξ k , η k is unitary.
The position of the point A of coordinates (x 0 , x 1 , ..., x n−1 ) can be also described with the aid of the distance d, Eq. (12), and of n − 1 angles defined further. Thus, in the plane of the axes v k ,ṽ k , the azimuthal angles φ k can be introduced by the relations where 0 ≤ φ k < 2π, so that there are [(n − 1)/2] azimuthal angles. If the projection of the point A on the plane of the axes v k ,ṽ k is A k , and the projection of the point A on the 4-dimensional space defined by the axes v 1 ,ṽ 1 , v k ,ṽ k is A 1k , the angle ψ k−1 between the line OA 1k and the 2-dimensional plane defined by the axes v k ,ṽ k is for 0 ≤ ψ k ≤ π/2, k = 2, ..., [(n − 1)/2], so that there are [(n − 3)/2] planar angles. Moreover, there is a polar angle θ + , which can be defined as the angle between the line OA 1+ and the axis v + , where A 1+ is the projection of the point A on the 3-dimensional space generated by the axes v 1 ,ṽ 1 , v + , where 0 ≤ θ + ≤ π , and in an even number of dimensions n there is also a polar angle θ − , which can be defined as the angle between the line OA 1− and the axis v − , where A 1− is the projection of the point A on the 3-dimensional space generated by the axes v 1 ,ṽ 1 , v − , where 0 ≤ θ − ≤ π . Thus, the position of the point A is described, in an even number of dimensions, by the distance d, by n/2 − 1 azimuthal angles, by n/2 − 2 planar angles, and by 2 polar angles. In an odd number of dimensions, the position of the point A is described by (n−1)/2 azimuthal angles, by (n−3)/2 planar angles, and by 1 polar angle. These angles are shown in Fig. 2. The variables ν, ρ, ρ k , tan θ + / √ 2, tan θ − / √ 2, tan ψ k are multiplicative and the azimuthal angles φ k are additive upon the multiplication of polar n-complex numbers.

The n-dimensional polar cosexponential functions
The exponential function of the polar n-complex variable u can be defined by the series exp u = 1 + u + u 2 /2! + u 3 /3! + · · · . It can be checked by direct multiplication of the series It can be seen with the aid of the representation in Fig. 1 that for p integer, k = 1, ..., n − 1. Then e h k y can be written as where the functions g nl , which will be called polar cosexponential functions in n dimensions, for l = 0, 1, ..., n − 1. If n is even, the polar cosexponential functions of even index k are even functions, g n,2l (−y) = g n,2l (y), and the polar cosexponential functions of odd index are odd functions, g n,2l+1 (−y) = −g n,2l+1 (y), l = 0, 1, ..., n/2 − 1. For odd values of n, the polar cosexponential functions do not have a definite parity. It can be checked that and, for even n, The expression of the polar n-dimensional cosexponential functions is for k = 0, 1, ..., n − 1. It can be shown from Eq. (23) that which does not contain exponential terms.
It can also be shown that The polar n-dimensional cosexponential functions are solutions of the n th -order differential equation whose solutions are of the form ζ(u) = A 0 g n0 (u) + A 1 g n1 (u) + · · · + A n−1 g n,n−1 (u). It can be checked that the derivatives of the polar cosexponential functions are related by .., dg n,n−2 du = g n,n−3 , dg n,n−1 du = g n,n−2 .

Exponential and trigonometric forms of polar n-complex numbers
In order to obtain the exponential and trigonometric forms of polar n-complex numbers, a new set of hypercomplex bases will be introduced for even n by the relations where k = 1, ..., [(n − 1)/2] and, if n is even, The multiplication relations for the new hypercomplex bases are e 2 + = e + , e 2 − = e − , e + e − = 0, e + e k = 0, e +ẽk = 0, e − e k = 0, e −ẽk = 0, where k, l = 1, ..., [(n − 1)/2]. It can be shown that, for even n, and for odd n The exponential form of the n-complex number u is where F (n) = 1 for even n and F (n) = 0 for odd n, and for even n, and for odd n.
The trigonometric form of the n-complex number u is (40)

Elementary functions of a polar n-complex variable
The logarithm u 1 of the polar n-complex number u, u 1 = ln u, can be defined as the solution of the equation u = e u 1 . For even n, ln u exists as an n-complex function with real components if v + > 0 and v − > 0. For odd n ln u exists as an n-complex function with real components if v + > 0. The expression of the logarithm is The function ln u is multivalued because of the presence of the termsẽ k φ k .
The power function u m of the polar n-complex variable u can be defined for real values of m as u m = e m ln u . It can be shown that For integer values of m, this expression is valid for any x 0 , ..., x n−1 . The power function is multivalued unless m is an integer.

Power series of polar n-complex numbers
A power series of the polar n-complex variable u is a series of the form a 0 + a 1 u + a 2 u 2 + · · · + a l u l + · · · .
Using the inequality which replaces the relation of equality extant for 2-dimensional complex numbers, it can be shown that the series (43) is absolutely convergent for |u| < c, where c = lim l→∞ |a l |/ √ n|a l+1 | .
The convergence of the series (43) can be also studied with the aid of the formulas (42), which for integer values of m are valid for any values of x 0 , ..., x n−1 . If a l = n−1 p=0 h p a lp , and for k = 1, ..., [(n − 1)/2], and for even n the series (43) can be written as The series in Eq. (43) is absolutely convergent for These relations show that the region of convergence of the series (43) is an n-dimensional cylinder.

Analytic functions of polar n-complex variables
The derivative of a function f (u) of the n-complex variables u is defined as a function f ′ (u) having the property that If the difference u − u 0 is not parallel to one of the nodal hypersurfaces, the definition in Eq.
(54) can also be written as The derivative of the function f (u) = u m , with m an integer, is f ′ (u) = mu m−1 , as can be and using the definition (54).
If the function f ′ (u) defined in Eq. (54) is independent of the direction in space along which u is approaching u 0 , the function f (u) is said to be analytic, analogously to the case of functions of regular complex variables. [5] The function u m , with m an integer, of the n-complex variable u is analytic, because the difference u m − u m 0 is always proportional to u − u 0 , as can be seen from Eq. (56). Then series of integer powers of u will also be analytic functions of the n-complex variable u, and this result holds in fact for any commutative algebra.
If the n-complex function f (u) of the polar n-complex variable u is written in terms of the real functions P k (x 0 , ..., x n−1 ), k = 0, 1, ..., n − 1 of the real variables x 0 , x 1 , ..., x n−1 as then relations of equality exist between the partial derivatives of the functions P k . The derivative of the function f can be written as where ∆u = n−1 k=0 h l ∆x l .
The relations between the partials derivatives of the functions P k are obtained by setting successively in Eq. (58) ∆u = h l ∆x l , for l = 0, 1, ..., n − 1, and equating the resulting expressions. The relations are for k = 0, 1, ..., n − 1. The relations (59) are analogous to the Riemann relations for the real and imaginary components of a complex function. It can be shown from Eqs. (59) that the components P k fulfil the second-order equations for k, l = 0, 1, ..., n − 1.

Integrals of polar n-complex functions
The singularities of polar n-complex functions arise from terms of the form 1/(u − u 0 ) m , with m > 0. Functions containing such terms are singular not only at u = u 0 , but also at all points of the hypersurfaces passing through u 0 and which are parallel to the nodal hypersurfaces.
The integral of a polar n-complex function between two points A, B along a path situated in a region free of singularities is independent of path, which means that the integral of an analytic function along a loop situated in a region free of singularities is zero, where it is supposed that a surface Σ spanning the closed loop Γ is not intersected by any of the hypersurfaces associated with the singularities of the function f (u). Using the expression, Eq. (57), for f (u) and the fact that du = n−1 k=0 h k dx k , the explicit form of the integral in Eq. (61) is The quantity du/(u − u 0 ) is Since ρ, ln( √ 2/ tan θ + ), ln( √ 2/ tan θ − ), ln(tan ψ k−1 ) are singlevalued variables, it follows that If f (u) is an analytic function of a polar n-complex variable which can be expanded in a series in the region of the curve Γ and on a surface spanning Γ, then where u 0ξ k η k and Γ ξ k η k are respectively the projections of the point u 0 and of the loop Γ on the plane defined by the axes ξ k and η k , as shown in Fig. 3.

Factorization of polar n-complex polynomials
A polynomial of degree m of the polar n-complex variable u has the form where a l , l = 1, ..., m, are in general polar n-complex constants. If a l = n−1 p=0 h p a lp , and with the notations of Eqs. (45)-(48) applied for l = 1, · · · , m, the polynomial P m (u) can be written as where the constants A l+ , A l− , A lk ,Ã lk are real numbers.

Representation of polar n-complex numbers by irreducible matrices
The polar n-complex number u ca be represented by the matrix The product u = u ′ u ′′ is represented by the matrix multiplication U = U ′ U ′′ . It can be shown that the irreducible form [6] of the matrix U in terms of matrices with real coefficients is, for even n, and, for odd n, where 3 Planar Hypercomplex Numbers in Even n Dimensions

Operations with planar n-complex numbers
A planar hypercomplex number in n dimensions is determined by its n components (x 0 , x 1 , ..., x n−1 ).
The planar n-complex numbers and their operations discussed in this paper can be represented by writing the n-complex number (x 0 , x 1 , ..., x n−1 ) as u = x 0 + h 1 x 1 + h 2 x 2 + · · · + h n−1 x n−1 , where h 1 , h 2 , · · · , h n−1 are bases for which the multiplication rules are for j, k, l = 0, 1, ..., n−1, where h 0 = 1. The rules for the planar bases differ from the rules for the polar bases by the minus sign which appears when n ≤ j + k ≤ 2n − 2. The significance of the composition laws in Eq. (74) can be understood by representing the bases h j , h k by points on a circle at the angles α j = πj/n, α k = πk/n, as shown in Fig. 4, and the product h j h k by the point of the circle at the angle π(j + k)/n. If π ≤ π(j + k)/n < 2π, the point is opposite to the basis h l of angle α l = π(j + k)/n − π.

Two n-complex numbers
The sum of the n-complex numbers u and u ′ is The product of the numbers u, u ′ is The product uu ′ can be written as If u, u ′ , u ′′ are n-complex numbers, the multiplication is associative, (uu ′ )u ′′ = u(u ′ u ′′ ), and commutative, uu ′ = u ′ u , because the product of the bases, defined in Eq. (74), is associative and commutative.
The inverse of the planar n-complex number u is the n-complex number u ′ having the property that uu ′ = 1. This equation has a solution provided that the corresponding determinant ν is not equal to zero, ν = 0. For planar n-complex numbers ν ≥ 0, and the quantity ρ = ν 1/n will be called amplitude of the n-complex number u. It can be shown that where Thus, a planar n-complex number has an inverse unless it lies on one of the nodal hypersurfaces ρ 1 = 0, or ρ 2 = 0, or ... or ρ n/2 = 0. It can also be shown that From this relation it results that if the product of two n-complex numbers is zero, uu ′ = 0, then ρ k ρ ′ k = 0, k = 1, ..., n/2, which means that either u = 0, or u ′ = 0, or u, u ′ belong to orthogonal hypersurfaces in such a way that the afore-mentioned products of components should be equal to zero.

Geometric representation of planar n-complex numbers
The planar n-complex number x 0 + h 1 x 1 + h 2 x 2 + · · · + h n−1 x n−1 can be represented by the point A of coordinates (x 0 , x 1 , ..., x n−1 ). If O is the origin of the n-dimensional space, the distance from the origin O to the point A of coordinates (x 0 , x 1 , ..., x n−1 ) has the expression written in Eq. (12). The quantity d will be called now modulus of the planar n-complex The modulus of an n-complex number u will be designated by d = |u|. The quantity ρ = ν 1/n will be called amplitude of the n-complex number u.
The exponential and trigonometric forms of the n-complex number u can be obtained conveniently in a rotated system of axes defined by the transformation v k = n/2ξ k ,ṽ k = n/2η k , for k = 1, ..., n/2. This transformation from the coordinates x 0 , ..., x n−1 to the variables ξ k , η k is unitary.
The position of the point A of coordinates (x 0 , x 1 , ..., x n−1 ) can be also described with the aid of the distance d, Eq. (12), and of n − 1 angles defined further. Thus, in the plane of the axes v k ,ṽ k , the radius ρ k and the azimuthal angle φ k can be introduced by the relations for 0 ≤ φ k < 2π, k = 1, ..., n/2, so that there are n/2 azimuthal angles. If the projection of the point A on the plane of the axes v k ,ṽ k is A k , and the projection of the point A on the 4-dimensional space defined by the axes v 1 ,ṽ 1 , v k ,ṽ k is A 1k , the angle ψ k−1 between the line OA 1k and the 2-dimensional plane defined by the axes v k ,ṽ k is where 0 ≤ ψ k ≤ π/2, k = 2, ..., n/2, so that there are n/2 − 1 planar angles. Thus, the position of the point A is described by the distance d, by n/2 azimuthal angles and by n/2 − 1 planar angles. The variables ν, ρ, ρ k , tan ψ k are multiplicative and the azimuthal angles φ k are additive upon the multiplication of polar n-complex numbers.

The planar n-dimensional cosexponential functions
The exponential function of the planar n-complex variable u can be defined by the series exp u = 1 + u + u 2 /2! + u 3 /3! + · · · . It can be checked by direct multiplication of the series that exp(u + u ′ ) = exp u · exp u ′ , so that exp u = exp x 0 · exp(h 1 x 1 ) · · · exp(h n−1 x n−1 ).
It can be seen with the aid of the representation in Fig. 4 that for p integer, k = 1, ..., n − 1. For k even, e h k y can be written as where g np are the polar n-dimensional cosexponential functions. For odd k, e h k y is where the functions f nk , which will be called planar cosexponential functions in n dimensions, for k = 0, 1, ..., n − 1.
The planar n-dimensional cosexponential function f nk (y) is related to the polar n-dimensional cosexponential function g nk (y) by the relation f nk (y) = e −iπk/n g nk e iπ/n y , for k = 0, ..., n − 1. The expression of the planar n-dimensional cosexponential functions is then f nk (y) = 1 n n l=1 exp y cos π(2l − 1) n cos y sin which does not contain exponential terms.
For odd n, the planar n-dimensional cosexponential function f nk (y) is related to the n-dimensional cosexponential function g nk (y) also by the relation as can be seen by comparing the series for the two classes of functions.
Addition theorems for the planar n-dimensional cosexponential functions can be obtained from the relation exp h 1 (y + z) = exp h 1 y · exp h 1 z, by substituting the expression of the exponentials as given by e h 1 y = n−1 p=0 h p f np (y), for k = 0, 1, ..., n − 1. It can also be shown that {f n0 (y) + h 1 f n1 (y) + · · · + h n−1 f n,n−1 (y)} l = f n0 (ly)+h 1 f n1 (ly)+· · ·+h n−1 f n,n−1 (ly). (96) The planar n-dimensional cosexponential functions are solutions of the n th -order differential equation whose solutions are of the form ζ(u) = A 0 f n0 (u) + A 1 f n1 (u) + · · · + A n−1 f n,n−1 (u). It can be checked that the derivatives of the planar cosexponential functions are related by For n = 2, the planar cosexponential functions are f 20 (y) = cos y and f 21 (y) = sin y.

Exponential and trigonometric forms of planar n-complex numbers
In order to obtain the exponential and trigonometric forms of planar n-complex numbers, a new set of hypercomplex bases will be introduced by the relations for k = 1, ..., n/2. The multiplication relations for the new hypercomplex bases are e 2 k = e k ,ẽ 2 k = −e k , e kẽk =ẽ k , e k e l = 0, e kẽl = 0,ẽ kẽl = 0, k = l, for k, l = 1, ..., n/2. It can be shown that The exponential form of the planar n-complex number u is where the amplitude is The trigonometric form of the planar n-complex number u is

Elementary functions of a planar n-complex variable
The The function ln u is multivalued because of the presence of the termsẽ k φ k .
The power function u m of the planar n-complex variable u can be defined for real values of m as u m = e m ln u . It can be shown that The power function is multivalued unless m is an integer.

Power series of planar n-complex numbers
A power series of the planar n-complex variable u is a series of the form a 0 + a 1 u + a 2 u 2 + · · · + a l u l + · · · .
Using the inequality which replaces the relation of equality extant for 2-dimensional complex numbers, it can be shown that the series (108) is absolutely convergent for |u| < c, where c = lim l→∞ |a l |/ n/2|a l+1 | .
The convergence of the series (108) can be also studied with the aid of the formulas (107), which for integer values of m are valid for any values of x 0 , ..., x n−1 . If a l = n−1 p=0 h p a lp , and for k = 1, ..., n/2, the series (108) can be written as The series in Eq. (108) is absolutely convergent for for k = 1, ..., n/2, where These relations show that the region of convergence of the series (108) is an n-dimensional cylinder.

Analytic functions of planar n-complex variables
If the n-complex function f (u) of the planar n-complex variable u is written in terms of the real functions P k (x 0 , ..., x n−1 ), k = 0, 1, ..., n − 1 of the real variables x 0 , x 1 , ..., x n−1 as then relations of equality exist between the partial derivatives of the functions P k , for k = 0, 1, ..., n − 1. The relations (116) are analogous to the Riemann relations for the real and imaginary components of a complex function. It can be shown from Eqs. (116) that the components P k fulfil the second-order equations for k, l = 0, 1, ..., n − 1.

Integrals of planar n-complex functions
The singularities of planar n-complex functions arise from terms of the form 1/(u − u 0 ) m , with m > 0. Functions containing such terms are singular not only at u = u 0 , but also at all points of the hypersurfaces passing through u 0 and which are parallel to the nodal hypersurfaces.
The integral of a planar n-complex function between two points A, B along a path situated in a region free of singularities is independent of path, which means that the integral of an analytic function along a loop situated in a region free of singularities is zero, where it is supposed that a surface Σ spanning the closed loop Γ is not intersected by any of the hypersurfaces associated with the singularities of the function f (u). Using the expression, Eq. (115), for f (u) and the fact that du = n−1 k=0 h k dx k , the explicit form of the integral in Eq. (118) is If the functions P k are regular on a surface Σ spanning the loop Γ, the integral along the loop Γ can be transformed in an integral over the surface Σ of terms of the form Since ρ and ln(tan ψ k−1 ) are singlevalued variables, it follows that Γ dρ/ρ = 0, and Γ d(ln tan ψ k−1 ) = 0. On the other hand, since φ k are cyclic variables, they may give contributions to the integral around the closed loop Γ.
If f (u) is an analytic function of a polar n-complex variable which can be expanded in a series which holds on the curve Γ and on a surface spanning Γ, then

Factorization of planar n-complex polynomials
A polynomial of degree m of the planar n-complex variable u has the form where a l , l = 1, ..., m, are in general planar n-complex constants. If a l = n−1 p=0 h p a lp , and with the notations of Eqs. (110)-(111) applied for l = 1, · · · , m, the polynomial P m (u) can be written as where the constants A lk ,Ã lk are real numbers.
These relations can be written with the aid of Eqs. (99) and (100) as where for p = 1, ..., m. For a given k, the roots e k v kp +ẽ kṽkp defined in Eq. (123) may be ordered arbitrarily. This means that Eq. (125) gives sets of m roots u 1 , ..., u m of the polynomial P m (u), corresponding to the various ways in which the roots e k v kp +ẽ kṽkp are ordered according to p for each value of k. Thus, while the planar hypercomplex components in Eq.
(123) taken separately have unique factorizations, the polynomial P n (u) can be written in many different ways as a product of linear factors.
3.10 Representation of planar n-complex numbers by irre-

ducible matrices
The planar n-complex number u ca be represented by the matrix The product u = u ′ u ′′ is, for even n, represented by the matrix multiplication U = U ′ U ′′ . It can be shown that the irreducible form [6] of the matrix U , in terms of matrices with real coefficients, is where for k = 1, ..., n/2. The relations between the variables v k ,ṽ k for the multiplication of planar

Conclusions
The polar and planar n-complex numbers described in this paper have a geometric represen-        Fig. 4