On l_p-Complex Numbers

Abstract

Dispensing with the common property of distributivity and replacing classical trigonometric functions with their $l_p$-counterparts in Euler’s trigonometric representation of complex numbers, classes of $l_p$-complex numbers are introduced and some of their basic properties are proved. The collection of all points that leave the $l_p$-absolute value of each $l_p$-complex number invariant under $l_p$-complex numbers multiplication is shown to be a group of elements that have $l_p$-absolute value one but not the symmetry group.

1. Introduction

The various analytical, geometric and algebraic aspects of a strong mathematical foundation of complex numbers, and their applications to physics and technique, having been a cornerstone in establishing these numbers, can be studied in different levels of exposition in [1,2,3,4,5].

Historically, the theory and applications of mathematics have not always developed in the same rhythm. As an example where complex numbers play a crucial role in technique we recall everyone’s school knowledge that an alternating current electrical generator may operate by turning a rotor within a magnetic stator. For everyday applications the rotor is a thick wound coil and voltage and current are dealt with as the real and imaginary parts of a complex number, respectively, or as the components of a vector that moves within a Euclidean circle in the complex plane with time. This appears to be reasonable because the continuous change of voltage and current can be satisfactorily modeled by sine and cosine functions. But in case the rotor is just a single turn of a wire it is a really nice experience to see that these trigonometric functions do not fit at all. Various types of functions may appear in dependence on how the single turn of a wire is moved within the magnetic stator.

In certain different technical solutions even appear sawtooth curves or rectangular curves, the latter being similar to $l_p$-trigonometric functions for large p, to show the behavior of a two-dimensional vector with time. One may say that in such cases a point moves through a ’generalized circle’ with time.

Applications in quantum mechanics such as for quaternions or octonions can make another application field of $l_p$-complex numbers, but this goes beyond the scope of the present work. This paper demonstrates the mathematical possibilities of using $l_{2,p}$-trigonometric functions under whatever conditions they may be useful.

A system of complex numbers may be considered to be an algebraic structure $(\mathbb{C},\oplus ,\odot )$ where $\mathbb{C}$ is a non-empty set and ⊕ and ⊙ are binary operations acting from $\mathbb{C}\times \mathbb{C}$ to $\mathbb{C}$ in a way such that $(\mathbb{C},\oplus )$ and $(\mathbb{C},\odot )$ are Abelian groups with neutral elements $\mathfrak{o}$ and $\mathfrak{e}$, respectively. Moreover, an element $\mathfrak{i}$ from $\mathbb{C}$ is assumed to satisfy the equality $\mathfrak{i}\odot \mathfrak{i}=-\mathfrak{e}$, and distributivity of the operations ⊕ and ⊙ is assumed to hold.

In the case of the well-known vector and matrix implementations of this structure,

$$\mathbb{C}=\{\left(\begin{array}{c}x\\ y\end{array}\right),x\in \mathbb{R},y\in \mathbb{R}\},\mathfrak{e}=\left(\begin{array}{c}1\\ 0\end{array}\right),\mathfrak{i}=\left(\begin{array}{c}0\\ 1\end{array}\right)$$

and

$$\mathbb{C}=\{\left(\begin{array}{cc}a& -b\\ b& a\end{array}\right),a\in \mathbb{R},b\in \mathbb{R}\},\mathfrak{e}=\left(\begin{array}{cc}1& 0\\ 0& 1\end{array}\right),\mathfrak{i}=\left(\begin{array}{cc}0& -1\\ 1& 0\end{array}\right),$$

$\mathfrak{o}$ is the zero vector or matrix, and common vector or matrix addition ⊕ and geometric vector multiplication or common matrix multiplication ⊙ are respectively assumed to hold. For parameter choice $p=2$, the notion of geometric vector multiplication is a particular case of Definition 1 below.

Classical representation of a complex structure is $\mathbb{C}=\{x+iy:x\in \mathbb{R},y\in \mathbb{R}\}$, combined with the understanding of separate addition of real and imaginary parts, multiplication $(x_1+i{y}_1)(x_2+i{y}_2)=x_1{x}_2-y_1{y}_2+i(x_1{y}_2+x_2{y}_1)$ and equality $\mathfrak{i}\odot \mathfrak{i}=-\mathfrak{e}$ being read as $i^2=-1$. The ’imaginary unit’ i is a symbol whose concrete definition depends on the realization of the complex structure, there is no clearly defined object that i stands for once and for all.

The multiplication allows an interpretation in terms of a geometric vector product. This circumstance motivates the generalization of the vector realization of the classical complex structure presented in the present paper. To this end, however, we must dispense with the distributive property from classical complex systems. On the other hand, the definition of a geometric vector product allows to introduce a new definition of a geometric exponential function being suitable for generalizing Euler’s famous trigonometric representation of complex numbers. In the classical vector realization, $\mathbb{C}$ is endowed with the absolute value function, $|x+iy|={(x^2+y^2)}^{1/2}$, and geometric vector multiplication with a complex number of absolute value one in particular means moving points from a circle $\{z\in \mathbb{C}:|z|=r\}$ without leaving this set, just like orthogonal transformations do. Here, we study the case that due to a different multiplication rule, these circles are replaced with $l_p$-unit circles where p is any positive real number.

We recall that there are several well-known generalizations of complex numbers which found far reaching applications to physics and electrical engineering. Quaternions or Hamilton numbers constitute such system, wherein, however, multiplication is not commutative. Octonions or Cayley numbers are an extension of quaternions. All these numbers can be understood being particular Clifford modules. The general notion of a Clifford algebra allows another representation called spinor which found basic application to the theory of elementary particles. For more details we refer to [6,7,8,9]. Another type of relatives of complex numbers are split-complex or hyperbolic-complex numbers.

The paper is structured as follows. The geometric definition of $l_p$-complex numbers multiplication and its analytical counterpart as well as the classes of $l_p$-complex numbers themselves are introduced and some of their basic properties are considered in Section 2. This section might also serve as a short introduction to ordinary complex numbers just choosing the parameter $p=2$. The focus of Section 3 is on transformations letting the $l_p$-absolute value of an $l_p$-complex number invariant while Section 4 deals with a generalization of Euler’s trigonometric representation. The paper ends with a discussion in Section 5.

2. Definition and Basic Properties of $l_p$-Complex Numbers

Let $\mathbb{R}$ be the real line, $\mathbb{C}=\{{(x,y)}^T:x\in \mathbb{R},y\in \mathbb{R}\}$ the two-dimensional Euclidean space and $z_i={(x_i,y_i)}^T,i=1,2$ two elements from $\mathbb{C}$. Endowed with common vector addition, $z_1\oplus z_2={(x_1+x_2,y_1+y_2)}^T,$ $(\mathbb{C},\oplus )$ is an Abelian group with the neutral element ${(0,0)}^T$ and the additive inverse element of ${(x,y)}^T$ being ${(-x,-y)}^T$.

For each real $p,p>0,\mathbb{C}$ can alternatively be written as

$$\mathbb{C}=\{{(r{cos}_p\phi ,r{sin}_p\phi )}^T:r>0,\phi \in [0,2\pi )\}$$

with the $l_{2,p}$-trigonometric functions being defined according to [10] as ${sin}_p\phi =\frac{sin\phi }{N_p\left(\phi \right)}$ and ${cos}_p\phi =\frac{cos\phi }{N_p\left(\phi \right)}$ where $N_p\left(\phi \right)={\left(\right|sin\phi |}^p+{|cos\phi |}^p{)}^{1/p}.$ If, vice versa, one is given $z={(x,y)}^T$ from $\mathbb{C}$ then, a.e., $r={\left|z\right|}_p={\left(\right|x|}^p+{\left|y\right|}^p{)}^{1/p}$ and $arctan\frac{y}{x}$ = $\phi \mathrm{in}{Q}_1,\pi -\phi \mathrm{in}{Q}_2,\phi -\pi \mathrm{in}{Q}_3,2\pi -\phi \mathrm{in}{Q}_4$ where $Q_1$ up to $Q_4$ denote the quadrants in ${\mathbb{R}}^2$ in the usual anticlockwise ordering.

Definition 1.

Letting $z_i={(x_i,y_i)}^T={(r_i{cos}_p{\phi }_i,r_i{sin}_p{\phi }_i)}^T,i=1,2,$ we define the geometric vector p-multiplication by

$$z_1{\odot }_p{z}_2=r_1{r}_2{({cos}_p({\phi }_1+{\phi }_2),{sin}_p({\phi }_1+{\phi }_2))}^T$$

where the angle ${\phi }_1+{\phi }_2$ is to be chosen modulo $2\pi$.

We recall that the two-dimensional Euclidean space, endowed with the $l_p$-norm $z\to {\left|z\right|}_p,p\ge 1$, is denoted $l_{2,p}$ and that $z\to {\left|z\right|}_p$ is an antinorm if $p\in (0,1]$. For the latter case, we refer to [11].

As a rule, for real $a_1,a_2$, we have

$$\left(a_1{z}_1\right){\odot }_p\left(a_2{z}_2\right)=\left(a_1{a}_2\right)\left(z_1{\odot }_p{z}_2\right).$$

By Definition 1,

$$z_1{\odot }_p{z}_2=\frac{|z_1{|}_p{\left|z_2\right|}_p}{N_p({\phi }_1+{\phi }_2)|z_1{|}_2{\left|z_2\right|}_2}{z}_1{\odot }_2{z}_2,$$

thus

$$N_p({\phi }_1+{\phi }_2)=\frac{|z_1{\odot }_2{z}_2{|}_p}{|z_1{|}_2{\left|z_2\right|}_2},$$

and

$$z_1{\odot }_p{z}_2=\frac{|z_1{|}_p{\left|z_2\right|}_p}{|z_1{\odot }_2{z}_2{|}_p}{z}_1{\odot }_2{z}_2.$$

That is

$$z_1{\odot }_p{z}_2={\left[\frac{\left(\right|x_1{|}^p+|y_1{|}^p\left)\right(|x_2{|}^p+|y_2{|}^p)}{|x_1{x}_2-y_1{y}_2{|}^p+{|x_1{y}_2+x_2{y}_1|}^p}\right]}^{\frac{1}{p}}\left(\begin{array}{c}{x}_1{x}_2-y_1{y}_2\\ x_1{y}_2+x_2{y}_1\end{array}\right).$$

(1)

For each $p>0$, $(\mathbb{C},{\odot }_p)$ is an Abelian group with neutral element ${(1,0)}^T$, and the $l_p$-multiplicative inverse element of ${(x,y)}^T$ is

$$z_{\left(p\right)}^{-1}={\left({\left|x\right|}^p+{\left|y\right|}^p\right)}^{-\frac{2}{p}}{(x,-y)}^T.$$

Definition 2.

We speak of ${\mathbb{C}}_p=(\mathbb{C},\oplus ,{\odot }_p)$ as of the plane of $l_p$-complex numbers, $p>0$, call the multiplication operation ${\odot }_p$ in (1) the $l_p$-complex numbers multiplication, and ${\left|z\right|}_p$ the $l_p$-absolute value of $z={(x,y)}^T$.

Subtraction of $l_p$-complex numbers, $z_1$ minus $z_2$, is defined as adding $z_1$ and the additive inverse element of $z_2$,

$$z_1\ominus z_2=z_1\oplus (-z_2)={(x_1-x_2,y_1-y_2)}^T.$$

Similarly, division of $l_p$-complex numbers, $z_1$ by $z_2$, is defined as multiplication of $z_1$ by the multiplicative inverse element of $z_2$, $z_1{\oslash }_p{z}_2=z_1{\odot }_p{\left(z_2\right)}_{\left(p\right)}^{-1}$, that is

$$\begin{array}{c}\hfill z_1{\oslash }_p{z}_2=\frac{\left(\right|x_1{|}^p+|y_1{{|}^p)}^{1/p}}{\left(\right|x_2{|}^p+|y_2{|}^p{)}^{1/p}·\left(\right|x_1{x}_2+y_1{y}_2{|}^p+|x_2{y}_1-y_2{x}_1{{|}^p)}^{1/p}}\\ \hfill \times {\left(x_1{x}_2+y_1{y}_2,x_2{y}_1-y_2{x}_1\right)}^T.\end{array}$$

(2)

For $z={(x,y)}^T\in \mathbb{C}$, we write alternatively $z=x{(1,0)}^T\oplus y{(0,1)}^T$ or, similarly like common complex numbers,

$$z=x+i_{p}y$$

where $i_p={(0,1)}^T$ is called $l_p$-imaginary unit. A property which share all $l_p$-complex number systems is that

$${(0,1)}^T{\odot }_p{(0,1)}^T={(-1,0)}^T,$$

(3)

or $i_p^2=-1,$ for short. For particular different properties ${(0,1)}^T$ has as an element of different spaces ${\mathbb{C}}_p$, see below. The existence of such properties motivates our notation $i_p$ instead of i, in the present framework.

Let $z_3={(x_3,y_3)}^T$ be another element of $\mathbb{C}.$ With view toward a possible distributivity rule, we note that

$$z_1{\odot }_p(z_2\oplus z_3)=\frac{|z_1{|}_p{|z_2\oplus z_3|}_p}{|z_1{\odot }_2(z_2\oplus z_3){|}_p}(z_1{\odot }_2{z}_2\oplus z_1{\odot }_2{z}_3).$$

(4)

Remark 1.

Unless for $p=2$, distributivity is missing in $(\mathbb{C},\oplus ,{\odot }_p)$.

Proof. 

This is shown by separately evaluating the expressions

$$\begin{array}{cc}\hfill \tilde{z}& =z_1{\odot }_p(z_2\oplus z_3)\hfill \\ =\frac{|z_1{|}_p{|z_2\oplus z_3|}_p}{|z_1{\odot }_2(z_2\oplus z_3){|}_p}(z_1{\odot }_2{z}_2\oplus z_1{\odot }_2{z}_3)\hfill \end{array}$$

and

$$\begin{array}{cc}\hfill \tilde{\tilde{z}}& =z_1{\odot }_p{z}_2\oplus z_1{\odot }_p{z}_3\hfill \\ =\frac{N_p\left({\phi }_1\right)N_p\left({\phi }_2\right)}{N_p({\phi }_1+{\phi }_2)}{z}_1{\odot }_2{z}_2\oplus \frac{N_p\left({\phi }_1\right)N_p\left({\phi }_3\right)}{N_p({\phi }_1+{\phi }_3)}{z}_1{\odot }_2{z}_3\hfill \end{array}$$

and equating them. □

Due to this remark, differently from what is true for complex numbers, ${\mathbb{C}}_p,p\ne 2$ is not a field. In the case $p=2$, the well known formula

$$z_1{\odot }_2(z_2\oplus z_3)=z_1{\odot }_2{z}_2\oplus z_1{\odot }_2{z}_3$$

follows from Equation (4) by choosing $p=2.$

We further remark that the $l_p$-absolute value of $z_1{\odot }_p{z}_2$, respectively the $l_{2,p}$-norm or antinorm of this vector, is $|z_1{|}_p{\left|z_2\right|}_p$ and its direction is the same as that of the common complex numbers product $z_1{\odot }_2{z}_2.$ It follows in particular that

$$\begin{array}{cc}\hfill z_1{\odot }_2{z}_2=& |z_1{|}_2{\left|z_2\right|}_2{(cos({\phi }_1+{\phi }_2),sin({\phi }_1+{\phi }_2))}^T\hfill \\ \hfill =& |z_1{|}_2{\left|z_2\right|}_2{(cos{\phi }_{1}cos{\phi }_2-sin{\phi }_{1}sin{\phi }_2,sin{\phi }_{1}cos{\phi }_2+cos{\phi }_{1}sin{\phi }_2)}^T\hfill \\ \hfill =& (x_1{x}_2-y_1{y}_2){(1,0)}^T\oplus (x_1{y}_2+x_2{y}_1){(0,1)}^T.\hfill \end{array}$$

Remark 2.

The latter relation is commonly written in the literature as

$$(x_1+i{y}_1)(x_2+i{y}_2)=(x_1{x}_2-y_1{y}_2)+i(x_1{y}_2+x_2{y}_1).$$

If $\overline{z}={(x,-y)}^T$ denotes the complex conjugate of $z={(x,y)}^T$ then

$$z{\odot }_p\overline{z}=r_p^2{(1,0)}^T$$

or $z{\odot }_p\overline{z}=r_p^2$, for short. It follows from (1) that

$$z{\odot }_{p}z=\frac{{\left(\right|x|}^p+{\left|y\right|}^p{)}^{2/p}}{\left(\right|x^2-y^2{|}^p+{\left|2xy\right|}^p{)}^{1/p}}\left(\begin{array}{c}{x}^2-y^2\\ 2xy\end{array}\right).$$

3. Invariant Transformations

It is well known that multiplication of elements from a Euclidean space by an orthogonal matrix leaves the Euclidean norm of the space elements invariant. Clockwise rotation of an element of $\mathbb{C}$ through an angle of magnitude $\alpha$ is described by multiplying it by the matrix

$$O^{+}\left(\alpha \right)=\left(\begin{array}{cc}cos\alpha & sin\alpha \\ -sin\alpha & cos\alpha \end{array}\right)$$

and can be expressed in terms of $l_2$-complex multiplication as

$$O^{+}\left(\alpha \right)\left[r\left(\begin{array}{c}cos\phi \\ sin\phi \end{array}\right)\right]=\left[r\left(\begin{array}{c}cos\phi \\ sin\phi \end{array}\right)\right]{\odot }_2\left(\begin{array}{c}cos(-\alpha )\\ sin(-\alpha )\end{array}\right)$$

(5)

where ${\left(cos(-\alpha ),sin(-\alpha )\right)}^T$ is an arbitrary Euclidean unit vector.

Theorem 1.

The $l_p$-absolute value of an $l_p$-complex number is not changed when the latter is $l_p$-complex multiplied by an $l_p$-complex number with the $l_p$-absolute value one.

Proof. 

By

$$\left[r\left(\begin{array}{c}{cos}_p\phi \\ {sin}_p\phi \end{array}\right)\right]{\odot }_p\left(\begin{array}{c}{cos}_p(-\alpha )\\ {sin}_p(-\alpha )\end{array}\right)=r\left(\begin{array}{c}{cos}_p(\phi -\alpha )\\ {sin}_p(\phi -\alpha )\end{array}\right)$$

where

$$|r\left(\begin{array}{c}{cos}_p\left(\phi \right)\\ {sin}_p\left(\phi \right)\end{array}\right){|}_p=r{|\left(\begin{array}{c}{cos}_p(\phi -\alpha )\\ {sin}_p(\phi -\alpha )\end{array}\right)|}_p=r>0$$

the statement of the theorem is proven. □

Remark 3.

The collection of $l_p$-complex numbers with the $l_p$-absolute value one is a group with respect to $l_p$-complex numbers multiplication which is, however, generally not the symmetry group.

4. Generalizing Euler’s Trigonometric Representation

Definition 3.

The k’th geometric power of vector $z={(x,y)}^T\in \mathbb{C}$ and the geometric exponential of z are defined by

$$z^k=z^{k-1}{\odot }_{2}z,k\ge 2,z^1=z,z^0={(1,0)}^{T}andexp\left(z\right)=\sum _{k=0}^{\infty }\frac{z^k}{k!},$$

respectively.

Since

$${\left(\begin{array}{c}0\\ 1\end{array}\right)}^{2k}={(-1)}^k\left(\begin{array}{c}1\\ 0\end{array}\right)\mathrm{and}{\left(\begin{array}{c}0\\ 1\end{array}\right)}^{2k+1}={(-1)}^k\left(\begin{array}{c}0\\ 1\end{array}\right)$$

(6)

it follows from the expansions

$$cosx\left(\begin{array}{c}1\\ 0\end{array}\right)=\left(\begin{array}{c}1\\ 0\end{array}\right)+\frac{x^2}{2!}{\left(\begin{array}{c}0\\ 1\end{array}\right)}^2+\frac{x^4}{4!}{\left(\begin{array}{c}0\\ 1\end{array}\right)}^4+\dots$$

and

$$sinx\left(\begin{array}{c}0\\ 1\end{array}\right)=x\left(\begin{array}{c}0\\ 1\end{array}\right)+\frac{x^3}{3!}{\left(\begin{array}{c}0\\ 1\end{array}\right)}^3+\frac{x^5}{5!}{\left(\begin{array}{c}0\\ 1\end{array}\right)}^5+\dots$$

that

$$cosx\left(\begin{array}{c}1\\ 0\end{array}\right)+sinx\left(\begin{array}{c}0\\ 1\end{array}\right)=\sum _{k=0}^{\infty }\frac{x^k}{k!}{\left(\begin{array}{c}0\\ 1\end{array}\right)}^k=exp\left(x\left(\begin{array}{c}0\\ 1\end{array}\right)\right).$$

The latter expression is written $e^{ix}$, for short. Therefore, the point $e^{ix}$ from ${\mathbb{C}}_2$, respectively the function $x\to e^{ix}$ satisfies the equalities

$$e^{ix}=cosx\left(\begin{array}{c}1\\ 0\end{array}\right)\oplus sinx\left(\begin{array}{c}0\\ 1\end{array}\right),x\in \mathbb{R}$$

and

$$cosx\left(\begin{array}{c}1\\ 0\end{array}\right)={\mathsf{\Pi }}_{\mathfrak{L}\left(\left\{{(1,0)}^T\right\}\right)}{e}^{ix}\mathrm{as}\mathrm{well}\mathrm{as}sinx\left(\begin{array}{c}0\\ 1\end{array}\right)={\mathsf{\Pi }}_{\mathfrak{L}\left(\left\{{(0,1)}^T\right\}\right)}{e}^{ix}$$

where ${\mathsf{\Pi }}_{\mathfrak{L}\left(\mathfrak{z}\right)}\mathfrak{y}$ means orthogonal projection of $\mathfrak{y}$ onto the linear space $\mathfrak{L}\left(\mathfrak{z}\right)$ spanned by $\mathfrak{z}\in \mathbb{C}$. One can further check that

$$e^{i(x+y)}=e^{ix}{\odot }_2{e}^{iy}$$

and

$$|e^{i\phi }{|}_p=N_p\left(\phi \right).$$

Remark 4.

Instead of writing the penultimate formula as we did, you usually write

$$e^{ix}=cosx+isinx\mathrm{and}{e}^{i(x+y)}=e^{ix}·e^{iy}.$$

The usual casual notation in these formulas, however, might not always have revealed the formulas’ actual meaning to the non-professional reader in a reliable way, in the past. Some philosophical misinterpretation of what the imaginary unit i is might have had its origin at this point.

Let $C\left(p\right)=\{{(x,y)}^T\in \mathbb{C}:|{(x,y)}^T{|}_p=1\},p>0$ denote the $l_{2,p}$-unit circle.

Definition 4.

The quantity $e^{i_{p}x}$ is defined as the central projection of the point $e^{ix}$ from the common circle $C\left(2\right)$ onto the $l_{2,p}$-circle $C\left(p\right)$.

According to this definition, the following generalization of Euler’s trigonometric representation of complex numbers is true:

$$e^{i_{p}x}=\frac{1}{|e^{ix}{|}_p}{e}^{ix}={cos}_p\left(x\right){(1,0)}^T\oplus {sin}_p\left(x\right){(0,1)}^T.$$

(7)

Theorem 2.

For arbitrary x and y from $\mathbb{R},$

$$e^{i_p(x+y)}=e^{i_{p}x}{\odot }_p{e}^{i_{p}y}.$$

Proof. 

On combining Definition 4 and Remark 4, we get

$$e^{i_p(x+y)}=\frac{1}{|e^{i(x+y)}{|}_p}{e}^{i(x+y)}={\left({cos}_p(x+y),{sin}_p(x+y)\right)}^T$$

from where it follows by Definition 1 and once again Definition 4 that

$$e^{i_p(x+y)}={\left({cos}_p\left(x\right),{sin}_p\left(x\right)\right)}^T{\odot }_p{\left({cos}_p\left(y\right),{sin}_p\left(y\right)\right)}^T=e^{i_{p}x}{\odot }_p{e}^{i_{p}y}$$

 □

In addition to the geometric approach to the quantity $e^{i_{p}x}$ in Definition 4, we now proceed to an analogy with a classical analytical method that uses a suitably adapted series expansion.

Definition 5.

The k’th $l_p$-geometric vector power of $z\in {\mathbb{C}}_p$ is defined by

$$z^{\left(0\right|p)}=\left(\begin{array}{c}1\\ 0\end{array}\right),z^{\left(k\right|p)}=z^{(k-1|p)}{\odot }_{p}z,k=1,2,\dots$$

With this notation, it follows from Theorem 2 that, for $z=r_p{({cos}_p\left(\phi \right),{sin}_p\left(\phi \right))}^T$,

$$z^{\left(n\right|p)}=r_p^n\left(\begin{array}{c}{cos}_p\left(n\phi \right)\\ {sin}_p\left(n\phi \right)\end{array}\right).$$

Sure, the k’th geometric vector power according to Definition 3 is just the k’th $l_2$-geometric vector power.

Definition 6.

The $l_p$-complex exponential function ${exp}_{\left(p\right)}:\mathbb{R}\to C\left(p\right)$ is defined by the $l_p$-geometric power series

$$ex{p}_{\left(p\right)}\left(x\right)=\sum _{k=0}^{\infty }\frac{x^k}{k!}{\left(\begin{array}{c}0\\ 1\end{array}\right)}^{\left(k\right|p)}.$$

Theorem 3.

The $l_p$-complex exponential function and the $l_p$-imaginary unit satisfy the equality $ex{p}_{\left(p\right)}\left(x\right)=N_p\left(x\right)e^{i_{p}x}$.

Proof. 

Because of the equations $N_p\left(0\right)=N_p(\pi /2)=N_p\left(\pi \right)=1$, we have that

$${\left(\begin{array}{c}1\\ 0\end{array}\right)}^{\left(2\right|p)}=\left(\begin{array}{c}{cos}_p\left(0\right)\\ {sin}_p\left(0\right)\end{array}\right){\odot }_p\left(\begin{array}{c}{cos}_p\left(0\right)\\ {sin}_p\left(0\right)\end{array}\right)=\left(\begin{array}{c}{cos}_p\left(0\right)\\ {sin}_p\left(0\right)\end{array}\right)=\left(\begin{array}{c}1\\ 0\end{array}\right)$$

and, similarly,

$${\left(\begin{array}{c}0\\ 1\end{array}\right)}^{\left(2\right|p)}=-\left(\begin{array}{c}1\\ 0\end{array}\right)\mathrm{as}\mathrm{well}\mathrm{as}\left(\begin{array}{c}1\\ 0\end{array}\right){\odot }_p\left(\begin{array}{c}0\\ 1\end{array}\right)=\left(\begin{array}{c}0\\ 1\end{array}\right).$$

Thus, Equation (6) hold still true if the k’th geometric vector power of z is replaced there with the k’th $l_p$-geometric vector power of z. It follows that

$$ex{p}_{\left(p\right)}\left(x\right)=\sum _{k=0}^{\infty }\frac{x^{2k}}{\left(2k\right)!}{\left(\begin{array}{c}0\\ 1\end{array}\right)}^{\left(2k\right|p)}+\sum _{k=0}^{\infty }\frac{x^{2k+1}}{(2k+1)!}{\left(\begin{array}{c}0\\ 1\end{array}\right)}^{\left(2k+1\right|p)},$$

hence,

$$N_p^{-1}\left(x\right)ex{p}_{\left(p\right)}\left(x\right)={cos}_p\left(x\right)\left(\begin{array}{c}1\\ 0\end{array}\right)\oplus {sin}_p\left(x\right)\left(\begin{array}{c}0\\ 1\end{array}\right).$$

The proof is completed by applying equality (7). □

Finally, we remark that the $l_p$-complex valued function $\gamma \left(x\right)=\left(ex{p}_{\left(p\right)}\left(x\right)\right)/N_p\left(x\right)$ has the properties $\gamma (x+y)=\gamma \left(x\right)\gamma \left(y\right)$ and $\gamma (-x)=\gamma \left(x\right)$ and is therefore a character of the set of real numbers, see for example (Definition 1.4.5. in [12]). Sure, $\gamma \left(0\right)=1,\left|\gamma \right(x\left)\right|=1,\forall x\in \mathbb{R}.$

5. Discussion

In a nutshell, we have introduced the plane of $l_p$-complex numbers, ${\mathbb{C}}_p=(\mathbb{C},\oplus ,{\odot }_p)$ where ⊕ means common vector addition in $\mathbb{C}$ and $l_p$-complex multiplication ${\odot }_p$ is defined by (1). In this setting, division of $l_p$-complex numbers is given by (2). The formula stated in Definition 1 may be considered to be a geometric interpretation of (1). We emphasize that although $i_p={(0,1)}^T$ looks the same for all $p>0$, and $i_p^2=-1$ for all $p>0$ according to (3), the $l_p$-imaginary unit $i_p$ has particularly different properties in dependence of which value p actually attains as is shown in Theorems 1–3.

If someone’s understanding of a set of numbers includes the idea of an ordered set then this approach fails when complex or $l_p$-complex numbers are under consideration. From such a point of view one could deny even using the notion of number for what we call complex or $l_p$-complex numbers because these numbers bring an additional dimension into play. Mainly having ideas of what properties a physical or technical object has along the two axes of the coordinate system leaves it open what the basic properties of the underlying two-dimensional space as a whole are. If the unit circle of this space is an $l_{2,p}$-circle then $l_p$-complex numbers may apply to making non-real (imaginary) calculations. Moreover, if the description of a physical or technical or whatever system makes it necessary to consider points on higher-dimensional (generalized) spheres then it may be of interest to extend the present work to such situation.