More Symmetrical Solutions of Quadratic Equations: The Accompanying Variables Method

Abstract

For polynomials of degree two that have no zeros, the method of accompanying variables is developed and zeros of associated vector polynomials are determined. Our flexible method uses a wide variety of possible vector-valued vector squaring methods and can be adopted to a wide variety of application situations. Known solutions are made much more precise by replacing the imaginary component and supplemented by introducing a whole class of new symmetric vector solutions. Circular, generalized circular and hyperbolic solutions are considered. As an application, we describe the way from considering complex signals to considering vector signals. Anyone who follows the approach of this work and considers equations of third or higher degree will come across further conclusions for the imaginary numbers usually used there.

1. Introduction

In his dissertation [1], C.F. Gauss proved that “Every Algebraic Rational Integral Function In One Variable can be Resolved into Real Factors of the First or the Second Degree”. It is well known when a real polynomial of degree two has no zeros. For such situations, the method of accompanying variables is developed in the present work.

In his work [2], Hamilton represented complex numbers as ordered pairs of real numbers and considered the real number $-1$ as the square of the particular pair $(0,1)$. He also considered quadratic equations in higher-dimensional situations, particularly for quaternions.

Gu [3] considered general polynomial equations and discussed, in particular, cubic, quartic and quintic equations. Öztürk [4] studied quadratic forms and their number systems, focussing in particular on quaternions, split quaternions and hybrid numbers.

In their paper [5], Siwach and Malhotra proposed a new numerical method to find complex and real roots of nonlinear equations, while Ahrend [6] deals with fast high-precision computation of complex square roots. A stochastic approach to solving quadratic equations is dealt with by Haldar, Chakraborty and Chaudhary [7].

In the present approach, two-dimensional polynomials are assigned to a univariate polynomial in an infinite number of possible ways and their vector zeros are described. The choice of accompanying variables can be made depending on an application problem at hand.

The use of so-called imaginary and thus formed complex numbers is avoided because there is no mathematical definition of these quantities that meets the strongest standards of mathematical rigor. However, there is a close connection between complex numbers and pairs of accompanying variables which have their values on Euclidean circles, as occurs, for example, with normalized pairs of sine and cosine signals in electrical engineering.

When signals in time follow generalized or modified sine and cosine curves as in medicine and various areas of technique, the introduction of accompanying variables, which take their values at a fixed time on generalized or modified circles, respectively, becomes interesting. We assume that the generalized circles considered enclose discs that are star-shaped sets and are generated by positively homogeneous functionals.

These functionals also serve to define a variety of vector-valued vector products. All these vector multiplications allow specific definitions of squaring in $R^2$. In this way, a large class of vector valued quadratic polynomials will be introduced whose zeros are subsequently studied. Known solutions are made much more precise by replacing the imaginary component and supplemented by introducing a whole class of new symmetric vector solutions.

The history of the d’Alembert–Gauss theorem begins as early as the 17th century and is described in detail in [8]. Even an extension of this theorem to quaternions dates back to the early middle of the 20th century, [9]. For a long time already, one might have therefore the impression that the theory of the fundamental theorem of algebra can be considered to be largely rounded off. While the present work approaches the sub-topic of quadratic equations from a new perspective, the search in [10] by Novikas and Macys for effective proofs of the theorem itself has also not yet come to an end. A link to the English translation of [1] can also be found there.

Thanks to the vector representations of all newly derived solutions of quadratic equations in this work, their evident symmetry properties are particularly well interpretable from the standpoint of vector analysis.

2. Ordinary Observations

This section deals with the real-valued function with real coefficients,

$$f\left(x\right)=x^2+px+q,x\in R,$$

which is strictly positive if

$$q-{\displaystyle \frac{p^2}{4}}>0.$$

(1)

In this case, the equation

$$f\left(x\right)=0$$

(2)

has no solution. In case

$${\displaystyle \frac{p^2}{4}}-q\ge 0,$$

(3)

Equation (2) has the well-known solutions

$$x_{1/2}=-{\displaystyle \frac{p}{2}}\stackrel{+}{-}\sqrt{{\displaystyle \frac{p^2}{4}}-q},$$

(4)

and for $x\in (-\infty ,x_2)\cup (x_1,\infty )$ holds $f\left(x\right)>0$ and for $x\in (x_1,x_2)$ inequality $f\left(x\right)<0$ holds true.

The following section is mainly devoted to the case where condition (1) is fulfilled.

3. Circular Accompanying Variables

We now assume that there is a real variable y accompanying the real variable x, possibly hidden, and consider the vector-valued function with real coefficients and the vector $\mathbf{1}=\left(\begin{array}{c}1\\ 0\end{array}\right)\in R^2,$

$$f_{⊛}\left(z\right)=z^{⊛2}\oplus p·z\oplus q·\mathbf{1},z=\left(\begin{array}{c}x\\ y\end{array}\right)\in R^2$$

where $p·z=\left(\begin{array}{c}px\\ py\end{array}\right)$ means multiplication of vector z by scalar $p\in R,$$\left(\begin{array}{c}{x}_1\\ y_1\end{array}\right)\oplus \left(\begin{array}{c}{x}_2\\ y_2\end{array}\right)=\left(\begin{array}{c}{x}_1+x_2\\ y_1+y_2\end{array}\right)$ means common vector addition and $z^{⊛2}=z⊛z$ denotes circular squaring of $z\in R^2$. Here, the circular vector-valued vector multiplication is defined according to [11] by

$$\left(\begin{array}{c}{x}_1\\ y_1\end{array}\right)⊛\left(\begin{array}{c}{x}_2\\ y_2\end{array}\right)=\left(\begin{array}{c}{x}_1{x}_2-y_1{y}_2\\ x_1{y}_2+x_2{y}_1\end{array}\right).$$

(5)

Obviously,

$$\mathbf{1}⊛z=z,z\in R^2.$$

Let $C\left(r\right)=\{\left(\begin{array}{c}x\\ y\end{array}\right)\in R^2:x^2+y^2=r^2\},r>0$ denote the Euclidean circle of radius r and let $z_2$ be an arbitrary element of $C\left(1\right)$. For $z_1\in C\left(r_1\right)$ then holds $z_1⊛z_2\in C\left(r_1\right)$; that is, $z_2$ moves $z_1$ on the Euclidean circle $C\left(r_1\right)$. Such a movement is complemented by a change from the circle $C\left(r_1\right)$ to the circle $C\left(r_1{r}_2\right)$ if the element from $C\left(1\right)$ is additionally multiplied by $r_2>0$ so that $z_2\in C\left(r_2\right),r_2>0.$ In particular, with $z=\left(\begin{array}{c}x\\ y\end{array}\right),$

$$z^{⊛2}=z⊛z=\left(\begin{array}{c}{x}^2-y^2\\ 2xy\end{array}\right).$$

(6)

The operations ⊕ and ⊛ are distributive. For the sake of simplicity, let $\lambda ·\mathbf{1}=\lambda \mathbf{1}\mathrm{and}(-1)·\mathbf{1}=-\mathbf{1}$. Obviously,

$${\mathbf{I}}^{⊛2}=-\mathbf{1},$$

(7)

where

$$\mathbf{I}=\left(\begin{array}{c}0\\ 1\end{array}\right)\in R^2$$

(8)

also has the property of causing an orthogonal rotation,

$$\left(\begin{array}{c}x\\ y\end{array}\right)⊛\mathbf{I}=\left(\begin{array}{c}-y\\ x\end{array}\right).$$

Let us now consider the accompanying vector equation

$$f_{⊛}\left(z\right)=\mathfrak{o},$$

(9)

where $\mathfrak{o}=\left(\begin{array}{c}0\\ 0\end{array}\right).$

Theorem 1.

(a) If condition (1) is fulfilled then the solution of Equation (9) is

$$z_{1/2}=-{\displaystyle \frac{p}{2}}\mathbf{1}\stackrel{+}{-}\sqrt{q-{\displaystyle \frac{p^2}{4}}}\mathit{I}.$$

(10)

(b) If condition (3) is fulfilled then with $x_{1/2}$ from (4) the solution of (9) is

$$z_{1/2}=\left(\begin{array}{c}{x}_{1/2}\\ 0\end{array}\right).$$

(11)

Proof. 

First, the vector Equation (9), that is

$$\left(\begin{array}{c}{x}^2-y^2\\ 2xy\end{array}\right)+p\left(\begin{array}{c}x\\ y\end{array}\right)+q\left(\begin{array}{c}1\\ 0\end{array}\right)=\left(\begin{array}{c}0\\ 0\end{array}\right),$$

is written as the system of univariate equations

$$x^2-y^2+px+q=0$$

(12)

and

$$2xy+py=0.$$

(13)

In the case $y=0$, Equation (13) is trivially satisfied and Equation (12) agrees with Equation (2), which under assumption (3), has the solutions $x_{1/2}$ out of (4), but under assumption (1), has none. This proves (b).

If $y\ne 0$, Equation (13) means $x=-{\displaystyle \frac{p}{2}}$ and Equation (12) reduces to $y^2=q-{\displaystyle \frac{p^2}{4}}$. Under assumption (3), Equation (12) has no solution, but under (1) holds $y=\stackrel{+}{-}\sqrt{q-{\displaystyle \frac{p^2}{4}}}$, so Equation (9) has the solutions out of (10).

In summary, Equation (9) under assumption (3) has the solutions (11) and under assumption (1) the solutions (10). □

Remark 1.

It is essential for the understanding of this work that one must not “identify” the vector in (11) with the scalar in (4). Also, the product ⊛, borrowed from the current theory of complex numbers, is consistently used here only for vectors from $R^2$.

In the following theorem we give a geometric description of the solution of Equation (9) that is equivalent to the anlytical description in Theorem 1.

Theorem 2.

If condition (1) is met, the solutions of the vector Equation (9) can be represented geometrically after the polar coordinates transformation $x=rcos\phi ,y=rsin\phi$ as follows:

$$r=\sqrt{q}\mathit{and}cos\phi =-{\displaystyle \frac{p}{2\sqrt{q}}},sin\phi =\stackrel{+}{-}\sqrt{1-{\displaystyle \frac{p^2}{4q}}}.$$

(14)

Proof. 

The Equation (9) can be written in the form of the following two equations:

$$r^2({cos}^2\phi -{sin}^2\phi )+prcos\phi +q=0$$

and

$$2r^{2}cos\phi sin\phi +prsin\phi =0,$$

or

$$r^2(2{cos}^2\phi -1)+prcos\phi +q=0$$

and

$$rsin\phi (2rcos\phi +p)=0.$$

Except for $r=0$ or $\phi =k\pi ,k\in \{0,\stackrel{+}{-}1,\stackrel{+}{-}2,...\},$ the last equation is satisfied for $rcos\phi =-{\displaystyle \frac{p}{2}}.$ The previous equation is then

$$2{\left({\displaystyle \frac{p}{2}}\right)}^2-r^2+p(-{\displaystyle \frac{p}{2}})+q=0$$

or $r^2=q.$ □

Remark 2.

The solution of Equation (9), that is the variable of interest x and the variable y, which is assumed to exist, jointly belong to the circle $C\left(\sqrt{q}\right).$ In other words, the vector ${(x,y)}^T$ of accompanying variables takes values in $C\left(\sqrt{q}\right)$.

4. Accompanying Variables in Generalized Circles with Respect to phs-Functionals

Let $(R^2,\oplus ,·)$ be the two-dimensional vector space of columns ${(x,y)}^T$ of real numbers where $\oplus :R^2\times R^2\to R^2$ and $·:R\times R^2\to R^2$ still denote the usual componentwise vector addition and scalar multiplication, respectively. The vector $\mathfrak{o}={(0,0)}^T$ is the additive neutral element of this space. Suppose that $\left|\right|.\left|\right|:R^2\to [0,\infty )$ is a positively homogeneous functional such that the set $B=\{z\in R^2:|\left|z\right||\le 1\}$ is star-shaped with respect to the inner point $\mathfrak{o}$. Such a functional is called a phs-functional. A particular element of this class of functionals can be a norm or an antinorm as defined in [12]. Furthermore, we call B the unit disc with respect to the functional $\left|\right|.\left|\right|$ and its boundary

$$S=\{z\in R^2:|\left|z\right||=1\}$$

the corresponding generalized unit circle.

The variety of accompanying variables made possible in the present work is essentially based on the introduction of numerous vector-valued vector products such as in [11,13]. The following definition of a generalized circular vector-valued vector product is taken from the literature.

Definition 1.

The vector-valued product of the vectors $z_l={(x_l,y_l)}^T,l=1,2$ from $R^2$ with respect to the phs-functional $\left|\right|.\left|\right|$ is defined by

$$z_1{\circ }_{\left|\right|.\left|\right|}{z}_2={\displaystyle \frac{\left|\right|z_1\left|\right|·\left|\right|z_2\left|\right|}{\left|\right|z_1⊛z_2\left|\right|}}·z_1⊛z_2$$

(15)

and abbreviated as $z_1{\circ }_{\left|\right|.\left|\right|}{z}_2=z_1\circ z_2$.

Table 1 provides an overview of several symbols used in this work and the places where they first appear.

Table 1. Vector-valued operations.

1	Third line of Section 3
⊕	Fifth line of Section 3
·	Fifth line of Section 3
⊛	(5)
I	(8)
o	(9)
${\circ }_{\left|\right|.\left|\right|}$	(15)
∘	Definition 1
ⓗ	(31)

Note that if the functional $\left|\right|.\left|\right|$ denotes the Euclidean norm ${|.|}_2$, then the multiplication operation ∘ agrees with the operation ⊛ and is called the circular vector multiplication or the vector-valued vector multiplication with respect to the functional ${|.|}_2,\mathrm{thus}{\circ }_{{|.|}_2}=⊛$. In addition, the product with respect to the arbitrary phs-functional $\left|\right|.\left|\right|$, ∘, is commutative and associative. Further, vector addition ⊕ and vector multiplication ∘ are not distributive, in general, but a weak distributivity property in the sense of Hankel [14] is valid. Moreover,

$$z_1\circ z_2=\mathfrak{o}\mathrm{if}\mathrm{and}\mathrm{only}\mathrm{if}{z}_1=\mathfrak{o}\mathrm{or}{z}_2=\mathfrak{o},$$

$${\displaystyle \frac{z_1}{\left|\right|z_1\left|\right|}}\circ {\displaystyle \frac{z_2}{\left|\right|z_2\left|\right|}}\in S\mathrm{if}{z}_1\ne \mathfrak{o}\mathrm{and}{z}_2\ne \mathfrak{o}$$

as well as

$$\left(\lambda z_1\right)\circ \left(\mu z_2\right)=\left(\lambda \mu \right)z_1\circ z_2\mathrm{for}\mathrm{all}{z}_1\mathrm{and}{z}_2\mathrm{from}{R}^2\mathrm{and}\mathrm{real}\mathrm{numbers}\lambda \mathrm{and}\mu .$$

Using the notations

$$z^{\circ 2}=z\circ z$$

for general vector-valued squaring of vector z, and

$$Q\left(z\right)={\displaystyle \frac{{\left|\right|z\left|\right|}^2}{\left|\right|z^{⊛2}\left|\right|}},$$

we now consider the vector-valued vector polynomial function

$$\begin{array}{cc}\hfill f\left(z\right)& =z^{\circ 2}+p·z+q·\mathbf{1}\hfill \\ & =Q\left(z\right)z^{⊛2}+p·z+q·\mathbf{1}\hfill \end{array}$$

and look for its vector-valued zeros that is for all $z={(x,y)}^T\in R^2$ satisfying equation

$$f\left(z\right)=\mathfrak{o}.$$

(16)

If we would introduce the notation $f=f_{\circ }$ and if ∘ would denote the product with respect to the Euclidean norm ${|.|}_2$ then we would have $f=f_{⊛}$.

The following theorem follows on from the statement in Theorem 2 and contains a first statement about where solutions of (16) are located. For the rest of this section, let

$$Q\left(z\right)\ge {\displaystyle \frac{p^2}{4q}}.$$

Theorem 3.

(a) Every solution $z={(x,y)}^T$ of Equation (16) satisfying $y\ne 0$ belongs to the circle

$$\{{(x,y)}^T\in R^2:({(x+{\displaystyle \frac{q}{p}})}^2+y^2={({\displaystyle \frac{q}{p}})}^2\}$$

(17)

and satisfies the equations

$$x=-{\displaystyle \frac{p}{2Q\left(z\right)}}\mathrm{and}{y}^2={\displaystyle \frac{q}{Q\left(z\right)}}-{({\displaystyle \frac{p}{2Q\left(z\right)}})}^2.$$

(18)

(b) If condition (3) is satisfied then every solution of (16) with $y=0$ can be represented as $z=\left(\begin{array}{c}x\\ 0\end{array}\right)$ where x solves (2) according to (4).

Proof. 

Equation (16) can be written as the equation system

$$Q\left(z\right)(x^2-y^2)+px+q=0,$$

(19)

$$2Q\left(z\right)xy+py=0.$$

(20)

Let $y\ne 0$. By the second equation,

$$x=-{\displaystyle \frac{p}{2Q\left(z\right)}}.$$

With the first equation this leads to

$$y^2={\displaystyle \frac{q}{Q\left(z\right)}}-{({\displaystyle \frac{p}{2Q\left(z\right)}})}^2$$

from which finally follows the first statement. The validity of the second one can be seen as follows. Every phs-functional $\left|\right|.\left|\right|$ satisfies

$$Q(\left(\begin{array}{c}x\\ 0\end{array}\right))={\displaystyle \frac{\left|\right|\left(\begin{array}{c}x\\ 0\end{array}\right){\left|\right|}^2}{\left|\right|\left(\begin{array}{c}{x}^2\\ 0\end{array}\right)\left|\right|}}=1,$$

thus

$${\left(\begin{array}{c}x\\ 0\end{array}\right)}^{\circ 2}+p\left(\begin{array}{c}x\\ 0\end{array}\right)+q\mathbf{1}=\left(\begin{array}{c}{x}^2+px+q\\ 0\end{array}\right)=\left(\begin{array}{c}0\\ 0\end{array}\right).$$

An application of Theorem 1(b) completes the proof. □

To move to a geometric representation of the solution of Equation (16), let

$$N\left(\phi \right)=\left|\right|\left(\begin{array}{c}cos\phi \\ sin\phi \end{array}\right)\left|\right|\mathrm{and}{Q}^{⊛}\left(\phi \right)={\displaystyle \frac{N^2\left(\phi \right)}{N\left(2\phi \right)}}.$$

Theorem 4.

In terms of usual polar coordinates r and φ, $\phi \notin \{0,\pi \}$, every solution $z={(x,y)}^T$ of the vector Equation (16) satisfies the following two equations:

$${\displaystyle \frac{N^2\left(\phi \right){cos}^2\phi }{N\left(2\phi \right)}}={\displaystyle \frac{p^2}{4q}}$$

(21)

and

$$r={\displaystyle \frac{\sqrt{qN\left(2\phi \right)}}{N\left(\phi \right)}}.$$

(22)

Proof. 

First, it should be noted that under polar coordinate transformation $Q\left(z\right)$ becomes

$${\displaystyle \frac{\left|\right|\left(\begin{array}{c}cos\phi \\ sin\phi \end{array}\right){\left|\right|}^2}{\left|\right|\left(\begin{array}{c}cos\phi \\ sin\phi \end{array}\right)⊛\left(\begin{array}{c}cos\phi \\ sin\phi \end{array}\right)\left|\right|}}=Q^{⊛}\left(\phi \right).$$

Now, the vector Equation (16) is written as

$$Q^{⊛}\left(\phi \right)r^2\left(\begin{array}{c}cos2\phi \\ sin2\phi \end{array}\right)+pr\left(\begin{array}{c}cos\phi \\ sin\phi \end{array}\right)+q\left(\begin{array}{c}1\\ 0\end{array}\right)=\left(\begin{array}{c}0\\ 0\end{array}\right),$$

or as the two individual equations

$$Q^{⊛}\left(\phi \right)r^2(2{cos}^2\phi -1)+prcos\phi +q=0$$

(23)

and

$$(2Q^{⊛}\left(\phi \right)r^{2}cos\phi +pr)sin\phi =0.$$

(24)

The first factor in (24) equals zero if

$$r{Q}^{⊛}\left(\phi \right)cos\phi =-{\displaystyle \frac{p}{2}}$$

or

$$r^2={\displaystyle \frac{p^2}{4{cos}^2\phi Q^{⊛2}\left(\phi \right)}}.$$

(25)

This inserted into (23) gives

$$q=r^2{Q}^{⊛}\left(\phi \right)={\displaystyle \frac{p^2}{4{cos}^2\phi Q^{⊛}\left(\phi \right)}}$$

or

$${cos}^2\phi Q^{⊛}\left(\phi \right)={\displaystyle \frac{p^2}{4q}}$$

(26)

from where the result follows. □

Remark 3.

If one considers the particular Equation (9) then the two conditions (21) and (22) reduce to (14).

5. From Complex-Valued Signals to Vector Signals

In many applied fields one is interested in signals that are functions of time. Trigonometric functions like cosine and sine are suitable to model most simple periodic signals. Let $x=x\left(t\right)=cost,t\in T$ be a signal that evolves over time, and may we be interested in knowing which values of x satisfy the equation

$$x^2\left(t\right)+px\left(t\right)+q=0.$$

(27)

According to the ordinary observations in Section 2, Equation (27) has no solution if $q>{\displaystyle \frac{p^2}{4}}$. In this case, following Gabor’s [15] classical heuristic or pragmatic approach, a so-called complex-valued signal $z\left(t\right)$ may be built from the real signal $x\left(t\right)$ by combining the latter with a phase-shifted version of it, say $z\left(t\right)=x\left(t\right)+i·y\left(t\right)=cost+i·sint.$ In this case,

$$x^2\left(t\right)+y^2\left(t\right)=1,$$

(28)

and the two-dimensional signal vector is located in an Euclidean circle at all times.

The formal or symbolic quantity i represents the so-called imaginary unit, of which it is only known that it is not a real number and that it allows squaring of a kind such that $i^2=-1$. It is then stated, purely heuristically or pragmatically, that if $q\ge {\displaystyle \frac{p^2}{4}}$ then $z\left(t\right)$ satisfies the equation

$$z^2\left(t\right)+pz\left(t\right)+q=0$$

(29)

if it attains one of the two values

$$z_1\left(t\right)=-{\displaystyle \frac{p}{2}}+i\sqrt{q-{\displaystyle \frac{p^2}{4}}}\mathrm{and}{z}_2\left(t\right)=-{\displaystyle \frac{p}{2}}-i\sqrt{q-{\displaystyle \frac{p^2}{4}}}.$$

However, as the following consideration shows, this statement does not meet the usual criteria of mathematical rigor.

The result of the usual transformations

$$\begin{array}{cc}\hfill {(x\left(t\right)+iy\left(t\right))}^2+p(x\left(t\right)+iy\left(t\right))+q& =x^2\left(t\right)+2ix\left(t\right)y\left(t\right)+i^2{y}^2\left(t\right)+px\left(t\right)+piy\left(t\right)+q\hfill \\ & =x^2\left(t\right)-y^2\left(t\right)+px\left(t\right)+q+i(2x\left(t\right)y\left(t\right)+py\left(t\right))\hfill \end{array}$$

shows a strong closeness with the left-hand terms of Equations (12) and (13), and could be reformulated in a vectorial language as

$$\left(\begin{array}{c}{x}^2\left(t\right)-y^2\left(t\right)\\ 2x\left(t\right)y\left(t\right)\end{array}\right)+p·\left(\begin{array}{c}x\left(t\right)\\ y\left(t\right)\end{array}\right)+q·\left(\begin{array}{c}1\\ 0\end{array}\right).$$

This could motivate squaring a vector according to Formula (6).

But the above consideration also raises the following questions:

• From which set $\mathrm{\Theta }$ of known mathematical objects does i originate?
• What range of values $\mathrm{\Psi }$ does the mapping $+:R\times \mathrm{\Theta }\to \mathrm{\Psi }$ posses, and how is the mapping itself defined?
• What does squaring of elements of $\mathrm{\Theta }$ mean?

Heuristic or pragmatic transformations can often be practically successful without meeting the usual standards of mathematical rigor.

Our consideration shows how the traditional complex-analytical approach to solving quadratic equations can successfully be replaced by a sound new two-dimensional real vector approach. Specifically, the usual complex squaring can be replaced by vector squaring (6).

Numerous practical situations are dealing with other types of signals. Let, for example, the signal $x\left(t\right)$ be the $l_p$-generalized trigonometric function ${cos}_p\left(t\right)$. Then, with $y\left(t\right)={sin}_p\left(t\right)$, at all times, the signal vector $z\left(t\right)=x\left(t\right)\mathbf{1}+y\left(t\right)\mathbf{I}$ belongs to the $l_p$-circle

$$S_p{=\{(x,y)}^T\in R^2{:\left|x\right|}^p+{\left|y\right|}^p=1\}.$$

According to Theorem 3, the solution vector of the quadratic Equation (16) belongs to the circle (17) and satisfies Equation (18) where

$$Q\left(z\right)={\displaystyle \frac{{\left(\right|x|}^p+{\left|y\right|}^p{)}^{2/p}}{\left(\right|x^2-y^2{|}^p+{\left|2xy\right|}^p{)}^{1/p}}}.$$

6. Hyperbolically Accompanying Variable

In this section, it is assumed that the real variables x and y come from the range

$$x\in R\mathrm{and}\left|y\right|<x$$

(30)

which can easily be restricted to a space-time model by putting $x>0$. A hyperbolic vector-valued vector product is now defined according to [13] as

$$\left(\begin{array}{c}{x}_1\\ y_1\end{array}\right)ⓗ\left(\begin{array}{c}{x}_2\\ y_2\end{array}\right)\left(\begin{array}{c}{x}_1{x}_2+y_1{y}_2\\ x_1{y}_2+x_2{y}_1\end{array}\right).$$

(31)

In general, such a multiplication describes a movement on a hyperbola and additionally the change between two hyperbolas. Let $C_h^{+}\left(r\right)=\{\left(\begin{array}{c}x\\ y\end{array}\right)\in R^2:x>0,x^2-y^2=r^2\}$ denote a “hyperbolic half circle of radius $r>0$” and let $z_2$ be an arbitrary element of $C_h^{+}\left(1\right)$. For $z_1\in C_h^{+}\left(r_1\right)$ then holds $z_1ⓗz_2\in C_h^{+}\left(r_1\right)$; that is $z_2$ moves $z_1$ along the “half circle” $C_h^{+}\left(r_1\right)$. Such a movement is complemented by a change from the “half circle” $C_h^{+}\left(r_1\right)$ to the “half circle” $C_h^{+}\left(r_1{r}_2\right)$ if the element from $C_h^{+}\left(1\right)$ is additionally multiplied by $r_2>0$ and so that $z_2\in C_h^{+}\left(r_2\right),r_2>0$. In particular, for $z=\left(\begin{array}{c}x\\ y\end{array}\right)$, we have the equations

$$z^{ⓗ2}=\left(\begin{array}{c}{x}^2+y^2\\ 2xy\end{array}\right)$$

(32)

and, in contrast to (7),

$${\mathbf{I}}^{ⓗ2}=\mathbf{1}.$$

(33)

In the following, the function

$$f_{ⓗ}\left(z\right)=z^{ⓗ2}\oplus p·z\oplus q·\mathbf{1}$$

and the equation

$$f_{ⓗ}\left(z\right)=\mathfrak{o}$$

(34)

will be considered.

Theorem 5.

If condition (1) is fulfilled then the Equation (34) has no solution. If assumption (3) is satisfied then Equation (34) with $x_{1/2}$ out of (4) has the solutions

$$z_{1/2}=\left(\begin{array}{c}{x}_{1/2}\\ 0\end{array}\right)=x_{1/2}·\mathbf{1}\mathit{and}{z}_{3/4}=-{\displaystyle \frac{p}{2}}·\mathbf{1}\stackrel{+}{-}\sqrt{{\displaystyle \frac{p^2}{4}}-q}·\mathit{I}.$$

Proof. 

First, the vector Equation (34) is written as the system of univariate equations

$$x^2+y^2+px+q=0$$

(35)

and

$$2xy+py=0.$$

(36)

Equation (35) can be rewritten as

$${(x+{\displaystyle \frac{p}{2}})}^2+y^2={\displaystyle \frac{p^2}{4}}-q$$

and has no solution if condition (1) is satisfied. Therefore, let from now on assumption (3) be fulfilled. Equation (36) reads as

$$y(2x+p)=0$$

and can be solved if $y=0$ or $x=-{\displaystyle \frac{p}{2}}.$

If $y=0$, then (35) coincides with (2) and therefore has solutions $x_{1/2}$. In this case, (34) is solved by $z_{1/2}$.

If $x=-{\displaystyle \frac{p}{2}}$, then Equation (35) means $y^2={\displaystyle \frac{p^2}{4}}-q$ or $y=\stackrel{+}{-}\sqrt{{\displaystyle \frac{p^2}{4}}-q}$, which leads to the solutions $z_{3/4}$. □

Remark 4.

The solution behavior of a quadratic equation with accompanying variables in a hyperbolic space differs significantly from that in a Euclidean space or a space that is provided with an arbitrarily chosen phs-functional. The role a semi-antinorm in the sense of [12] plays in a corresponding hyperbolic space is discussed in [13].

7. Comments

In contrast to the standard literature on quadratic equations, this work does not make use of the so-called imaginary unit but instead makes use of vector-valued vector multiplication. In [11], the author consistently replaced the imaginary unit with a well-defined element of a vector space. The first consequences for solving general quadratic equations were drawn in [16] and a series of subsequent papers by the same author and are continued here in much greater generality.

Regarding the works [2,17] by Hamilton, Saric and Sriraman [18] write about the purpose of their work: “The main purpose of this chapter is to reaffirm Hamilton’s idea to make the solutions of polynomial equations completely real and autonomous from complex numbers.”

In [19], the author aims to present a complete analogy of complex analysis and real analysis. To this end, he chooses ordered pairs $(1,i0)$ and $(0,i)$ as the basis of a two-dimensional real vector space where i is an imaginary unit.

In case condition (1) is fulfilled, the well-known complex analysis on the solution of Equation (2) states that it has the form

$$x_{1/2}=-{\displaystyle \frac{p}{2}}\stackrel{+}{-}i\sqrt{q-{\displaystyle \frac{p^2}{4}}},$$

(37)

and the following is said very nebulously about quantity i:

P1

The quantity i is not a real number.

P2

The quantity i satisfies the equation $i^2=-1.$

Sometimes i is not said to be just a quantity but to be a number with the properties P1 and P2. But what, if the complex numbers have not yet been introduced, are numbers other than a real number? For example, if i ( in harmony with property P1) denotes the “moon-number”, how is the squaring of the “moon-number” defined, as in property P2? Moreover, how is it defined so that the same squaring-operation is also applicable to real numbers?

When graphically representing complex numbers, some authors start with a coordinate system whose axes they do not label. Figures of complex numbers shown might occasionally give the impression that they are figures in $R^2.$ However, figures formed with imaginary numbers (or “moon-numbers”) are imaginary by nature and actually therefore cannot be graphically represented. The fortunate resolution of this conflict is based on a consistent vector space approach to complex numbers as used in [11,13,16] and this study.

In a certain part of the standard literature, i is said to be the vector $\mathbf{I}=\left(\begin{array}{c}0\\ 1\end{array}\right)$, but in later steps, when introducing complex numbers, for example, it is said, in violation of mathematical correctness, that “obviously, $1=\mathbf{1}$ holds”.

Moreover, what can one imagine by the sum of the real number $-p/2$ and the nebulous expression $i\sqrt{q-{\displaystyle \frac{p^2}{4}}}$ as in (37)? In comparison to (37), Equation (10) describes a situation that is mathematically completely clarified on the basis of vector calculus.

To the best of author’s knowledge, complex numbers are always introduced in the standard literature in one way or another with such a serious lack of mathematical rigor.

The argument of unobserved contradictions when using imaginary numbers put forward as justification by some authors should from now on be recognized as inadequacy.

If we consider the algebraic structure

$$\mathfrak{C}=(R^2,\oplus ,⊛,·,\mathfrak{o},\mathbf{1},\mathbf{I})$$

to be the complex plane then $\mathfrak{C}$ is not an extension field of the real numbers because R is isomorphic to a subspace of $R^2$ but not a subset. However, note that we then have

$${\mathbf{I}}^2=-\mathbf{1}\mathrm{instead}\mathrm{of}“i^2=-1”.$$

It was thanks to the genius of Gauss that he interpreted complex numbers as points in the complex number plane. At the time of writing, for example [1], the theory of vector spaces was not yet available to him. How cautious Gauss was towards the field of complex numbers is expressed, among other things, in his following words from [1]: “Again, since the time when the analysts found out that there are infinitely many equations which had no roots at all unless quantities of the form $a+b\sqrt{-1}$ were admitted as a peculiar kind of quantities called imaginaries distinct from reals, these supposed quantities have been studied, and have been introduced in all of analysis: with what justification, I shall not discuss here. I shall free my proof from any help of imaginary quantities, although I might avail myself of that liberty of which all those dealing with analysis make use.”

8. Conclusions

Replacing imaginary and thus complex quantities with two-dimensional vectors allows for a new description of the solution behavior of quadratic equations, as shown in this work. This raises, for example, the following questions:

• What are the related statements when several accompanying variables are included?
• What new formulations for solving higher-degree equations result?

Answering such questions will lead to further significant extensions of the context of the fundamental theorem of algebra.