New Approach to Neutrosophic Numbers and Neutrosophic Complex Numbers

Abstract

In this study, we introduced non-Newtonian neutrosophic numbers and non-Newtonian neutrosophic complex numbers by combining two recently popular approaches and examined some of their properties. Furthermore, we presented the non-Newtonian neutrosophic triangle inequality and some properties of the non-Newtonian neutrosophic norm, which can be frequently used in analysis and geometry. Thus, compared to existing studies, we provided a broader perspective for fields such as artificial intelligence, quantum mechanics, medicine, analysis, and geometry.

1. Introduction

The non-Newtonian calculus introduced by Grossman and Katz in 1972 provides an alternative perspective to classical Newtonian and Leibnizian calculus, thus establishing a new domain in mathematics [1]. The non-Newtonian calculus encompasses a variety of systems, such as geometric calculus, bigeometric calculus, and harmonic calculus, each tailored to specific types of functional relationships and growth patterns. These alternative calculations have been applied in various fields, including biology, economics, and information theory, where traditional methods may not produce satisfactory results. For example, they have been used in modeling biological growth processes and financial mathematics to understand market dynamics better. Some studies based on non-Newtonian calculus in different fields, such as medicine, economics, and engineering, are as follows: [2,3,4,5,6,7,8,9]. Also, recently, based on non-Newtonian calculus, non-Newtonian quaternions, non-Newtonian Cayley numbers, and non-Newtonian number sequences have been defined and their properties have been investigated [10,11,12].

Neutrosophic numbers, a concept with roots in neutrosophy, were first introduced by Florentin Smarandache in the early 21st century [13]. Neutrosophic numbers expand classical number systems by incorporating ambiguity. They allow a more comprehensive representation of uncertain data by combining accuracy, uncertainty, and inaccuracy degrees. This is especially useful when dealing with complex and uncertain data in fields such as artificial intelligence, decision-making, and quantum mechanics. Neutrosophic numbers have facilitated advances in these fields by providing a mathematical framework that can handle incomplete, inconsistent, and ambiguous information. Additionally, they are used to develop new algorithms for image processing and to increase the accuracy of medical diagnoses. Some studies conducted in medicine, artificial intelligence, data analysis, applied mathematics and psychology using the neutrosophic approach are as follows: [14,15,16,17,18,19].

These concepts introduced by Grossman, Katz and Smarandache have expanded the scope of mathematical analysis and effectively solved challenges in various scientific disciplines. As research on these concepts progresses, discoveries and technological innovations will likely pave the way.

In this study, we define non-Newtonian neutrosophic real numbers and non-Newtonian neutrosophic complex numbers by combining these two concepts and examine some of their properties.

The paper is organized as follows: Section 2 provides fundamental information about non-Newtonian and neutrosophic numbers.

Section 3 introduces non-Newtonian neutrosophic numbers based on non-Newtonian calculus. Then, by defining the addition and multiplication operations of these new numbers, we examine some of their properties and state that $\left(N\left({\mathbb{R}}_{\alpha }\right),\widehat{\oplus },\widehat{\odot }\right)$ is a field. We also provide several examples.

Section 4 introduces non-Newtonian neutrosophic complex numbers using non-Newtonian neutrosophic numbers and defines the addition and multiplication operations of these numbers, examining the properties of these operations. We also provide some examples. Subsequently, we define the conjugate and inverse of a non-Newtonian neutrosophic complex number, examine their properties, and state that $\left(C_T,\oplus ,\odot \right)$ is a field. We also define the ∇-distance and ∇-norm, present the non-Newtonian neutrosophic triangle inequality, and discuss some properties of the ∇-norm. Based on these properties and the definition of a normed space, which is frequently used in the analysis, we conclude that $\left(C_T,\ddot{\parallel }.\ddot{\parallel }\right)$ is a normed space.

2. Preliminaries

This section presents fundamental information about non-Newtonian and neutrosophic numbers. The symbol I is used to represent the indeterminacy of any concept, relation, or thought. Specifically, when no relationship can be established between two concepts, or when a concept cannot be defined, the symbol I is employed to indicate this indeterminacy.

The set of real neutrosophic numbers is defined as

$$N\left(\mathbb{R}\right)=\left\{q_I=a+bI|a,b\in \mathbb{R}\right\},$$

where $I^n=I$, $0.I=0$, I represents indeterminacy.

More detailed information about neutrosophic numbers can be found in [13]. Also, using this definition, Alhasan defined the general exponential form of a neutrosophic complex number [20]. A neutrosophic complex number is represented as

$$\left(x_1+y_{1}I\right)+\left(x_2+y_{2}I\right)i,$$

where $x_1,x_2,y_1,y_2\in \mathbb{R},i^2=-1$ and I is the indeterminacy element.

For a neutrosophic complex number $z=\left(x_1+y_{1}I\right)+\left(x_2+y_{2}I\right)i$, $x_1+y_{1}I$ is called the neutrosophic scalar part of z and $x_2+y_{2}I$ is called the neutrosophic imaginary part of z, denoted by $S_I\left(z\right)$ and $V_I\left(z\right)$, respectively, [21]. Accordingly, a neutrosophic complex number z can be written as

$$z=S_I\left(z\right)+V_I\left(z\right)i.$$

A completely ordered field is called arithmetic if its realm is a subset of $\mathbb{R}$. A generator is a one-to-one function whose domain $\mathbb{R}$ and whose range is a subset of $\mathbb{R}$. Let $\alpha$ be a generator with range A. We denote by ${\mathbb{R}}_{\alpha }$, which are called non-Newtonian real numbers.

Let $\alpha$ be an arbitrarily chosen generator that images the set $\mathbb{R}$ to A and let ∗-calculus be the ordered pairs of arithmetic operations. For more details, see [1,22].

The following notations will be used:

$$\stackrel{.}{a}\dot{+}\stackrel{.}{b}=\alpha \left\{{\alpha }^{-1}\left(\stackrel{.}{a}\right)+{\alpha }^{-1}\left(\stackrel{.}{b}\right)\right\},$$

$$\stackrel{.}{a}\dot{-}\stackrel{.}{b}=\alpha \left\{{\alpha }^{-1}\left(\stackrel{.}{a}\right)-{\alpha }^{-1}\left(\stackrel{.}{b}\right)\right\},$$

$$\stackrel{.}{a}\dot{\times }\stackrel{.}{b}=\alpha \left\{{\alpha }^{-1}\left(\stackrel{.}{a}\right)\times {\alpha }^{-1}\left(\stackrel{.}{b}\right)\right\},$$

$$\stackrel{.}{a}\dot{/}\stackrel{.}{b}=\alpha \left\{{\alpha }^{-1}\left(\stackrel{.}{a}\right)/{\alpha }^{-1}\left(\stackrel{.}{b}\right)\right\},\stackrel{.}{b}\ne \stackrel{.}{0},$$

$$\stackrel{.}{a}\dot{\le }\stackrel{.}{b}\iff {\alpha }^{-1}\left(\stackrel{.}{a}\right)\le {\alpha }^{-1}\left(\stackrel{.}{b}\right).$$

$\alpha$-zero and $\alpha$-one numbers are denoted by $\dot{0}=\alpha \left(0\right)$ and $\dot{1}=\alpha \left(1\right)$. The set of non-Newtonian real numbers is a field with operations defined on it [1].

Using generator I, defined by $\alpha \left(x\right)=x$, we obtain classical arithmetic, as follows:

$$\begin{array}{ccc}\hfill x\dot{+}y& =& \alpha \left[{\alpha }^{-1}\left(x\right)+{\alpha }^{-1}\left(y\right)\right]\hfill \\ & =& x+y\mathrm{classic}\mathrm{addition};\hfill \\ \hfill x\dot{-}y& =& \alpha \left[{\alpha }^{-1}\left(x\right)-{\alpha }^{-1}\left(y\right)\right]\hfill \\ & =& x-y\mathrm{classic}\mathrm{subtraction};\hfill \\ \hfill x\dot{\times }y& =& \alpha \left[{\alpha }^{-1}\left(x\right)\times {\alpha }^{-1}\left(y\right)\right]\hfill \\ & =& x\times y\mathrm{classic}\mathrm{multiplication};\hfill \\ \hfill x\dot{/}y& =& \alpha \left[{\alpha }^{-1}\left(x\right)/{\alpha }^{-1}\left(y\right)\right]\hfill \\ & =& x/y\mathrm{classic}\mathrm{division}.\hfill \end{array}$$

Also, by choosing the generator exp defined by $\alpha \left(x\right)=e^x$, we obtain geometric arithmetic, as follows [23]:

$$\begin{array}{ccc}\hfill x\dot{+}y& =& \alpha \left[{\alpha }^{-1}\left(x\right)+{\alpha }^{-1}\left(y\right)\right]=e^{(lnx+lny)}\hfill \\ & =& x·y\mathrm{geometric}\mathrm{addition};\hfill \\ \hfill x\dot{-}y& =& \alpha \left[{\alpha }^{-1}\left(x\right)-{\alpha }^{-1}\left(y\right)\right]\hfill \\ & =& e^{(lnx-lny)}={\displaystyle \frac{x}{y}}\left(y\ne 0\right)\mathrm{geometric}\mathrm{subtraction};\hfill \\ \hfill x\dot{\times }y& =& \alpha \left[{\alpha }^{-1}\left(x\right)\times {\alpha }^{-1}\left(y\right)\right]\hfill \\ & =& e^{(lnx+lny)}=x^y\mathrm{geometric}\mathrm{multiplication};\hfill \\ \hfill x\dot{/}y& =& \alpha \left[{\alpha }^{-1}\left(x\right)/{\alpha }^{-1}\left(y\right)\right]\hfill \\ & =& e^{(lnx-lny)}=x^{1/y}\left(y\ne 0\right)\mathrm{geometric}\mathrm{division}.\hfill \end{array}$$

The isomorphism from $\alpha$-arithmetic to $\beta$-arithmetic is the unique function $\iota$ (iota) which has the following three properties [22]:

(i)

$\iota$ is one to one.

(ii)

$\iota$ is on A and onto B.

(iii)

For any numbers x and y in A

$$\begin{array}{ccc}\hfill \iota \left(x\dot{+}y\right)& =& \iota \left(x\right)\ddot{+}\iota \left(y\right);\hfill \\ \hfill \iota \left(x\dot{-}y\right)& =& \iota \left(x\right)\ddot{-}\iota \left(y\right);\hfill \\ \hfill \iota \left(x\dot{\times }y\right)& =& \iota \left(x\right)\ddot{\times }\iota \left(y\right);\hfill \\ \hfill \iota \left(x\dot{/}y\right)& =& \iota \left(x\right)\ddot{/}\iota \left(y\right),y\ne \dot{0};\hfill \\ \hfill x\dot{\le }y& ⟺& \iota \left(x\right)\ddot{\le }\iota \left(y\right).\hfill \end{array}$$

It turns out that $\iota \left(x\right)=\beta \left\{{\alpha }^{-1}\left(x\right)\right\}$ for all x in A and that $\iota \left(\dot{n}\right)=\ddot{n}$ for every integer n.

Since, for example, $x\dot{+}y={\iota }^{-1}\left\{\iota \left(x\right)\ddot{+}\iota \left(y\right)\right\}$, it should be clear that any statement in $\alpha$-arithmetic can readily be transformed into a statement in $\beta$-arithmetic [23].

The n-th non-Newtonian exponent $\left(x^{\dot{n}}\right)$ and n-th non-Newtonian root $\left(\sqrt[\dot{n}]{x}\right)$ of $x\in {\mathbb{R}}_{\alpha }$ are defined as

$$\begin{array}{ccc}\hfill x^{\dot{2}}& =& x\dot{\times }x=\alpha \left\{{\alpha }^{-1}\left(x\right)\times {\alpha }^{-1}\left(x\right)\right\}=\alpha \left\{{\left[{\alpha }^{-1}\left(x\right)\right]}^2\right\},\hfill \\ \hfill x^{\dot{3}}& =& x^{\dot{2}}\dot{\times }x=\alpha \left\{{\alpha }^{-1}\left(\alpha \left({\alpha }^{-1}\left(x\right)\times {\alpha }^{-1}\left(x\right)\right)\right)\times {\alpha }^{-1}\left(x\right)\right\}=\alpha \left\{{\left[{\alpha }^{-1}\left(x\right)\right]}^3\right\},\hfill \\ & ⋮& \\ \hfill x^{\dot{n}}& =& x^{\dot{\left(n-1\right)}}\dot{\times }x=\alpha \left\{{\left[{\alpha }^{-1}\left(x\right)\right]}^n\right\}\hfill \end{array}$$

and $\sqrt[\dot{n}]{x}=\alpha \left\{\sqrt[n]{{\alpha }^{-1}\left(x\right)}\right\}$, respectively.

The non-Newtonian absolute value of $x\in {\mathbb{R}}_{\alpha }$ is defined as $\alpha \left\{{\alpha }^{-1}\left(\right|x\left|\right)\right\}$ and is denoted by $\dot{|}x\dot{|}$.

Let $\alpha$ and $\beta$ be arbitrarily chosen generators, where the ordered pair ( $\alpha$-arithmetic, $\beta$-arithmetic) represents their respective arithmetic structures. Table 1 provides a useful notation reference for understanding the concepts of $\alpha$-arithmetic and $\beta$-arithmetic.

Table 1. Notation in $\alpha$-arithmetic and $\beta$-arithmetic.

	$\mathit{\alpha }$-Arithmetic	$\mathit{\beta }$-Arithmetic
Realm	A	B
Addition	$\dot{+}$	$\ddot{+}$
Subtraction	$\dot{-}$	$\ddot{-}$
Multiplication	$\dot{\times }$	$\ddot{\times }$
Division	$\dot{/}$	$\ddot{/}$
Ordering	$\dot{\le }$	$\ddot{\le }$

Also, the definitions for $\alpha$-arithmetic are also valid for $\beta$-arithmetic.

3. Non-Newtonian Neutrosophic Numbers

This section presents non-Newtonian neutrosophic numbers based on neutrosophic numbers, providing their definition, followed by the definition of addition and multiplication operations.

Definition 1. 

The set of non-Newtonian neutrosophic numbers is defined as

$$N\left({\mathbb{R}}_{\alpha }\right)=\left\{q_N=a\dot{+}b\dot{\times }{I}^{*}|a,b\in {\mathbb{R}}_{\alpha }\right\},$$

where $I^{*}=\alpha \left(I\right)$, ${\left(I^{*}\right)}^n=I^{*}$, $0\times I^{*}=0$, $I^{*}$ represents the indeterminacy and $n\in \mathbb{R}$.

Definition 2. 

The addition and multiplication of non-Newtonian neutrosophic numbers are defined as follows:

$$\begin{array}{ccc}\hfill \widehat{\oplus }& :& N\left({\mathbb{R}}_{\alpha }\right)\times N\left({\mathbb{R}}_{\alpha }\right)\to N\left({\mathbb{R}}_{\alpha }\right)\hfill \\ \hfill \left(q_N,p_N\right)& \to & q_N\widehat{\oplus }{p}_N=\left(a\dot{+}b\dot{\times }{I}^{*}\right)\widehat{\oplus }\left(c\dot{+}d\dot{\times }{I}^{*}\right)\hfill \\ & =& \left(a\dot{+}c\right)\dot{+}\left(b\dot{+}d\right)\dot{\times }{I}^{*}\hfill \end{array}$$

and

$$\begin{array}{ccc}\hfill \widehat{\odot }& :& N\left({\mathbb{R}}_{\alpha }\right)\times N\left({\mathbb{R}}_{\alpha }\right)\to N\left({\mathbb{R}}_{\alpha }\right)\hfill \\ \hfill \left(q_N,p_N\right)& \to & q_N\widehat{\odot }{p}_N=\left(a\dot{\times }c\right)\dot{+}\left(a\dot{\times }d\dot{+}b\dot{\times }c\dot{+}b\dot{\times }d\right)\dot{\times }{I}^{*}.\hfill \end{array}$$

Definition 3. 

Let $q_N=a\dot{+}b\dot{\times }{I}^{*}$, $p_N=c\dot{+}d\dot{\times }{I}^{*}\in N\left({\mathbb{R}}_{\alpha }\right)$. $q_N=p_N$ if and only if $a=c$ and $b=d$.

Proposition 1. 

Let $q_N,p_N,r_N\in N\left({\mathbb{R}}_{\alpha }\right)$. We have the following:

i. 

$q_N\widehat{\oplus }{p}_N=p_N\widehat{\oplus }{q}_N$;

ii. 

$q_N\widehat{\oplus }\left(p_N\widehat{\oplus }{r}_N\right)=\left(q_N\widehat{\oplus }{p}_N\right)\widehat{\oplus }{r}_N$;

iii. 

$q_N\widehat{\odot }{p}_N=p_N\widehat{\odot }{q}_N$;

iv. 

$q_N\widehat{\odot }\left(p_N\widehat{\odot }{r}_N\right)=\left(q_N\widehat{\odot }{p}_N\right)\widehat{\odot }{r}_N$;

v. 

$q_N\widehat{\odot }\left(p_N\widehat{\oplus }{r}_N\right)=\left(q_N\widehat{\odot }{p}_N\right)\widehat{\oplus }\left(q_N\widehat{\odot }{r}_N\right)$.

Proof. 

Let $q_N=a\dot{+}b\dot{\times }{I}^{*},p_N=c\dot{+}d\dot{\times }{I}^{*},r_N=e\dot{+}f\dot{\times }{I}^{*}\in N\left({\mathbb{R}}_{\alpha }\right)$.

i.

$$\begin{array}{ccc}\hfill q_N\widehat{\oplus }{p}_N& =& \left(a\dot{+}c\right)\dot{+}\left(b\dot{+}d\right)\dot{\times }{I}^{*}\hfill \\ & =& \alpha \left({\alpha }^{-1}\left(a\right)+{\alpha }^{-1}\left(c\right)\right)\dot{+}\alpha \left({\alpha }^{-1}\left(\alpha \left({\alpha }^{-1}\left(b\right)+{\alpha }^{-1}\left(d\right)\right)\right)\times I\right)\hfill \\ & =& \alpha \left({\alpha }^{-1}\left(c\right)+{\alpha }^{-1}\left(a\right)\right)\dot{+}\alpha \left({\alpha }^{-1}\left(\alpha \left({\alpha }^{-1}\left(d\right)+{\alpha }^{-1}\left(b\right)\right)\right)\times I\right)\hfill \\ & =& \left(c\dot{+}a\right)\dot{+}\alpha \left({\alpha }^{-1}\left(d\dot{+}b\right)\times I\right)\hfill \\ & =& \left(c\dot{+}a\right)\dot{+}\left(d\dot{+}b\right)\dot{\times }{I}^{*}\hfill \\ & =& p_N\widehat{\oplus }{q}_N.\hfill \end{array}$$

ii.

$$\begin{array}{ccc}\hfill q_N\widehat{\oplus }\left(p_N\widehat{\oplus }{r}_N\right)& =& \left(a\dot{+}b\dot{\times }{I}^{*}\right)\widehat{\oplus }\left(\left(c\dot{+}e\right)\dot{+}\left(d\dot{+}f\right)\dot{\times }{I}^{*}\right)\hfill \\ & =& \left(a\dot{+}\left(c\dot{+}e\right)\right)\dot{+}\left(b\dot{+}\left(d\dot{+}f\right)\right)\dot{\times }{I}^{*}\hfill \\ & =& \alpha \left\{{\alpha }^{-1}\left(a\right)+{\alpha }^{-1}\left(\alpha \left({\alpha }^{-1}\left(c\right)+{\alpha }^{-1}\left(e\right)\right)\right)\right\}\hfill \\ & & \dot{+}\alpha \left\{{\alpha }^{-1}\left(\alpha \left({\alpha }^{-1}\left(b\right)+{\alpha }^{-1}\left(\alpha \left({\alpha }^{-1}\left(d\right)+{\alpha }^{-1}\left(f\right)\right)\right)\right)\right)\times I\right\}\hfill \\ & =& \alpha \left\{\begin{array}{c}{\alpha }^{-1}\left(\alpha \left({\alpha }^{-1}\left(a\right)+\left({\alpha }^{-1}\left(c\right)+{\alpha }^{-1}\left(e\right)\right)\right)\right)\\ +{\alpha }^{-1}\left(\alpha \left(\left({\alpha }^{-1}\left(b\right)+\left({\alpha }^{-1}\left(d\right)+{\alpha }^{-1}\left(f\right)\right)\right)\times I\right)\right)\end{array}\right\}\hfill \\ & =& \alpha \left\{\begin{array}{c}{\alpha }^{-1}\left(a\right)+\left({\alpha }^{-1}\left(c\right)+{\alpha }^{-1}\left(e\right)\right)\\ +\left({\alpha }^{-1}\left(b\right)+\left({\alpha }^{-1}\left(d\right)+{\alpha }^{-1}\left(f\right)\right)\right)\times I\end{array}\right\}\hfill \\ & =& \alpha \left\{\begin{array}{c}\left({\alpha }^{-1}\left(a\right)+{\alpha }^{-1}\left(c\right)\right)+{\alpha }^{-1}\left(e\right)\\ +\left(\left({\alpha }^{-1}\left(b\right)+{\alpha }^{-1}\left(d\right)\right)+{\alpha }^{-1}\left(f\right)\right)\times I\end{array}\right\}\hfill \\ & =& \alpha \left\{\begin{array}{c}{\alpha }^{-1}\left(\alpha \left(\left({\alpha }^{-1}\left(a\right)+{\alpha }^{-1}\left(c\right)\right)+{\alpha }^{-1}\left(e\right)\right)\right)\\ +{\alpha }^{-1}\left(\alpha \left(\left(\left({\alpha }^{-1}\left(b\right)+{\alpha }^{-1}\left(d\right)\right)+{\alpha }^{-1}\left(f\right)\right)\times I\right)\right)\end{array}\right\}\hfill \\ & =& \alpha \left\{{\alpha }^{-1}\left(\alpha \left({\alpha }^{-1}\left(a\right)+{\alpha }^{-1}\left(c\right)\right)\right)+{\alpha }^{-1}\left(e\right)\right\}\hfill \\ & & \dot{+}\alpha \left\{{\alpha }^{-1}\left(\alpha \left({\alpha }^{-1}\left(\alpha \left({\alpha }^{-1}\left(b\right)+{\alpha }^{-1}\left(d\right)\right)\right)+{\alpha }^{-1}\left(f\right)\right)\right)\times I\right\}\hfill \\ & =& \left(\left(a\dot{+}c\right)\dot{+}e\right)\dot{+}\left(\left(b\dot{+}d\right)\dot{+}f\right)\dot{\times }{I}^{*}\hfill \\ & =& \left(\left(a\dot{+}c\right)\dot{+}\left(b\dot{+}d\right)\dot{\times }{I}^{*}\right)\widehat{\oplus }\left(e\dot{+}f\dot{\times }{I}^{*}\right)\hfill \\ & =& \left(q_N\widehat{\oplus }{p}_N\right)\widehat{\oplus }{r}_N.\hfill \end{array}$$

iii.

$$\begin{array}{ccc}\hfill q_N\widehat{\odot }{p}_N& =& \left(a\dot{\times }c\right)\dot{+}\left(a\dot{\times }d\dot{+}b\dot{\times }c\dot{+}b\dot{\times }d\right)\dot{\times }{I}^{*}\hfill \\ & =& \alpha \left\{{\alpha }^{-1}\left(a\right)\times {\alpha }^{-1}\left(c\right)\right\}\hfill \\ & & \dot{+}\alpha \left\{\left(\left({\alpha }^{-1}\left(a\right)\times {\alpha }^{-1}\left(d\right)\right)+\left({\alpha }^{-1}\left(b\right)\times {\alpha }^{-1}\left(c\right)\right)+\left({\alpha }^{-1}\left(b\right)\times {\alpha }^{-1}\left(d\right)\right)\right)\times I\right\}\hfill \\ & =& \alpha \left\{\begin{array}{c}{\alpha }^{-1}\left(\alpha \left({\alpha }^{-1}\left(a\right)\times {\alpha }^{-1}\left(c\right)\right)\right)+\\ {\alpha }^{-1}\left(\alpha \left(\left({\alpha }^{-1}\left(a\right)\times {\alpha }^{-1}\left(d\right)+{\alpha }^{-1}\left(b\right)\times {\alpha }^{-1}\left(c\right)+{\alpha }^{-1}\left(b\right)\times {\alpha }^{-1}\left(d\right)\right)\times I\right)\right)\end{array}\right\}\hfill \\ & =& \alpha \left\{\begin{array}{c}{\alpha }^{-1}\left(a\right)\times {\alpha }^{-1}\left(c\right)+\\ \left(\left({\alpha }^{-1}\left(a\right)\times {\alpha }^{-1}\left(d\right)\right)+\left({\alpha }^{-1}\left(b\right)\times {\alpha }^{-1}\left(c\right)\right)+\left({\alpha }^{-1}\left(b\right)\times {\alpha }^{-1}\left(d\right)\right)\right)\times I\end{array}\right\}\hfill \\ & =& \alpha \left\{\begin{array}{c}{\alpha }^{-1}\left(c\right)\times {\alpha }^{-1}\left(a\right)+\\ \left(\left({\alpha }^{-1}\left(d\right)\times {\alpha }^{-1}\left(a\right)\right)+\left({\alpha }^{-1}\left(c\right)\times {\alpha }^{-1}\left(b\right)\right)+\left({\alpha }^{-1}\left(d\right)\times {\alpha }^{-1}\left(b\right)\right)\right)\times I\end{array}\right\}\hfill \\ & =& \alpha \left\{{\alpha }^{-1}\left(c\right)\times {\alpha }^{-1}\left(a\right)\right\}\hfill \\ & & \dot{+}\alpha \left\{\left(\left({\alpha }^{-1}\left(d\right)\times {\alpha }^{-1}\left(a\right)\right)+\left({\alpha }^{-1}\left(c\right)\times {\alpha }^{-1}\left(b\right)\right)+\left({\alpha }^{-1}\left(d\right)\times {\alpha }^{-1}\left(b\right)\right)\right)\times I\right\}\hfill \\ & =& \left(c\dot{\times }a\right)\dot{+}\left(d\dot{\times }a\dot{+}c\dot{\times }b\dot{+}d\dot{\times }b\right)\dot{\times }{I}^{*}\hfill \\ & =& p_N\widehat{\odot }{q}_N\hfill \end{array}$$

iv.

This can be easily proven since $\left({\mathbb{R}}_{\alpha },\dot{+},\dot{\times }\right)$ is a field.

v.

$$\begin{array}{ccc}\hfill q_N\widehat{\odot }\left(p_N\widehat{\oplus }{r}_N\right)& =& \left(a\dot{+}b\dot{\times }{I}^{*}\right)\widehat{\odot }\left(\left(c\dot{+}e\right)\dot{+}\left(d\dot{+}f\right)\dot{\times }{I}^{*}\right)\hfill \\ & =& \left(a\dot{\times }\left(c\dot{+}e\right)\right)\dot{+}\left(a\dot{\times }\left(d\dot{+}f\right)\dot{+}b\dot{\times }\left(c\dot{+}e\right)\dot{+}b\dot{\times }\left(d\dot{+}f\right)\right)\dot{\times }{I}^{*}\hfill \\ & =& a\dot{\times }c\dot{+}a\dot{\times }e\dot{+}\left(a\dot{\times }d\dot{+}a\dot{\times }f\dot{+}b\dot{\times }c\dot{+}b\dot{\times }e\dot{+}b\dot{\times }d\dot{+}b\dot{\times }f\right)\dot{\times }{I}^{*}\hfill \\ & =& \left(a\dot{\times }c\dot{+}\left(a\dot{\times }d\dot{+}b\dot{\times }c\dot{+}b\dot{\times }d\right)\dot{\times }{I}^{*}\right)\widehat{\oplus }\left(a\dot{\times }e\dot{+}\left(a\dot{\times }f\dot{+}b\dot{\times }e\dot{+}b\dot{\times }f\right)\dot{\times }{I}^{*}\right)\hfill \\ & =& \left(q_N\widehat{\odot }{p}_N\right)\widehat{\oplus }\left(q_N\widehat{\odot }{r}_N\right)\hfill \end{array}$$

□

Corollary 1. 

$\left(N\left({\mathbb{R}}_{\alpha }\right),\widehat{\oplus },\widehat{\odot }\right)$ is a field.

Example 1. 

Let $q_N=\dot{1}\dot{+}\dot{3}\dot{\times }{I}^{*},p_N=\dot{2}\dot{+}{I}^{*},r_N=I^{*}\in N\left({\mathbb{R}}_{\alpha }\right)$.

$$\begin{array}{ccc}\hfill q_N\widehat{\odot }\left(p_N\widehat{\oplus }{r}_N\right)& =& \left(\dot{1}\dot{+}\dot{3}\dot{\times }{I}^{*}\right)\widehat{\otimes }\left(\left(\dot{2}\dot{+}{I}^{*}\right)\widehat{\oplus }{I}^{*}\right)\hfill \\ & =& \left(\dot{1}\dot{+}\dot{3}\dot{\times }{I}^{*}\right)\widehat{\otimes }\left(\dot{2}\dot{+}\dot{2}\dot{\times }{I}^{*}\right)\hfill \\ & =& \dot{2}\dot{+}\dot{14}\dot{\times }{I}^{*}\hfill \end{array}$$

and

$$\begin{array}{ccc}\hfill \left(q_N\widehat{\odot }{p}_N\right)\widehat{\oplus }\left(q_N\widehat{\odot }{r}_N\right)& =& \left(\left(\dot{1}\dot{+}\dot{3}\dot{\times }{I}^{*}\right)\widehat{\otimes }\left(\dot{2}\dot{+}{I}^{*}\right)\right)\widehat{\oplus }\left(\left(\dot{1}\dot{+}\dot{3}\dot{\times }{I}^{*}\right)\widehat{\otimes }{I}^{*}\right)\hfill \\ & =& \left(\dot{2}\dot{+}\dot{10}\dot{\times }{I}^{*}\right)\widehat{\oplus }\left(\dot{0}\dot{+}\dot{4}\dot{\times }{I}^{*}\right)\hfill \\ & =& \dot{2}\dot{+}\dot{14}\dot{\times }{I}^{*}.\hfill \end{array}$$

Therefore, we can easily see that $q_N\widehat{\odot }\left(p_N\widehat{\oplus }{r}_N\right)=\left(q_N\widehat{\odot }{p}_N\right)\widehat{\oplus }\left(q_N\widehat{\odot }{r}_N\right)=\dot{2}\dot{+}\dot{14}\dot{\times }{I}^{*}$.

4. Non-Newtonian Neutrosophic Complex Numbers

This section presents non-Newtonian neutrosophic complex numbers based on non-Newtonian neutrosophic numbers. It defines the addition and multiplication operations of these numbers and examines their properties.

Let $\dot{a}\in \left(A,\widehat{\oplus },\widehat{\ominus },\widehat{\otimes },\widehat{\oslash },\widehat{\le }\right)$ and $\ddot{b}\in \left(B,\widehat{\widehat{\oplus }},\widehat{\widehat{\ominus }},\widehat{\widehat{\otimes }},\widehat{\widehat{\oslash }},\widehat{\widehat{\le }}\right)$ be arbitrarily chosen elements from corresponding arithmetic. Then, the ordered pair $\left(\dot{a},\ddot{b}\right)$ is called a *-neutrosophic point. The set of all *-neutrosophic points is called the set of *-neutrosophic complex numbers and is denoted by

$$C_T=\left\{z^{*}=\left(\dot{a},\ddot{b}\right)|\dot{a}\in A\subseteq N\left({\mathbb{R}}_{\alpha }\right),\ddot{b}\in B\subseteq N\left({\mathbb{R}}_{\beta }\right)\right\}.$$

Define the binary operations addition $\left(\oplus \right)$ and multiplication $\left(\odot \right)$ of *-neutrosophic complex numbers $z_1^{*}=\left({\dot{a}}_1,{\ddot{b}}_1\right)$ and $z_2^{*}=\left({\dot{a}}_2,{\ddot{b}}_2\right)$ as follows:

$$\begin{array}{ccc}\hfill \oplus & :& C_T\times C_T⟶C_T\hfill \\ \hfill \left(z_1^{*},z_2^{*}\right)& \to & z_1^{*}\oplus z_2^{*}=\left({\dot{a}}_1\widehat{\oplus }{\dot{a}}_2,{\ddot{b}}_1\widehat{\widehat{\oplus }}{\ddot{b}}_2\right)\hfill \\ & =& \left(\alpha \left(m_1+m_2+\left(n_1+n_2\right)I\right),\beta \left(k_1+k_2+\left(l_1+l_2\right)I\right)\right)\hfill \end{array}$$

$$\begin{array}{ccc}\hfill \odot & :& C_T\times C_T⟶C_T\hfill \\ \hfill \left(z_1^{*},z_2^{*}\right)& \to & z_1^{*}\odot z_2^{*}=\left(\alpha \left(m_1{m}_2-k_1{k}_2+cI\right),\beta \left(m_1{k}_2+k_1{m}_2+dI\right)\right),\hfill \end{array}$$

where $c=m_1{n}_2+n_1{m}_2+n_1{n}_2-k_1{l}_2-l_1{k}_2-l_1{l}_2$, $d=m_1{l}_2+n_1{k}_2+n_1{l}_2+k_1{n}_2+l_1{m}_2+l_1{n}_2$, and $m_1,m_2,n_1,n_2,k_1,k_2,l_1,l_2\in \mathbb{R}$.

Note 1. 

The identity of ⊕ is ${\theta }^{*}=\left(\dot{0},\ddot{0}\right)$, the identity of ⊙ is $1^{*}=\left(\dot{1},\ddot{0}\right)$, and for a non-Newtonian neutrosophic complex number $z^{*}=\left(\dot{a},\ddot{b}\right)$, its additive counterpart is $\ominus z^{*}=\left(\widehat{\ominus }\dot{a},\widehat{\widehat{\ominus }}\ddot{b}\right)$.

Example 2. 

For $z^{*}=\left(\dot{7},\ddot{5}\right)\in C_T$, its additive counterpart is $\ominus z^{*}=\left(\widehat{\ominus }\dot{7},\widehat{\widehat{\ominus }}\ddot{5}\right)\in C_T$.

Definition 4. 

The ∇-distance $d^{\nabla }$ between any two elements $z_1^{*}=\left({\dot{a}}_1,{\ddot{b}}_1\right)$ and $z_2^{*}=\left({\dot{a}}_2,{\ddot{b}}_2\right)$ of the set $C_T$ is defined by

$$\begin{array}{ccc}\hfill d^{\nabla }& :& C_T\times C_T⟶\stackrel{..}{[}0,\infty \stackrel{..}{)}\hfill \\ \hfill \left(z_1^{*},z_2^{*}\right)& \to & d^{\nabla }\left(z_1^{*},z_2^{*}\right)=\sqrt[··]{{\left[\iota \left({\dot{a}}_1\widehat{\ominus }{\dot{a}}_2\right)\right]}^{\ddot{2}}\widehat{\widehat{\oplus }}{\left({\ddot{b}}_1\widehat{\widehat{\ominus }}{\ddot{b}}_2\right)}^{\ddot{2}}}\hfill \\ & =& \beta \left(\sqrt{{\left(m_1+m_2+\left(n_1+n_2\right)I\right)}^2+{\left(k_1+k_2+\left(l_1+l_2\right)I\right)}^2}\right).\hfill \end{array}$$

Definition 5. 

$d^{\nabla }\left(z^{*},{\theta }^{*}\right)$ is called ∇-norm of $z^{*}\in C_T$ and is denoted by

$$\begin{array}{ccc}\hfill \ddot{\parallel }{z}^{*}\ddot{\parallel }& =& d^{\nabla }\left(z^{*},{\theta }^{*}\right)=\sqrt[··]{{\left[\iota \left({\dot{a}}_1\widehat{\ominus }\dot{0}\right)\right]}^{\ddot{2}}\widehat{\widehat{\oplus }}{\left({\ddot{b}}_1\widehat{\widehat{\ominus }}\ddot{0}\right)}^{\ddot{2}}}\hfill \\ & =& \beta \left(\sqrt{{\left(m_1+n_{1}I\right)}^2+{\left(k_1+l_{1}I\right)}^2}\right),\hfill \end{array}$$

where ${\theta }^{*}=\left(\dot{0},\ddot{0}\right)$, $\dot{0}=\alpha \left(0+0I\right)$, $\ddot{0}=\beta \left(0+0I\right)$.

Proposition 2. 

Let $z_1^{*},z_2^{*},z_3^{*}\in C_T$. We have the following:

i.

$z_1^{*}\oplus z_2^{*}=z_2^{*}\oplus z_1^{*}$;

ii.

$z_1^{*}\oplus \left(z_2^{*}\oplus z_3^{*}\right)=\left(z_1^{*}\oplus z_2^{*}\right)\oplus z_3^{*}$;

iii.

$z_1^{*}\odot z_2^{*}=z_2^{*}\odot z_1^{*}$;

iv.

$z_1^{*}\odot \left(z_2^{*}\odot z_3^{*}\right)=\left(z_1^{*}\odot z_2^{*}\right)\odot z_3^{*}$;

v.

$z_1^{*}\odot \left(z_2^{*}\oplus z_3^{*}\right)=\left(z_1^{*}\odot z_2^{*}\right)\oplus \left(z_1^{*}\odot z_3^{*}\right)$.

Proof. 

Let $z_1^{*}=\left({\dot{a}}_1,{\ddot{b}}_1\right)=\left(\alpha \left(m_1+n_{1}I\right),\beta \left(k_1+l_{1}I\right)\right)$, $z_2^{*}=\left({\dot{a}}_2,{\ddot{b}}_2\right)=\left(\alpha \left(m_2+n_{2}I\right),\beta \left(k_2+l_{2}I\right)\right)$, $z_3^{*}=\left({\dot{a}}_3,{\ddot{b}}_3\right)=\left(\alpha \left(m_3+n_{3}I\right),\beta \left(k_3+l_{3}I\right)\right)\in C_T$.

i.

$$\begin{array}{ccc}\hfill z_1^{*}\oplus z_2^{*}& =& \left(\alpha \left(m_1+m_2+\left(n_1+n_2\right)I\right),\beta \left(k_1+k_2+\left(l_1+l_2\right)I\right)\right)\hfill \\ & =& \left(\alpha \left(m_2+m_1+\left(n_2+n_1\right)I\right),\beta \left(k_2+k_1+\left(l_2+l_1\right)I\right)\right)\hfill \\ & =& z_2^{*}\oplus z_1^{*}.\hfill \end{array}$$

ii.

$$\begin{array}{ccc}\hfill z_1^{*}\oplus \left(z_2^{*}\oplus z_3^{*}\right)& =& \left(\begin{array}{c}\alpha \left(m_1+\left(m_2+m_3\right)+\left(n_1+\left(n_2+n_3\right)\right)I\right),\\ \beta \left(k_1+\left(k_2+k_3\right)+\left(l_1+\left(l_2+l_3\right)\right)I\right)\end{array}\right)\hfill \\ & =& \left(\begin{array}{c}\alpha \left(\left(m_1+m_2\right)+m_3+\left(\left(n_1+n_2\right)+n_3\right)I\right),\\ \beta \left(\left(k_1+k_2\right)+k_3+\left(\left(l_1+l_2\right)+l_3\right)I\right)\end{array}\right)\hfill \\ & =& \left(z_1^{*}\oplus z_2^{*}\right)\oplus z_3^{*}.\hfill \end{array}$$

iii.

$$\begin{array}{ccc}\hfill z_1^{*}\odot z_2^{*}& =& \left(\begin{array}{c}\alpha \left(m_1{m}_2-k_1{k}_2+\left(m_1{n}_2+n_1{m}_2+n_1{n}_2-k_1{l}_2-l_1{k}_2-l_1{l}_2\right)I\right),\\ \beta \left(m_1{k}_2+k_1{m}_2+\left(m_1{l}_2+n_1{k}_2+n_1{l}_2+k_1{n}_2+l_1{m}_2+l_1{n}_2\right)I\right)\end{array}\right)\hfill \\ & =& \left(\begin{array}{c}\alpha \left(m_2{m}_1-k_2{k}_1+\left(n_2{m}_1+m_2{n}_1+n_2{n}_1-l_2{k}_1-k_2{l}_1-l_2{l}_1\right)I\right),\\ \beta \left(k_2{m}_1+m_2{k}_1+\left(l_2{m}_1+k_2{n}_1+l_2{n}_1+n_2{k}_1+m_2{l}_1+n_2{l}_1\right)I\right)\end{array}\right)\hfill \\ & =& z_2^{*}\odot z_1^{*}.\hfill \end{array}$$

iv.

$$\begin{array}{ccc}\hfill \left(z_1^{*}\odot z_2^{*}\right)\odot z_3^{*}& =& \left(\begin{array}{c}\alpha \left(\left(m_1{m}_2-k_1{k}_2\right)m_3-\left(m_1{k}_2+k_1{m}_2\right)k_3+c_{1}I\right),\\ \beta \left(\left(m_1{m}_2-k_1{k}_2\right)k_3+\left(m_1{k}_2+k_1{m}_2\right)m_3+d_{1}I\right)\end{array}\right)\hfill \\ & =& \left(\begin{array}{c}\alpha \left(m_1\left(m_2{m}_3-k_2{k}_3\right)-k_1\left(m_2{k}_3+k_2{m}_3\right)+c_{1}I\right),\\ \beta \left(m_1\left(m_2{k}_3+k_2{m}_3\right)+k_1\left(m_2{m}_3-k_2{k}_3\right)+d_{1}I\right)\end{array}\right)\hfill \\ & =& z_1^{*}\odot \left(z_2^{*}\odot z_3^{*}\right).\hfill \end{array}$$

where

$$\begin{array}{ccc}\hfill c_1& =& \left(m_1{m}_2-k_1{k}_2\right)n_3+\left(m_1{n}_2+n_1{m}_2+n_1{n}_2-k_1{l}_2-l_1{k}_2-l_1{l}_2\right)m_3\hfill \\ & & +\left(m_1{n}_2+n_1{m}_2+n_1{n}_2-k_1{l}_2-l_1{k}_2-l_1{l}_2\right)n_3-\left(m_1{k}_2+k_1{m}_2\right)l_3\hfill \\ & & -\left(m_1{l}_2+n_1{k}_2+n_1{l}_2+k_1{n}_2+l_1{m}_2+l_1{n}_2\right)k_3\hfill \\ & & -\left(m_1{l}_2+n_1{k}_2+n_1{l}_2+k_1{n}_2+l_1{m}_2+l_1{n}_2\right)l_3,\hfill \\ \hfill d_1& =& \left(m_1{m}_2-k_1{k}_2\right)l_3+\left(m_1{n}_2+n_1{m}_2+n_1{n}_2-k_1{l}_2-l_1{k}_2-l_1{l}_2\right)k_3\hfill \\ & & +\left(m_1{n}_2+n_1{m}_2+n_1{n}_2-k_1{l}_2-l_1{k}_2-l_1{l}_2\right)l_3+\left(m_1{k}_2+k_1{m}_2\right)n_3\hfill \\ & & +\left(m_1{l}_2+n_1{k}_2+n_1{l}_2+k_1{n}_2+l_1{m}_2+l_1{n}_2\right)m_3\hfill \\ & & +\left(m_1{l}_2+n_1{k}_2+n_1{l}_2+k_1{n}_2+l_1{m}_2+l_1{n}_2\right)n_3.\hfill \end{array}$$

v.

$$\begin{array}{ccc}\hfill z_1^{*}\odot \left(z_2^{*}\oplus z_3^{*}\right)& =& \left(\begin{array}{c}\alpha \left\{m_1\left(m_2+m_3\right)-k_1\left(k_2+k_3\right)\right\}\\ \widehat{\oplus }\alpha \left\{\left(\begin{array}{c}{m}_1\left(n_2+n_3\right)+n_1\left(m_2+m_3\right)+n_1\left(n_2+n_3\right)\\ -k_1\left(l_2+l_3\right)-l_1\left(k_2+k_3\right)-l_1\left(l_2+l_3\right)\end{array}\right)I\right\},\\ \beta \left\{m_1\left(k_2+k_3\right)+k_1\left(m_2+m_3\right)\right\}\\ \widehat{\widehat{\oplus }}\beta \left\{\left(\begin{array}{c}{m}_1\left(l_2+l_3\right)+n_1\left(k_2+k_3\right)+n_1\left(l_2+l_3\right)\\ +k_1\left(n_2+n_3\right)+l_1\left(m_2+m_3\right)+l_1\left(n_2+n_3\right)\end{array}\right)I\right\}\end{array}\right)\hfill \\ & =& \left(\begin{array}{c}\alpha \left\{m_1{m}_2+m_1{m}_3-k_1{k}_2-k_1{k}_3\right\}\\ \widehat{\oplus }\alpha \left\{\left(\begin{array}{c}{m}_1{n}_2+m_1{n}_3+n_1{m}_2+n_1{m}_3+n_1{n}_2\\ +n_1{n}_3-k_1{l}_2-k_1{l}_3-l_1{k}_2-l_1{k}_3-l_1{l}_2-l_1{l}_3\end{array}\right)I\right\},\\ \beta \left\{m_1{k}_2+m_1{k}_3+k_1{m}_2+k_1{m}_3\right\}\\ \widehat{\widehat{\oplus }}\beta \left\{\left(\begin{array}{c}{m}_1{l}_2+m_1{l}_3+n_1{k}_2+n_1{k}_3+n_1{l}_2+n_1{l}_3\\ +k_1{n}_2+k_1{n}_3+l_1{m}_2+l_1{m}_3+l_1{n}_2+l_1{n}_3\end{array}\right)I\right\}\end{array}\right).\hfill \end{array}$$

$$\left(z_1^{*}\odot z_2^{*}\right)\oplus \left(z_1^{*}\odot z_3^{*}\right)=\left(\begin{array}{c}\alpha \left\{m_1{m}_2-k_1{k}_2+m_1{m}_3-k_1{k}_3\right\}\\ \widehat{\oplus }\alpha \left\{\left(\begin{array}{c}{m}_1{n}_2++n_1{m}_2+n_1{n}_2-k_1{l}_2-l_1{k}_2\\ -l_1{l}_2+m_1{n}_3+n_1{m}_3+n_1{n}_3-k_1{l}_3-l_1{k}_3-l_1{l}_3\end{array}\right)I\right\},\\ \beta \left\{m_1{k}_2+k_1{m}_2+m_1{k}_3+k_1{m}_3\right\}\\ \widehat{\widehat{\oplus }}\beta \left\{\left(\begin{array}{c}{m}_1{l}_2+n_1{k}_2+n_1{l}_2+k_1{n}_2+l_1{m}_2+l_1{n}_2\\ +m_1{l}_3+n_1{k}_3+n_1{l}_3+k_1{n}_3+l_1{m}_3+l_1{n}_3\end{array}\right)I\right\}\end{array}\right).$$

Thus, we can easily see that $z_1^{*}\odot \left(z_2^{*}\oplus z_3^{*}\right)=\left(z_1^{*}\odot z_2^{*}\right)\oplus \left(z_1^{*}\odot z_3^{*}\right)$.

□

Example 3. 

Let $z_1^{*}=\left(\alpha \left(1+3I\right),\beta \left(3+2I\right)\right),z_2^{*}=\left(\alpha \left(2+I\right),\beta \left(2-2I\right)\right)$, $z_3^{*}=\left(\alpha \left(3-I\right),\beta \left(-2-4I\right)\right)$.

$$\begin{array}{ccc}\hfill z_1^{*}\odot \left(z_2^{*}\odot z_3^{*}\right)& =& z_1^{*}\odot \left(\left(\alpha \left(2+I\right),\beta \left(2-2I\right)\right)\odot \left(\alpha \left(3-I\right),\beta \left(-2-4I\right)\right)\right)\hfill \\ & =& z_1^{*}\odot \left(\begin{array}{c}\alpha \left(6+4+\left(-2+3-1+8-4-8\right)I\right),\\ \beta \left(-4+6+\left(-8-2-4-2-6+2\right)I\right)\end{array}\right)\hfill \\ & =& \left(\alpha \left(1+3I\right),\beta \left(3+2I\right)\right)\odot \left(\alpha \left(10-4I\right),\beta \left(2-20I\right)\right)\hfill \\ & =& \left(\begin{array}{c}\alpha \left(10-6+\left(-4+30-12+60-4+40\right)I\right),\\ \beta \left(2+30+\left(-20+6-60-12+20-8\right)I\right)\end{array}\right)\hfill \\ & =& \left(\alpha \left(4-110I\right),\beta \left(32-74I\right)\right)\hfill \end{array}$$

and

$$\begin{array}{ccc}\hfill \left(z_1^{*}\odot z_2^{*}\right)\odot z_3^{*}& =& \left(\left(\alpha \left(1+3I\right),\beta \left(3+2I\right)\right)\odot \left(\alpha \left(2+I\right),\beta \left(2-2I\right)\right)\right)\odot z_3^{*}\hfill \\ & =& \left(\begin{array}{c}\alpha \left(2-6+\left(1+6+3+6-4+4\right)I\right),\\ \beta \left(2+6+\left(-2+6-6+3+4+2\right)I\right)\end{array}\right)\odot z_3^{*}\hfill \\ & =& \left(\alpha \left(-4+16I\right),\beta \left(8+7I\right)\right)\odot \left(\alpha \left(3-I\right),\beta \left(-2-4I\right)\right)\hfill \\ & =& \left(\begin{array}{c}\alpha \left(-12+16+\left(4+48-16+32+14+28\right)I\right),\\ \beta \left(8+24+\left(16-32-64-8+21-7\right)I\right)\end{array}\right)\hfill \\ & =& \left(\alpha \left(4-110I\right),\beta \left(32-74I\right)\right).\hfill \end{array}$$

Therefore, we can easily see that $z_1^{*}\odot \left(z_2^{*}\odot z_3^{*}\right)=\left(z_1^{*}\odot z_2^{*}\right)\odot z_3^{*}=\left(\alpha \left(4-110I\right),\beta \left(32-74I\right)\right)$.

Definition 6. 

Conjugate of a non-Newtonian neutrosophic complex number $z_1^{*}=\left({\dot{a}}_1,{\ddot{b}}_1\right)=\left(\alpha \left(m_1+n_{1}I\right),\beta \left(k_1+l_{1}I\right)\right)$ is $\overline{z_1^{*}}=\left({\dot{a}}_1,\widehat{\widehat{\ominus }}{\ddot{b}}_1\right)=\left(\alpha \left(m_1+n_{1}I\right),\beta \left(-k_1-l_{1}I\right)\right)$.

Example 4. 

For $z_1^{*}=\left(\alpha \left(-7+11I\right),\beta \left(3-8I\right)\right)$, its conjugate is $\overline{z_1^{*}}=\left(\alpha \left(-7+11I\right),\beta \left(-3+8I\right)\right)$.

Note 2. 

We can denote a non-Newtonian neutrosophic complex number as follows:

$$\begin{array}{ccc}\hfill z^{*}& =& \left(\dot{a},\ddot{b}\right)=\iota \left(\dot{a}\right)\widehat{\widehat{\oplus }}\ddot{b}\widehat{\widehat{\otimes }}{i}^{*}\hfill \\ & =& \iota \left(\alpha \left(m+nI\right)\right)\widehat{\widehat{\oplus }}\beta \left(k+lI\right)\widehat{\widehat{\otimes }}{i}^{*}.\hfill \end{array}$$

Definition 7. 

A non-Newtonian neutrosophic complex number can be determined as follows:

$$z^{*}=\iota \left(\dot{a}\right)\widehat{\widehat{\oplus }}\ddot{b}\widehat{\widehat{\otimes }}{i}^{*},$$

where $\iota \left(\dot{a}\right)=\iota \left(\alpha \left(m+nI\right)\right)$ is called the non-Newtonian neutrosophic scalar part of $z^{*}$, and $\ddot{b}=\beta \left(k+lI\right)$ is called the non-Newtonian neutrosophic imaginary part of $z^{*}$. These are denoted by $S_I^N\left(z^{*}\right)$ and $V_I^N\left(z^{*}\right)$, respectively.

Consequently, a non-Newtonian neutrosophic complex number $z^{*}$ can be written as

$$z^{*}=S_I^N\left(z^{*}\right)\widehat{\widehat{\oplus }}{V}_I^N\left(z^{*}\right)\widehat{\widehat{\otimes }}i.$$

Definition 8. 

The inverse of a non-Newtonian neutrosophic complex number is defined as ${\left(z^{*}\right)}^{-1}={\displaystyle \frac{\overline{z^{*}}}{{∥z^{*}∥}^2}}$.

Example 5. 

Let $z^{*}=\iota \left(\alpha \left(1+3I\right)\right)\widehat{\widehat{\oplus }}\beta \left(2+4I\right)\widehat{\widehat{\otimes }}{i}^{*}$; then, ${\left(z^{*}\right)}^{-1}={\displaystyle \frac{1+3I}{5+47I}}-{\displaystyle \frac{2+4I}{5+47I}}i$.

Proposition 3. 

Let $z_1^{*},z_2^{*}\in C_T$. We have

i.

$\overline{z_1^{*}\oplus z_2^{*}}=\overline{z_1^{*}}\oplus \overline{z_2^{*}}$, $\overline{z_1^{*}\ominus z_2^{*}}=\overline{z_1^{*}}\ominus \overline{z_2^{*}}$,

ii.

$\overline{z_1^{*}\odot z_2^{*}}=\overline{z_1^{*}}\odot \overline{z_2^{*}}$,

iii.

$\overline{\overline{z_1^{*}}}=z_1^{*}$.

Proof. 

For $m_1,m_2,n_1,n_2,k_1,k_2,l_1,l_2\in \mathbb{R}$, let $z_1^{*}=\left({\dot{a}}_1,{\ddot{b}}_1\right)=\left(\alpha \left(m_1+n_{1}I\right),\beta \left(k_1+l_{1}I\right)\right)$ and $z_2^{*}=\left({\dot{a}}_2,{\ddot{b}}_2\right)=\left(\alpha \left(m_2+n_{2}I\right),\beta \left(k_2+l_{2}I\right)\right)$ be any two elements of $C_T$.

i.

From the definition of the conjugate of a non-Newtonian neutrosophic complex number and the addition of non-Newtonian neutrosophic complex numbers, we can easily see that

$$\begin{array}{ccc}\hfill \overline{z_1^{*}\oplus z_2^{*}}& =& \overline{\left({\dot{a}}_1\widehat{\oplus }{\dot{a}}_2,{\ddot{b}}_1\widehat{\widehat{\oplus }}{\ddot{b}}_2\right)}\hfill \\ & =& \left({\dot{a}}_1\widehat{\oplus }{\dot{a}}_2,\widehat{\widehat{\ominus }}\left({\ddot{b}}_1\widehat{\widehat{\oplus }}{\ddot{b}}_2\right)\right)\hfill \\ & =& \left({\dot{a}}_1,\widehat{\widehat{\ominus }}{\ddot{b}}_1\right)\oplus \left({\dot{a}}_2,\widehat{\widehat{\ominus }}{\ddot{b}}_2\right)\hfill \\ & =& \overline{z_1^{*}}\oplus \overline{z_2^{*}}.\hfill \end{array}$$

ii.

$$\begin{array}{ccc}\hfill \overline{z_1^{*}}\odot \overline{z_2^{*}}& =& \left(\alpha \left(m_1+n_{1}I\right),\beta \left(-k_1-l_{1}I\right)\right)\odot \left(\alpha \left(m_2+n_{2}I\right),\beta \left(-k_2-l_{2}I\right)\right)\hfill \\ & =& \left(\alpha \left(m_1{m}_2-k_1{k}_2+cI\right),\beta \left(-m_1{k}_2-k_1{m}_2-dI\right)\right)\hfill \\ & =& \overline{z_1^{*}\odot z_2^{*}},\hfill \end{array}$$

where $c=m_1{n}_2+n_1{m}_2+n_1{n}_2-k_1{l}_2-l_1{k}_2-l_1{l}_2$ and $d=m_1{l}_2+n_1{k}_2+n_1{l}_2+k_1{n}_2+l_1{m}_2+l_1{n}_2$.

iii.

$$\begin{array}{ccc}\hfill \overline{\overline{z_1^{*}}}& =& \overline{\overline{\left(\alpha \left(m_1+n_{1}I\right),\beta \left(k_1+l_{1}I\right)\right)}}\hfill \\ & =& \overline{\left(\alpha \left(m_1+n_{1}I\right),\beta \left(-k_1-l_{1}I\right)\right)}\hfill \\ & =& \left(\alpha \left(m_1+n_{1}I\right),\beta \left(k_1+l_{1}I\right)\right)\hfill \\ & =& z_1^{*}\hfill \end{array}$$

□

Corollary 2. 

$\left(C_T,\oplus ,\odot \right)$ is a field.

Example 6. 

Let $z_1^{*}=\left(\alpha \left(-2-5I\right),\beta \left(8-I\right)\right),z_2^{*}=\left(\alpha \left(4+3I\right),\beta \left(-7+2I\right)\right)\in C_T$.

$$\begin{array}{ccc}\hfill \overline{z_1^{*}\oplus z_2^{*}}& =& \overline{\left(\alpha \left(-2-5I\right),\beta \left(8-I\right)\right)\oplus \left(\alpha \left(4+3I\right),\beta \left(-7+2I\right)\right)}\hfill \\ & =& \overline{\left(\alpha \left(2-2I\right),\beta \left(1+I\right)\right)}\hfill \\ & =& \left(\alpha \left(2-2I\right),\beta \left(-1-I\right)\right)\hfill \end{array}$$

and

$$\begin{array}{ccc}\hfill \overline{z_1^{*}}\oplus \overline{z_2^{*}}& =& \overline{\left(\alpha \left(-2-5I\right),\beta \left(8-I\right)\right)}\oplus \overline{\left(\alpha \left(4+3I\right),\beta \left(-7+2I\right)\right)}\hfill \\ & =& \left(\alpha \left(-2-5I\right),\beta \left(-8+I\right)\right)\oplus \left(\alpha \left(4+3I\right),\beta \left(7-2I\right)\right)\hfill \\ & =& \left(\alpha \left(2-2I\right),\beta \left(-1-I\right)\right).\hfill \end{array}$$

Then, $\overline{z_1^{*}\oplus z_2^{*}}=\overline{z_1^{*}}\oplus \overline{z_2^{*}}$, $\overline{z_1^{*}\ominus z_2^{*}}=\overline{z_1^{*}}\ominus \overline{z_2^{*}}=\left(\alpha \left(2-2I\right),\beta \left(-1-I\right)\right)$.

Proposition 4. 

Let $z_1^{*},z_2^{*}\in C_T$. We have

i.

$\ddot{\parallel }{z}_1^{*}\oplus z_2^{*}\ddot{\parallel }\widehat{\widehat{\le }}\ddot{\parallel }{z}_1^{*}\ddot{\parallel }\widehat{\widehat{\oplus }}\ddot{\parallel }{z}_2^{*}\ddot{\parallel }$ (non-Newtonian neutrosophic triangle inequality);

ii.

$\ddot{\parallel }{z}_1^{*}\odot z_2^{*}\ddot{\parallel }=\ddot{\parallel }{z}_1^{*}\ddot{\parallel }\widehat{\widehat{\otimes }}\ddot{\parallel }{z}_2^{*}\ddot{\parallel }$;

iii.

${\ddot{\parallel }{z}_1^{*}\ddot{\parallel }}^{\ddot{2}}=z_1^{*}\odot z_1^{*}$;

iv.

$\ddot{\parallel }{z}_1^{*}\ddot{\parallel }=\ddot{\parallel }\overline{z_1^{*}}\ddot{\parallel }$.

Proof. 

For $m_1,m_2,n_1,n_2,k_1,k_2,l_1,l_2\in \mathbb{R}$, let $z_1^{*}=\left({\dot{a}}_1,{\ddot{b}}_1\right)=\left(\alpha \left(m_1+n_{1}I\right),\beta \left(k_1+l_{1}I\right)\right)$ and $z_2^{*}=\left({\dot{a}}_2,{\ddot{b}}_2\right)=\left(\alpha \left(m_2+n_{2}I\right),\beta \left(k_2+l_{2}I\right)\right)$ be any two elements of $C_T$.

i.

$$\begin{array}{ccc}\hfill \ddot{\parallel }{z}_1^{*}\oplus z_2^{*}\ddot{\parallel }& =& \ddot{\parallel }\left(\alpha \left(m_1+m_2+\left(n_1+n_2\right)I\right),\beta \left(k_1+k_2+\left(l_1+l_2\right)I\right)\right)\ddot{\parallel }\hfill \\ & =& \ddot{\parallel }\left(\alpha \left(m_1+m_2\right)\widehat{\oplus }\alpha \left(n_1+n_2\right)\widehat{\otimes }{I}^{*},\beta \left(k_1+k_2\right)\widehat{\widehat{\oplus }}\beta \left(l_1+l_2\right)\widehat{\widehat{\otimes }}{I}^{*}\right)\ddot{\parallel }\hfill \\ & =& \beta \left(\sqrt{{\left(m_1+m_2+\left(n_1+n_2\right)I\right)}^2+{\left(k_1+k_2+\left(l_1+l_2\right)I\right)}^2}\right)\hfill \\ & \widehat{\widehat{\le }}& \beta \left(\sqrt{{\left(m_1+n_{1}I\right)}^2+{\left(k_1+l_{1}I\right)}^2}+\sqrt{{\left(m_2+n_{2}I\right)}^2+{\left(k_2+l_{2}I\right)}^2}\right)\hfill \\ & =& \beta \left\{\begin{array}{c}{\beta }^{-1}\left(\beta \left(\sqrt{{\left(m_1+n_{1}I\right)}^2+{\left(k_1+l_{1}I\right)}^2}\right)\right)\\ +{\beta }^{-1}\left(\beta \left(\sqrt{{\left(m_2+n_{2}I\right)}^2+{\left(k_2+l_{2}I\right)}^2}\right)\right)\end{array}\right\}\hfill \\ & =& \beta \left\{{\beta }^{-1}\left(\ddot{\parallel }{z}_1^{*}\ddot{\parallel }\right)+{\beta }^{-1}\left(\ddot{\parallel }{z}_2^{*}\ddot{\parallel }\right)\right\}\hfill \\ & =& \ddot{\parallel }{z}_1^{*}\ddot{\parallel }\widehat{\widehat{\oplus }}\ddot{\parallel }{z}_2^{*}\ddot{\parallel }.\hfill \end{array}$$

ii.

$$\begin{array}{ccc}\hfill \ddot{\parallel }{z}_1^{*}\odot z_2^{*}\ddot{\parallel }& =& \beta \left(\sqrt{\begin{array}{c}{\left[\left(m_1+n_{1}I\right)\left(m_2+n_{2}I\right)-\left(k_1+l_{1}I\right)\left(k_2+l_{2}I\right)\right]}^2\\ +{\left[\left(m_1+n_{1}I\right)\left(k_2+l_{2}I\right)+\left(k_1+l_{1}I\right)\left(m_2+n_{2}I\right)\right]}^2\end{array}}\right)\hfill \\ & =& \beta \left(\sqrt{{\left(m_1+n_{1}I\right)}^2+{\left(k_1+l_{1}I\right)}^2}\times \sqrt{{\left(m_2+n_{2}I\right)}^2+{\left(k_2+l_{2}I\right)}^2}\right)\hfill \\ & =& \beta \left\{\begin{array}{c}{\beta }^{-1}\left(\beta \left(\sqrt{{\left(m_1+n_{1}I\right)}^2+{\left(k_1+l_{1}I\right)}^2}\right)\right)\\ \times {\beta }^{-1}\left(\beta \left(\sqrt{{\left(m_2+n_{2}I\right)}^2+{\left(k_2+l_{2}I\right)}^2}\right)\right)\end{array}\right\}\hfill \\ & =& \beta \left\{{\beta }^{-1}\left(\ddot{\parallel }{z}_1^{*}\ddot{\parallel }\right)\times {\beta }^{-1}\left(\ddot{\parallel }{z}_2^{*}\ddot{\parallel }\right)\right\}\hfill \\ & =& \ddot{\parallel }{z}_1^{*}\ddot{\parallel }\widehat{\widehat{\otimes }}\ddot{\parallel }{z}_2^{*}\ddot{\parallel }.\hfill \end{array}$$

iii.

$$\begin{array}{ccc}\hfill z_1^{*}\odot z_1^{*}& =& \left(\alpha \left({\left(m_1+n_{1}I\right)}^2+{\left(k_1+l_{1}I\right)}^2\right),\beta \left(0\right)\right)\hfill \\ & =& \beta \left\{{\left(m_1+n_{1}I\right)}^2+{\left(k_1+l_{1}I\right)}^2\right\}\hfill \\ & =& \beta \left\{{\left({\beta }^{-1}\left(\beta \left(\sqrt{{\left(m_1+n_{1}I\right)}^2+{\left(k_1+l_{1}I\right)}^2}\right)\right)\right)}^2\right\}\hfill \\ & =& {\ddot{\parallel }{z}_1^{*}\ddot{\parallel }}^{\ddot{2}}.\hfill \end{array}$$

iv.

This can be easily proven using the definition of the conjugate of a non-Newtonian neutrosophic complex number and the ∇-norm of a non-Newtonian neutrosophic complex number.

□

Example 7. 

Let $z_1^{*}=\left(\alpha \left(2+4I\right),\beta \left(3-I\right)\right)\in C_T$. The conjugate of $z_1^{*}$ is $\overline{z_1^{*}}=\left(\alpha \left(2+4I\right),\beta \left(-3+I\right)\right)$.

$$\begin{array}{ccc}\hfill \ddot{\parallel }{z}_1^{*}\ddot{\parallel }& =& \beta \left(\sqrt{{\left(2+4I\right)}^2+{\left(3-I\right)}^2}\right)\hfill \\ & =& \beta \left(\sqrt{{\left(2+4I\right)}^2+{\left(-3+I\right)}^2}\right)\hfill \\ & =& \ddot{\parallel }\overline{z_1^{*}}\ddot{\parallel }.\hfill \end{array}$$

Definition 9. 

Let $\mathbf{X}$ be a vector space over the field $\mathbb{F}$. The map $\parallel .\parallel$ is called a norm and $\mathbf{X}$ is called a normed space, if the map $\parallel .\parallel$ satisfies the following properties [24,25]:

(i) 

$\parallel x\parallel \ge 0;\parallel x\parallel =0\iff x=0$;

(ii) 

$\parallel \lambda x\parallel =\left|\lambda \right|\parallel x\parallel$;

(iii) 

$\parallel x+y\parallel \le \parallel x\parallel +\parallel y\parallel$.

These properties are known as the norm axioms.

Theorem 3. 

$\left(C_T,\ddot{\parallel }.\ddot{\parallel }\right)$ is a normed space.

Proof. 

Using the definitions and properties above, it can be easily seen that $C_T$ satisfies the norm conditions. □

5. Conclusions

In this study, we developed a new approach to neutrosophic numbers and neutrosophic complex numbers based on non-Newtonian calculus. The non-Newtonian numbers and non-Newtonian complex numbers obtained through this approach possess a more general algebraic structure compared to existing studies. By using these numbers, a new perspective can be introduced to different algebraic structures such as quaternions and number sequences. Furthermore, these numbers can be applied in various fields, such as artificial intelligence, quantum mechanics, medicine, analysis, and geometry, providing a novel approach to solving problems in these areas.

In our future studies, we will introduce non-Newtonian neutrosophic bicomplex numbers and non-Newtonian quaternions and discuss their advantages by presenting their applications in the field of geometry.