Mathematics constantly discovers new forms of numbers, expanding our understanding of space and symmetry. This paper considers the next step—the introduction of a special unit that allows us to construct a new four-dimensional space with natural multiplication rules and a rich geometric interpretation.

Euler's formula [1] is well known, where the exponent contains an imaginary unit in the exponent. This unit, in general, can be used to organize 2D space over \(\mathbb{R}\): \[ e^{ix} = \cos x + i \sin x \] where: \(i\) is the imaginary unit [2].

In this paper, we obtain a 4D space over \(\mathbb{R}\) using a new imaginary unit, which has several advantages over similar algebras. Among others, the algebra is commutative and has a mathematically well-founded basis, something that, for example, quaternions cannot boast [3]. But let's start with the hyperbolic analogue of Euler's formula for 4D space: \[ e^{\it x} = a + \is^1 b + \is^2 c + \is^3 d \]

The algebra presented here is isomorphic to \(\mathbb{C} \oplus \mathbb{C}\) and can be factored over subalgebras, for example, into a pair of complex numbers, and manipulated independently. It easily introduces conjugations, norms, and works with series of the form \(e^{\iota x}\). Here, reflections and more exotic permutations between powers can be constructed. That is, there are potentially more symmetries than quaternions.

We'll call the new imaginary number "iota" — after this symbol. A more formal name for it is hypercomplex imaginary unit of order four.

It should be noted that obtaining the value of such a square root in the domain of real or even complex numbers is impossible. Describing it requires an extension to hypercomplex numbers. Let's show why this is so.

• In the real number domain, \(\sqrt{\mathstrut \i}\) is impossible, because \(\i\) doesn't exist there (after all, \(\i^2 = +1\), but \(\i \ne \pm 1\)).
• In the complex number domain, it's also impossible: if \(\i\) belonged to \(\mathbb{C}\), then from \(\is^2 = \i\) it would follow that \(\i \in \mathbb{C}\). But in the complex numbers, there's no element whose square is \(+1\), except \(\pm 1\). And we need a new one, independent of them.
• Consequently, \(\it = \sqrt{\mathstrut \i}\,\) is a new hypercomplex unit, which requires an extension of the algebra \(\mathbb{C}\).

In modern terms, this effectively introduces a new generator \(\it\) for the algebra \(\mathbb{R}[t]/(t^4 - 1)\). The new algebra has a basis: \[\tag{2} \{1, \is^1, \is^2, \is^3\} \]

We will outline some indirect ways of understanding the new number below. For now, let's find the powers of this new unit: \[\tag{3} \is^1 = \is, \quad \is^2 = \i, \quad \is^3 = \i \is, \quad \is^4 = 1 \] It's clear that these values ​​will repeat every four steps. We'll be actively using this property later.

In the power series, which is the Maclaurin expansion of the exponential function [5], \[\tag{4} \exp x = \sumn0 {x^n \over n!}, \] we substitute \[\tag{5} x = \ia \] We obtain the following series with repeating imaginary coefficients at its terms: \[\tag{6} \exp(\ia) = 1 + \it {\a^1 \over 1!} + \ik {\a^2 \over 2!} + \i\it {\a^3 \over 3!} + 1 {\a^4 \over 4!} + \it {\a^5 \over 5!} + \ldots \]

The graphs of such partial sum functions are shown in the following figure:

You can check this by adding these partial sums without imaginary coefficients.ents. As a result, we again obtain the exponential function: \[\tag{8} S_0 + S_1 + S_2 + S_3 = \exp\a \] Partial sums can also be checked this way: \[\tag{9} S_0 - S_1 + S_2 - S_3 = \exp(-\a) \] Also, some combinations of squares of partial sums may be interesting: \[\tag{10} S_0^2 + S_1^2 + S_2^2 + S_3^2 = \ch\!^2 \a \\ S_0^2 - S_1^2 + S_2^2 - S_3^2 = \cos^2\!\a \] You can also get some completely non-obvious combinations: \[\tag{11} S_1^2 + S_3^2 = 2 S_0 S_2 \\ S_0^2 + S_2^2 = 1 - 2 S_1 S_3 \] Such identities can be useful when working with modules. They will be presented in more detail in the next section of this paper.

The conjugate function of \(\exp(\ia)\) is conveniently defined by replacing \(\it \mapsto -\it\). Then, expansion (7) takes the form: \[\tag{12} \overline{\exp(\ia)} = \exp(-\ia) = S_0 - \is^1 S_1 + \is^2 S_2 - \is^3 S_3 \] The algebraic modulus is defined similarly to the complex case— as the product of a function and its conjugate: \[\tag{13} |\exp(\ia)| = \sqrt{\,\overline{\exp(\ia)} \cdot \exp(\ia)} = 1 \] At the same time, the Euclidean norm in the coefficient space \((S_0, S_1, S_2, S_3)\) is calculated by the formula: \[\tag{14} \sqrt{S_0^2 + S_1^2 + S_2^2 + S_3^2} = \ch a \] Thus, the algebraic modulus and the Euclidean length are defined differently and have different values.

The algebraic modulus (13) shows that the exponential with imaginary unit \(\it\) always remains on the "algebraic circle" of radius 1, similar to how \(e^{i a}\) describes the unit circle in the complex plane. The Euclidean norm (14), on the other hand, characterizes the geometry of the coefficient space and defines the surface of a hyperbolic sphere in four-dimensional space. Thus, the modulus and the Euclidean length describe different structures: the former is associated with the internal symmetry of the algebra, the latter with the geometry of real 4D space.

Definition. Let \(\it\) be a hypercomplex unit such that \[\tag{15} \is^2 = \i, \quad \i^2 = 1, \quad \is^4 = 1 \] Then the set of all expressions of the form \[\tag{16} z = S_0 + \is^1 S_1 + \is^2 S_2 + \is^3 S_3, \quad S_l \in \mathbb{R}, \] forms a linear space over \(\mathbb{R}\), which we will denote by \(\mathbb{R}_{\it}^4\).

Properties:

• The dimension of the space is 4, and its basis is: \(\{1, \is^1, \is^2, \is^3\}\).
• Multiplication in \(\mathbb{R}_{\it}^4\) is defined by the power rule \(\it\): \[\is^k \cdot \is^m = \is^{(k+m)\, \mathbb{mod}\, 4}. \]
• Unlike quaternions, multiplication here is commutative and associative.
• The set \(\mathbb{R}_{\it}^4\) is also an algebra, i.e., it defines linear operations and compatible multiplication.

In essence, this can be called a new 4D hypercomplex space based on \(\it\), since it does not coincide with quaternions, ordinary complex numbers, or double numbers.

For ordinary complex numbers, the exponential function has the form \[ e^{i x} = \cos x + i \sin x ,\] and its absolute value is \[|e^{i x}| = \sqrt{e^{i x} \cdot e^{-i x}} = 1 .\] In the case of the hypercomplex unit \(\it\), the structure is preserved: the algebraic absolute value \(|\exp(\ia)|\) is also equal to one, see (13). However, the Euclidean norm (14) is different and is expressed in terms of the hyperbolic cosine: \(\ch \a\). Thus, the analogy with the complex case preserves the "unit circle" but is supplemented by a new hyperbolic layer in the coefficient space. Furthermore, instead of 2D, we obtain a 4D space.

Quaternions have three imaginary units \(i,j,k\) and are a non-commutative algebra. Their modulus is determined by the Euclidean norm \[|q| = \sqrt{a^2 + b^2 + c^2 + d^2}, \] and is preserved under all rotations of the automorphism group SO(3). In contrast, the hypercomplex system with unity \(\it\) remains commutative and has a richer set of conjugations. The new algebra combines the properties of complex numbers (preservation of modulus) and extended hyperbolic symmetries, which quaternions lack.

Unlike quaternions, where imaginary units \(i,j,k\) and the rules for their multiplication are introduced as postulates (e.g., \[ ij = k, \quad ijk = -1, \] in the case of a hypercomplex unit, the basis and the rules for multiplying all its powers follow from strictly mathematical relations. Thus, from the definition of \(\is^2 = \i\) it follows \[ \it \cdot \is^2 = \is^3, \quad \is \cdot \is^3 = 1, \quad \is^2 \cdot \is^3 = \it, \] and so on—that is, the entire algebra, and its basis, is constructed naturally through powers and their properties.

This paper introduces a new fourth-order hypercomplex imaginary unit \(\it\), defined as the square root of the hyperbolic unit. The algebra \(\mathbb{R}_{\it}^4\) constructed from it forms a 4-dimensional space over \(\mathbb{R}\), distinct from complex numbers, quaternions, and doubles.

The main properties of the new space are:

• commutativity and associativity of multiplication;
• a natural basis arising from powers of \(\it\), rather than being postulated, as in the case of quaternions;
• the presence of several consistent definitions of modulus and conjugacy;
• a developed symmetry structure, including both complex analogs and new hyperbolic transformations.

Thus, the proposed construction opens the possibility of considering exponential functions, series, and geometric interpretations in the new hypercomplex space. Its further study may be useful both in purely algebraic studies and in applications requiring new ways to describe multidimensional symmetries.

A natural next step is to introduce another new hypercomplex number: \(\sqrt{\it}\) and obtain the iota-octanion space of 8D. Using the same principle, more powerful spaces with dimension \(2^N\) can be obtained. All of them will have the advantages of the algebra presented here, such as commutativity and associativity. Such structures may find application in modeling multidimensional symmetries, quantum physics, string theory, and information theory, where natural handling of high-dimensional numbers is required.