In the theory of unit space, there is a need for a special algebra [1] to correctly describe all of its properties. This is due to the fact that all actions there, in fact, occur under the square root, which arises automatically after transformation of a scalar function into its vector analogue. The use of this method does not allow the use of actions with the classical imaginary unit [2], and makes them unsuitable for our purposes.

For example, if the function is given \[ f(x) = \sqrt{ -\sin(x) } \tag{1}\] then in classical algebra it could be written as follows: \[ f(x) = i\, \sqrt{ \sin(x) } \tag{2}\] This will lead to the fact that when negative values ​​\(x\) appear, we get \[ f(x) = i\, \sqrt{ \sin(-x) } = i\, \sqrt{- \sin(x) } = i^2 \sqrt{\sin(x) } = - \sqrt{ \sin(x) } \tag{3}\] where: \(i\) -- imaginary unit, whose square is equal to minus one [2].

But in fact we need to get \[ f(x) = \sqrt{ -\sin(-x) } = \sqrt{ --\sin(x) } = \sqrt{ \sin(x) } \tag{4}\] which is quite logical, since no one forced us to take the minus out of the root sign :)

As we can see, there is a significant difference in the sign before the root between (3) and (4). The same problem automatically arises in the case of transforming scalar functions into vector ones according to the following theorem. The problem is completely solved if, along with vectors, we introduce hyperbolic numbers [3].

These numbers also have other names: paracomplex, split-complex, "Plucker numbers" and even dual or split. It will be convenient for us to use them specifically for case (4), when operations with a minus are not taken out of the radical expression. Let's talk a little about the rules of such algebra.

Definition
The number is of the form \[ z = a + \i b, \quad a,b \in \mathbb{R}, \quad \i^2 = 1 \tag{5}\] Here we introduce the symbol "\(\i\)" -- the hyperbolic unit, to distinguish it from the classical imaginary unit -- \(i\), which can also be used in this algebra, in general.

Basic Rules

1. Addition \[ (a + \i b) + (c + \i d) = (a + c) + \i (b + d) \]
2. Multiplication \[ (a + \i b) (c + \i d) = (ac + bd) + \i (ad + bc) \]
3. Conjugation \[ \overline{a + \i b} = a - \i b \]
4. "Norm" (by analogy with complex) \[ z \cdot \overline{z} = (a + \i b) (a - \i b) = a^2 - b^2 \] The norm is not always a positive number, and can be equal to zero, even at \(z \ne 0\).
5. Some useful operations with the hyperbolic unit \[ \i = -\oi \] \[ \oi\kern1pt^n = (-1)^n \i^n, \quad n \in \mathbb{Z} \] \[ \i^n \cdot \oi\kern1pt^n = (-1)^n \]

Let's compare these two algebras in the following table.

The difference between these two algebras can also be seen in the following example. Two roots of negative one can be multiplied, in general, in two ways: \[ \sqrt{-1} \cdot \sqrt{-1} = \sqrt{-1}^2 = -1 \tag{6}\] and \[ \sqrt{-1} \cdot \sqrt{-1} = \sqrt{-1 \cdot -1} = 1 \tag{7}\] Obviously, hyperbolic algebra uses the second possibility, and therefore it can be called «radicandescent», since all multiplications are performed here under the square root. Incidentally, this point has great philosophical significance.

The goal of this paper is to connect vector algebra with the algebra of hyperbolic numbers. Together, they will give us a tool that describes the formalism of unit space well, which can be called "hyperbolic vector algebra".

In this algebra, the scalar product of two vectors containing \(\i\) will be slightly different from the classical one: \[ \A \cdot \B = \sum_n \overline{a}_n\, b_n \tag{8}\] This is the so-called Hermitian scalar multiplication, where \(\sum \overline{a}_n b_n\) is the usual sum of coordinate products, but \(\overline{a}_n\) must be complex conjugate. That is, if \[ a_n = c_n + \i\kern1pt d_n \tag{9}\] then the complex conjugate value will be: \[ \overline{a}_n = c_n - \i\kern1pt d_n \tag{10}\] This implies the non-commutativity of such an algebra \[ \A \cdot \B \ne \B \cdot \A \tag{11}\] which leads to some restrictions, but only in cases involving the hyperbolic unit in the transformations (example). But hyperbolic functions [4] in a vector space do not require a hyperbolic unit, no matter how strange it may sound! There, the property of commutativity works quite well \[ \A \cdot \B = \B \cdot \A, \tag{12}\] and scalar multiplication is performed in the classical way. That is, hyperbolic vector algebra can be adapted to a specific problem. Its application can be seen in this work.

We will use a ready-made transformation of a scalar cosine into a vector one, which we will write in the following form: \[ \cos(x/2) \to \j_0 + \summa \j_{2n} (\i)^n X_{2n} \tag{13}\] In formula (13) on the right we got a vector cosine, in which \[ X_{2n} = \sqrt{ \frac12 {x^{2n} \over (2n)!} } \tag{14}\] Here: \(\j_n\) - unit vectors, \(n\) - integers.

Now let's try to multiply the vector cosine by itself according to the rules of hyperbolic vector algebra: \[ \left( \j_0 + \summa \j_n (\i)^n X_n \right) \cdot \left( \j_0 + \summa \j_n (\oi)^n X_n \right) = 1 + \summa (\i)^n (\oi)^n X_n^2 = \\ = 1 + \summa (-1)^n X_n^2 = {1 + \cos(x) \over 2} = \cos^2(x/2) \tag{15}\] This is what we needed to get. Multiplication of a hyperbolic unit by its conjugate, raised to the power \(n\), can be seen in the Basic Rules presented a little earlier. We take the sum of the series from the standard expansion in the Maclaurin series [5].

This paper proposed a formalism that combines vector algebra and the algebra of hyperbolic numbers, which provides a rigorous mathematical apparatus for describing the theory of a unit space. The introduction of a hyperbolic unit eliminates ambiguities in operations with expressions under the square root and allows for a natural interpretation of hyperbolic geometry and relativistic transformations. The constructed algebra has extended properties compared to the classical complex algebra, including the ability to describe non-classical metrics and identify the non-commutativity of a number of operations. The presented approach forms the basis for further research into the analytical properties of vector functions and the development of geometric models within the framework of the theory of unit space.