2025-08-29
Hyperbolic numbers and vector algebra in the theory of unit space
Complex numbers have significantly expanded the scope of mathematics: the imaginary unit allows us to work with negative squares and to describe rotations, oscillations, and many other processes. However, this does not mean that the only possible additional unit must be a quantity whose square equals \(-1\). There are problems that require preserving a sign within a transformation and splitting an expression into two independent algebraic components. To achieve this, a different unit the hyperbolic unit is employed.
In this article, we will first examine a simple example illustrating the limitations of the standard transformation involving the imaginary unit; then, we will introduce the rules governing hyperbolic numbers and apply them to vector algebra in a unit space. Finally, we will demonstrate how the hyperbolic unit naturally leads to two idempotent components.
Why the imaginary unit is sometimes insufficient
Consider the function
\[ f(x)=\sqrt{-\sin x}.\tag{1} \] Let us first choose a value of \(x\) such that \(\sin x>0\). Then, the negative sign can be factored out from under the square root using the imaginary unit:
\[ f(x)=i\sqrt{\sin x},\qquad i^2=-1.\tag{2} \] Now, let us replace \(x\) with \(-x\). If we do this in the original function (1), we obtain
\[ f(-x)=\sqrt{-\sin(-x)}=\sqrt{\sin x}.\tag{3} \] However, formally substituting \(-x\) into the right-hand side of (2) leads to a different result:
\[ i\sqrt{\sin(-x)} =i\sqrt{-\sin x} =i^2\sqrt{\sin x} =-\sqrt{\sqrt{\sin x}}.\tag{4} \] The difference between (3) and (4) does not constitute a contradiction in complex algebra. Formula (2) was derived under the condition \(\sin x>0\), whereas the sign of the sine changes upon switching to \(-x\). Consequently, the previous form of the expression cannot simply be extended automatically into the new domain. In complex algebra, the function should be written in a piecewise manner:
\[ f(x)= \begin{cases} i\sqrt{\sin x}, & \sin x\geqslant0,\\ \sqrt{-\sin x}, & \sin x<0. \end{cases}\tag{5} \] The imaginary unit effectively represents a negative square, but once the sign is factored out from under the root, the domain of applicability of the transformation must be monitored separately. Problems involving a unitary space require a different algebraic element that allows the splitting to be preserved within the transformation itself.
The hyperbolic unit \(\j\) serves as such an element. It does not replace the imaginary unit \(i\), nor is it another value of \(\sqrt{-1}\). This is an independent element of a different algebra.
Hyperbolic numbers
Hyperbolic numbers are also called paracomplex or split-complex numbers. Unlike the imaginary unit, the square of the hyperbolic unit equals positive one, although the unit itself is neither \(+1\) nor \(-1\):
\[ \j^2=1,\qquad \j\ne\pm1.\tag{6} \] A hyperbolic number has the form
\[ z=a+\j b,\qquad a,b\in\mathbb{R}.\tag{7} \] Hyperbolic numbers should not be confused with dual numbers. For the dual unit, \(\varepsilon^2=0\) holds, so it forms a different algebra.
Basic rules
1. Addition
\[ (a+\j b)+(c+\j d)=(a+c)+\j(b+d).\tag{8} \] 2. Multiplication
\[ (a+\j b)(c+\j d)=(ac+bd)+\j(ad+bc).\tag{9} \] 3. Conjugation
\[ \overline{z}=\overline{a+\j b}=a-\j b.\tag{10} \] 4. Quadratic form
\[ N(z)=z\overline z=(a+\j b)(a-\j b)=a^2-b^2.\tag{11} \] The quantity \(N(z)\) is often called the norm by analogy with complex numbers, but it can be positive, negative, or zero. Therefore, strictly speaking, it is not an ordinary positive-definite norm, but a quadratic form.
5. Conjugate powers of the hyperbolic unit
\[ \overline{\j}=-\j,\qquad \overline{\j}^{\,n}=(-1)^n\j^n,\qquad \j^n\overline{\j}^{\,n}=(-1)^n, \quad n\in\mathbb{Z}.\tag{12} \] Comparison of complex and hyperbolic algebras
Let us compare the roles of the two units and the geometry of the corresponding number systems.
| Type | Square | Number form | Norm | Geometry / meaning |
|---|---|---|---|---|
| \(i\) | \(i^2=-1\) | \(a+ib\) | \(a^2+b^2\geqslant0\) | Ordinary plane, rotations and trigonometry. Basis of classical complex analysis |
| \(\j\) | \(\j^2=+1\) \(\j \ne 1\) | \(a+\j b\) | \(a^2-b^2\), can be greater than, less than, or equal to zero | Hyperbolic plane, transformationsLorentz (relativistic "rotations") |
Thus, the complex unit and the hyperbolic unit are intended for different algebraic operations. The complex unit is associated with a sign change upon repeated multiplication, since \(i^2=-1\). The hyperbolic unit yields a positive one, since \(j^2=1\), yet it maintains an independent direction within the two-dimensional algebra.
Zero divisors
Hyperbolic numbers possess an unusual property: the product of two non-zero numbers can equal zero. For example,
\[ (1+j)(1-j)=1-j^2=0.\tag{13} \] Such numbers are called zero divisors. Consequently, division is not always possible in hyperbolic algebra. For a number \(z=a+jb\), an inverse element exists only if
\[ a^2-b^2\ne0.\tag{14} \] This property is not a flaw of the algebra. On the contrary, it indicates the presence of two special, mutually annihilating directions, which can subsequently be used to split the space.
Connection to the theory of unitary space
In the theory of unitary space, a scalar function is transformed into a vector analogue whose coordinates may contain square roots. To manage the sign alternation in the subsequent scalar product, it is convenient to combine vector algebra with the algebra of hyperbolic numbers.
For vectors with hyperbolic coordinates, we introduce the conjugate scalar product
\[ \langle\A,\B\rangle =\sum_n\overline{a_n}b_n.\tag{15} \] If \(a_n=c_n+\j d_n\), then
\[ \overline{a_n}=c_n-\j d_n.\tag{16} \] Hyperbolic numbers themselves form a commutative algebra: \(zw=wz\). However, the conjugate scalar product is not symmetric in the general case. Instead of ordinary symmetry, conjugate symmetry holds:
\[ \langle\B,\A\rangle =\overline{\langle\A,\B\rangle}.\tag{17} \] Therefore, the difference between \(\langle\A,\B\rangle\) and \(\langle\B,\A\rangle\) does not imply non-commutativity of the hyperbolic numbers themselves. It arises from the conjugation of the first argument.
Example involving the vector cosine
Let us consider the transformation of the scalar cosine into a vector cosine, described earlier in the theory of unit space:
\[ \cos\frac{x}{2} \;\longrightarrow\; \mathbf{j}_0+\sum \mathbf{j}_{2n} j^n X_{2n},\tag{18} \] where \(\mathbf{j}_{2n}\) are mutually orthogonal unit vectors, and
\[ X_{2n}=\sqrt{\frac12\frac{x^{2n}}{(2n)!}}.\tag{19} \] Let us multiply the resulting vector by its conjugate. Due to the orthogonality of the basis directions, the mixed products vanish:
\[ \begin{aligned} &\left(\mathbf{j}_0+\sum \mathbf{j}_{2n} j^n X_{2n}\right) \cdot \left(\mathbf{j}_0+\sum \mathbf{j}_{2n} \overline{j}^{\,n} X_{2n}\right)={}\\ &=1+\sum j^n \overline{j}^{\,n} X_{2n}^2 =1+\frac12\sum (-1)^n\frac{x^{2n}}{(2n)!}={}\\ &=1+\frac12(\cos x-1) =\frac{1+\cos x}{2} =\cos^2\frac{x}{2}. \end{aligned}\tag{20} \] In this example, the hyperbolic unit does not play the role of the square root of \(-1\). Its task is to combine with the conjugate to form the factor \((-1)^n\), thereby restoring the correct sign alternation of the Maclaurin series.
Idempotent representation
One important application of the hyperbolic unit is the construction of two idempotent elements:
\[ \e=\frac{1+\j}{2},\qquad \eb=\frac{1-\j}{2}.\tag{21} \] They remain unchanged when squared and annihilate each other:
\[ \e^2=\e,\qquad \eb^2=\eb,\qquad \e\eb=0.\tag{22} \] Therefore, any hyperbolic number can be represented as the sum of two independent components:
\[ a+\j b=\e(a+b)+\eb(a-b).\tag{23} \] It is precisely this property that allows the hyperbolic unit to be used to construct split spaces. The idempotent basis and its geometric application are discussed in greater detail in subsequent works on the Wave Electricity.
Conclusion
Hyperbolic numbers do not correct complex algebra, nor do they replace the imaginary unit. They complement the mathematical toolkit with a distinct unit, \(\j^2=1\), designed for a different type of transformation. Their key features include an indefinite quadratic form, the presence of zero divisors, and the ability to decompose into two idempotent components.
In conjunction with vector algebra, these properties make it possible to control the signs of individual coordinates while maintaining them within a single transformation. Thus, the hyperbolic unit becomes a connecting element between the vector representation of functions, the unit space, and the resulting split geometry.