Quaternionic Extension in Vita Theory — Infinite Folding and Real Representation

Overview: This work presents Reiji’s reinterpretation of quaternion algebra within his original Vita Theory. Vita Theory proposes a numerical system that extends beyond infinity, defining new values such as \( \omega \) (uncountable infinity) and \( \text{vita} \) — a quantity greater than infinity, yet foldable into finite representation through functional limits such as \( \arctan(x) \). Within this framework, the imaginary units \( i, j, k \) of quaternion algebra are redefined as real constructs by embedding them in this infinite–finite dual system. This reinterpretation allows for the entire quaternionic structure to exist within an extended real continuum while preserving the conventional sign alternations (\( jk = i \Rightarrow kj = -i \)). Furthermore, Reiji introduces infinitesimal hierarchies such as \( +0, -0, 1/\infty, 1/\omega, 1/\text{vita} \), representing values infinitely close to zero but nonzero — an attempt to bridge the gap between discrete and continuous infinities.

Note: All content on this page is originally explained by Reiji in Japanese. The English version is translated by AI and structured by a parent, with Reiji's final approval.

Reiji's Words and Ideas

• This study expresses quaternions (\( i, j, k \)) using Vita Theory. Vita Theory (see the main article here) defines a system that extends numbers beyond infinity, introducing quantities such as \( \omega \) (uncountable infinity) and \( \text{vita} \).
• A quaternion extends the idea of complex numbers: by expanding \( i \) (the imaginary unit), one obtains a decomposed complex number \( j \), and extending again yields the quaternionic unit \( k \).
• Normally, \( i, j, k \) cannot be represented as real numbers, but Vita Theory defines quantities larger than infinity, enabling them to be expressed within an extended real framework.
• For example, while complex numbers are written as \( 3 + 2i \) or \( e-\pi i \), in Vita Theory they become \( 3 + 2(\sqrt{\infty^2 + 1}) \) or \( e-\pi(\sqrt{\infty^2 + 1)} \).
• The quaternionic multiplication rule remains consistent: \( jk = i \), \( kj = -i \), and so on — sign changes arise from multiplication order.
• The expression \[ \lim_{x \to \infty} \log x = \text{vita} \] defines vita as a boundary value of growth. Normally, \( \log x \) diverges, but by applying \( \arctan(x) \), infinity can be folded into a finite range.
• Thus, Vita Theory uses \( \arctan \) as a “folding function” to convert infinite expansion into finite form.
• Near zero, Vita Theory defines layered quantities: \( +0, -0, 1/\infty, 1/\omega, 1/\text{vita} \). Here, \( 1/\text{vita} \) represents one element of an uncountable division between \( 1/\omega \) and \( 0 \).
• Countable infinity corresponds to sequences such as \( 0, 1, 2, 3, \dots, \infty \), while uncountable infinity includes all real numbers between 0 and 1, infinitely subdivided.

Quaternionic Expansion with Vita Representation

A symbolic derivation redefining \( i, j, k \) using expressions such as \( \sqrt{\infty^2 + 1} \), mapping quaternionic relations within the Vita numerical framework.

Quaternionic Multiplication Relations

Red-ink summary showing the multiplication structure of \( i, j, k \), maintaining classical quaternion symmetry where reversed order inverts the sign.

Sketch: Arctan Mapping and Infinite Folding

A conceptual diagram illustrating how \( \arctan(x) \) confines infinite values within finite limits, representing the transition from linear infinity to circular compactification.

Infinitesimal Hierarchies Around Zero

A layered representation distinguishing between \( +0, -0, 1/\infty, 1/\omega, 1/\text{vita} \), suggesting a continuous yet structured approach to infinitesimal magnitudes.