A Note on the Measure of Vector and Pythagorean Theorem
Abstract
Why the square? We present a geometry-axiom-free derivation of the Pythagorean theorem and the square at its core, establishing their algebraic origin from within the bare vector-space framework. Such concepts as the (right) angle, rotation, inner product, orthogonality etc also emerge as a logical construct rather than taken as given. They are necessitated by the square, and the ensuing theory, in turn, canonically stems from a single definitional primitive − the (ℝvcenter\scriptscriptstyle+-quantitative) invariant 𝒬-measure of a vector. This provides the core of an algebraic justification for Euclidean geometry. Equally important, these findings account (also canonically) for the complex modulus-squared p = |a|2 − the quantum Born rule − and point out what is even admissible for being quantitatively interpreted. The linear structure and its endomorphisms are rigid in the sense that the well−defined interpretable turns out to be, up to gauge 𝒬 → const vcenter\scriptstyle\times𝒬, the unique gauge-invariant measure 𝒬=|hspace−0.18em| ··· |hspace−0.18em|2; independently of the field ℝ or ℂ.
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BibTeX
@misc{brezhnev2026note,
title = {{A Note on the Measure of Vector and Pythagorean Theorem}},
author = {Yu. V. Brezhnev},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2608.24943},
eprint = {2608.24943},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.24943},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - A Note on the Measure of Vector and Pythagorean Theorem
AU - Yu. V. Brezhnev
PY - 2026
DA - 2026/09/22
PB - arXiv
DO - 10.48550/arxiv.2608.24943
UR - https://arxiv.org/abs/2608.24943
AB - Why the square? We present a geometry-axiom-free derivation of the Pythagorean theorem and the square at its core, establishing their algebraic origin from within the bare vector-space framework. Such concepts as the (right) angle, rotation, inner product, orthogonality etc also emerge as a logical construct rather than taken as given. They are necessitated by the square, and the ensuing theory, in turn, canonically stems from a single definitional primitive − the (ℝ^vcenter\scriptscriptstyle+-quantitative) invariant 𝒬-measure of a vector. This provides the core of an algebraic justification for Euclidean geometry. Equally important, these findings account (also canonically) for the complex modulus-squared p = |a|^2 − the quantum Born rule − and point out what is even admissible for being quantitatively interpreted. The linear structure and its endomorphisms are rigid in the sense that the well−defined interpretable turns out to be, up to gauge 𝒬 → const vcenter\scriptstyle\times𝒬, the unique gauge-invariant measure 𝒬=|hspace−0.18em| ··· |hspace−0.18em|^2; independently of the field ℝ or ℂ.
ER -