Infinite Repeating Decimals Are Transcendental and Irrational

Based on the transcendence criterion proved by Qi Jin, it can be shown that all infinite repeating decimals are transcendental numbers, and hence irrational. Consequently, infinite repeating decimals are not rational numbers. Treating repeating decimals as rational numbers is a logical error. The infinite repeating decimal 0.(9) and the number 1 are two distinct numbers, even though the difference between them tends to zero.

Authors

Publication Details

Journal
Zenodo (CERN European Organization for Nuclear Research)
Published
2026-10-28
DOI
https://doi.org/10.5281/zenodo.23726132
Primary Topic
Analytic Number Theory Research
Type
preprint
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preprint

Infinite Repeating Decimals Are Transcendental and Irrational

Qi Jin
2 links
Zenodo (CERN European Organization for Nuclear Research)
Analytic Number Theory Research
preprint

Infinite Repeating Decimals Are Transcendental and Irrational

Qi Jin
preprint en

Abstract

Based on the transcendence criterion proved by Qi Jin, it can be shown that all infinite repeating decimals are transcendental numbers, and hence irrational. Consequently, infinite repeating decimals are not rational numbers. Treating repeating decimals as rational numbers is a logical error. The infinite repeating decimal 0.(9) and the number 1 are two distinct numbers, even though the difference between them tends to zero.

Zenodo (CERN European Organization for Nuclear Research)
Analytic Number Theory Research
DOIOpen Access PDFOpenAlex
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Infinite Repeating Decimals Are Transcendental and Irrational — Qi Jin · Zenodo (CERN European Organization for Nuclear Research) (2026) | TGRS Research Map | TGRS