On the Insufficiency of the Complex Field for Inverse Hyperoperations
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| Format: | Recurso digital |
| Langue: | anglais |
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Zenodo
2025
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| _version_ | 1866901731252633600 |
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| author | Bradley, Thomas |
| author_facet | Bradley, Thomas |
| contents | <p>The historical development of number systems has been driven by the need to establish closure under inverse operations. This paper investigates whether this pattern continues for the hyperoperation of tetration. We present a formal proof that the field of complex numbers, C, is not closed under the inverse operation of tetration (the super-root). This is demonstrated by proving that an equation of the form x^x = w has no solution in C for certain hypercomplex values of w, with x^x = j serving as the canonical example. This "definitional hole" in complex analysis provides a rigorous justification for the necessity of higher-dimensional number systems, such as the quaternions, to achieve closure for the inverses of hyperoperations.</p> |
| format | Recurso digital |
| id | zenodo_https___doi_org_10_5281_zenodo_15814084 |
| institution | Zenodo |
| language | eng |
| publishDate | 2025 |
| publisher | Zenodo |
| record_format | zenodo |
| spellingShingle | On the Insufficiency of the Complex Field for Inverse Hyperoperations Bradley, Thomas tetration complex numbers quaternions hyperoperations super-root mathematical analysis <p>The historical development of number systems has been driven by the need to establish closure under inverse operations. This paper investigates whether this pattern continues for the hyperoperation of tetration. We present a formal proof that the field of complex numbers, C, is not closed under the inverse operation of tetration (the super-root). This is demonstrated by proving that an equation of the form x^x = w has no solution in C for certain hypercomplex values of w, with x^x = j serving as the canonical example. This "definitional hole" in complex analysis provides a rigorous justification for the necessity of higher-dimensional number systems, such as the quaternions, to achieve closure for the inverses of hyperoperations.</p> |
| title | On the Insufficiency of the Complex Field for Inverse Hyperoperations |
| topic | tetration complex numbers quaternions hyperoperations super-root mathematical analysis |
| url | https://doi.org/10.5281/zenodo.15814084 |