{"id":"https://openalex.org/W7214094401","doi":"https://doi.org/10.48550/arxiv.2609.25096","title":"A new proof of the irrationality of $e$","display_name":"A new proof of the irrationality of $e$","publication_year":2026,"publication_date":"2026-09-19","ids":{"openalex":"https://openalex.org/W7214094401","doi":"https://doi.org/10.48550/arxiv.2609.25096"},"language":"en","primary_location":{"id":"pmh:oai:arXiv.org:2609.25096","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2609.25096","pdf_url":"https://arxiv.org/pdf/2609.25096","source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":"2331-8422","issn":["2331-8422"],"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":"https://openalex.org/I205783295","host_organization_name":"Cornell University","host_organization_lineage":["https://openalex.org/I205783295"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"text"},"volume":null,"type":"preprint","is_traditional":true,"indexed_in":["arxiv","datacite"],"open_access":{"is_oa":true,"oa_status":"green","oa_url":"https://arxiv.org/pdf/2609.25096","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5010554782","display_name":"Achraf Ben Said","orcid":"https://orcid.org/0009-0008-1923-7020","observed_orcids":["https://orcid.org/0009-0008-1923-7020"]},"institutions":[{"id":"https://openalex.org/I121748325","display_name":"Universidad Complutense de Madrid","ror":"https://ror.org/02p0gd045","country_code":"ES","type":"education","lineage":["https://openalex.org/I121748325"]}],"countries":["ES"],"is_corresponding":true,"raw_author_name":"Said, Achraf Ben","raw_affiliation_strings":["Department of Analysis and Applied Mathematics , Complutense University of Madrid , Madrid 28040"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Analysis and Applied Mathematics , Complutense University of Madrid , Madrid 28040","institution_ids":["https://openalex.org/I121748325"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":1,"corresponding_author_ids":["https://openalex.org/A5010554782"],"corresponding_institution_ids":["https://openalex.org/I121748325"],"apc_list":null,"apc_paid":null,"fwci":null,"has_fulltext":true,"cited_by_count":0,"citation_normalized_percentile":null,"cited_by_percentile_year":null,"biblio":{"volume":null,"issue":null,"first_page":null,"last_page":null},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T11166","display_name":"Analytic Number Theory Research","score":0.9380000233650208,"subfield":{"id":"https://openalex.org/subfields/2602","display_name":"Algebra and Number Theory"},"field":{"id":"https://openalex.org/fields/26","display_name":"Mathematics"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},"topics":[{"id":"https://openalex.org/T11166","display_name":"Analytic Number Theory Research","score":0.9380000233650208,"subfield":{"id":"https://openalex.org/subfields/2602","display_name":"Algebra and Number Theory"},"field":{"id":"https://openalex.org/fields/26","display_name":"Mathematics"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T11428","display_name":"Advanced Mathematical Identities","score":0.0215000007301569,"subfield":{"id":"https://openalex.org/subfields/2602","display_name":"Algebra and Number Theory"},"field":{"id":"https://openalex.org/fields/26","display_name":"Mathematics"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T12170","display_name":"History and Theory of Mathematics","score":0.02019999921321869,"subfield":{"id":"https://openalex.org/subfields/1207","display_name":"History and Philosophy of Science"},"field":{"id":"https://openalex.org/fields/12","display_name":"Arts and Humanities"},"domain":{"id":"https://openalex.org/domains/2","display_name":"Social Sciences"}}],"keywords":[{"id":"https://openalex.org/keywords/irrationality-of-e","display_name":"irrationality of e","score":0.9750000238418579},{"id":"https://openalex.org/keywords/euler-s-number","display_name":"Euler's number","score":0.9919999837875366},{"id":"https://openalex.org/keywords/irrationality-proofs","display_name":"irrationality proofs","score":0.8109999895095825},{"id":"https://openalex.org/keywords/continued-fractions","display_name":"continued fractions","score":0.6480000019073486},{"id":"https://openalex.org/keywords/taylor-expansion","display_name":"Taylor series expansion","score":0.8339999914169312}],"concepts":[{"id":"https://openalex.org/C2777019708","wikidata":"https://www.wikidata.org/wiki/Q1412340","display_name":"Irrationality","level":3,"score":0.8432999849319458},{"id":"https://openalex.org/C2780586882","wikidata":"https://www.wikidata.org/wiki/Q7520643","display_name":"Simple (philosophy)","level":2,"score":0.6128000020980835},{"id":"https://openalex.org/C144237770","wikidata":"https://www.wikidata.org/wiki/Q747534","display_name":"Mathematical economics","level":1,"score":0.6031000018119812},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.5680000185966492},{"id":"https://openalex.org/C143724316","wikidata":"https://www.wikidata.org/wiki/Q312468","display_name":"Series (stratigraphy)","level":2,"score":0.5252000093460083},{"id":"https://openalex.org/C2777686260","wikidata":"https://www.wikidata.org/wiki/Q144037","display_name":"Calculus (dental)","level":2,"score":0.5027999877929688},{"id":"https://openalex.org/C108710211","wikidata":"https://www.wikidata.org/wiki/Q11538","display_name":"Mathematical proof","level":2,"score":0.2777000069618225},{"id":"https://openalex.org/C111472728","wikidata":"https://www.wikidata.org/wiki/Q9471","display_name":"Epistemology","level":1,"score":0.27399998903274536},{"id":"https://openalex.org/C28901747","wikidata":"https://www.wikidata.org/wiki/Q177571","display_name":"Decision theory","level":2,"score":0.2694000005722046},{"id":"https://openalex.org/C149782125","wikidata":"https://www.wikidata.org/wiki/Q160039","display_name":"Econometrics","level":1,"score":0.26820001006126404},{"id":"https://openalex.org/C162324750","wikidata":"https://www.wikidata.org/wiki/Q8134","display_name":"Economics","level":0,"score":0.25380000472068787},{"id":"https://openalex.org/C18762648","wikidata":"https://www.wikidata.org/wiki/Q42213","display_name":"Work (physics)","level":2,"score":0.25290000438690186},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.25220000743865967},{"id":"https://openalex.org/C138885662","wikidata":"https://www.wikidata.org/wiki/Q5891","display_name":"Philosophy","level":0,"score":0.250900000333786}],"mesh":[],"locations_count":2,"locations":[{"id":"pmh:oai:arXiv.org:2609.25096","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2609.25096","pdf_url":"https://arxiv.org/pdf/2609.25096","source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":"2331-8422","issn":["2331-8422"],"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":"https://openalex.org/I205783295","host_organization_name":"Cornell University","host_organization_lineage":["https://openalex.org/I205783295"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"text"},{"id":"doi:10.48550/arxiv.2609.25096","doi":"https://doi.org/10.48550/arxiv.2609.25096","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.2609.25096","pdf_url":null,"source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":"2331-8422","issn":["2331-8422"],"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":"https://openalex.org/I205783295","host_organization_name":"Cornell University","host_organization_lineage":["https://openalex.org/I205783295"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":null,"is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"Preprint"}],"best_oa_location":{"id":"pmh:oai:arXiv.org:2609.25096","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2609.25096","pdf_url":"https://arxiv.org/pdf/2609.25096","source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":"2331-8422","issn":["2331-8422"],"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":"https://openalex.org/I205783295","host_organization_name":"Cornell University","host_organization_lineage":["https://openalex.org/I205783295"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"text"},"sustainable_development_goals":[],"study_designs":[],"awards":[],"funders":[{"id":"https://openalex.org/F4320335598","display_name":"Agencia Estatal de Investigaci\u00f3n","ror":"https://ror.org/003x0zc53"}],"has_content":{"pdf":true,"grobid_xml":true},"content_urls":{"pdf":"https://content.openalex.org/works/W7214094401.pdf","grobid_xml":"https://content.openalex.org/works/W7214094401.grobid-xml"},"referenced_works_count":0,"referenced_works":[],"related_works":["https://openalex.org/W4386934239","https://openalex.org/W4413926529","https://openalex.org/W2995735249","https://openalex.org/W2798230879","https://openalex.org/W2943884522","https://openalex.org/W2952817937","https://openalex.org/W3156810895","https://openalex.org/W2952243148","https://openalex.org/W2014286092","https://openalex.org/W2158234582"],"abstract_inverted_index":{"In":[0],"this":[1],"short":[2],"note,":[3],"we":[4],"provide":[5],"a":[6],"simple":[7],"proof":[8],"of":[9,12],"the":[10,18],"irrationality":[11],"Euler's":[13],"number":[14],"$e$,":[15],"without":[16],"using":[17],"classical":[19],"estimates":[20],"based":[21],"on":[22],"continued":[23],"fractions":[24],"or":[25],"Taylor":[26],"series":[27],"expansions.":[28]},"counts_by_year":[],"updated_date":"2026-10-28T10:33:24.094632","created_date":"2026-09-24T00:00:00"}