
irrtümlich
mistakenly
Based on an error or mistake; happening by accident.
Ich habe Ihnen irrtümlich die falsche Rechnung geschickt.
I mistakenly sent you the wrong invoice.
mistaken
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mistakenly
Based on an error or mistake; happening by accident.
Ich habe Ihnen irrtümlich die falsche Rechnung geschickt.
I mistakenly sent you the wrong invoice.
'Irrtümlich' (mistakenly) is derived from the noun 'der Irrtum' (the mistake, the error). When something happens 'irrtümlich' (mistakenly), it is based on a mistake.
Seine Annahme basierte auf einem Irrtum.
His assumption was based on a mistake.
'Irrtümlich' (mistaken) can stand as an adjective before a noun or as an adverb that describes a verb. The adverbial use is very common.
Das war eine irrtümliche Überweisung.
That was a mistaken transfer.
The word 'irrtümlich' (mistakenly) is often used in written language and in more formal contexts, for example in business correspondence or with public authorities.
Der Betrag wurde irrtümlich von Ihrem Konto abgebucht.
The amount was mistakenly debited from your account.
crazy / awesome
Irre and irrtümlich can both refer to something false or resulting from a mistake. Irtümlich is the normal word in present-day German for this meaning, while irre sounds elevated or outdated. In a current factual report, irrtümlich is therefore the better choice; irre is more likely to appear in older texts.

Der aktuelle Bericht enthält eine irrtümliche Angabe.
The current report contains an incorrect statement.
Der alte Bericht nennt die falsche Angabe irre.
The old report calls the incorrect information mistaken.