
der Irrtum
error
A false assumption, opinion, or action based on an incorrect assessment of reality.
Es war ein Irrtum zu glauben, dass der Test einfach werden würde.
It was an error to believe that the test would be easy.
error / misconception
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error
A false assumption, opinion, or action based on an incorrect assessment of reality.
Es war ein Irrtum zu glauben, dass der Test einfach werden würde.
It was an error to believe that the test would be easy.

to be mistaken
To be mistaken; to have a false idea about something.
Wenn du denkst, dass ich dir helfe, nachdem du mich belogen hast, bist du im Irrtum.
If you think I'm going to help you after you lied to me, you're mistaken.

error
A legal term for an incorrect understanding of reality that may entitle someone to contest a legal transaction.
Der Kaufvertrag konnte wegen eines wesentlichen Irrtums über die Eigenschaft der Sache angefochten werden.
The purchase contract could be contested because of a material error regarding the characteristics of the item.
'Der Irrtum' is derived from the verb 'sich irren', which means 'to be wrong' or 'to be mistaken'. An 'Irrtum' is therefore the result of having been mistaken.
Entschuldigung, da habe ich mich wohl geirrt.
Sorry, I guess I was mistaken there.
'Irrtum' is often used in written language or in formal situations to describe a false assumption or assessment.
Die Annahme, die Erde sei eine Scheibe, war ein großer wissenschaftlicher Irrtum.
The assumption that the Earth was a disk was a major scientific error.
Common formal expressions are 'einem Irrtum unterliegen' (to be mistaken) and 'ein Irrtum ist jemandem unterlaufen' (someone made an error).
Der Wissenschaftler erkannte, dass er einem Irrtum unterlegen war.
The scientist realized that he had been mistaken.
The phrase 'im Irrtum sein' is a fixed expression. It is formed with the preposition 'in' and the dative.
Ich fürchte, Sie sind da im Irrtum.
I'm afraid you're mistaken there.
mistake / error
An Irrtum is a false idea or assumption about reality. A Fehler is more often an incorrect decision, statement, or action, which can also result from an Irrtum. Use Irrtum for believing something incorrectly, and usually Fehler for acting incorrectly.

Die Annahme, der Termin beginne erst um zehn Uhr, war ein Irrtum.
The assumption that the appointment would not begin until ten was a misconception.
Wegen der falschen Uhrzeit war es ein Fehler, erst um zehn Uhr zum Termin zu kommen.
Because the time was wrong, arriving at the appointment only at ten was a mistake.