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.

'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
'der Fehler' and 'der Irrtum' both have to do with something being wrong. 'Fehler' is more general: it can be an incorrect action, a wrong answer, or a bad decision. 'Irrtum' is especially appropriate when someone believes something false or misjudges a situation. Learners may confuse the words because an 'Irrtum' often leads to a 'Fehler'. In practical terms: 'Irrtum' is more the false idea in your head, while 'Fehler' is more the wrong result or the incorrect action.

Sie war im Irrtum, denn der Termin ist nicht am Montag, sondern am Dienstag.
She was mistaken, because the appointment is not on Monday but on Tuesday.
Im Bericht ist ein Fehler, deshalb muss ich die Zahl noch einmal prüfen.
There is a mistake in the report, so I need to check the number again.