however / again
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however
Expresses a contrast, a different perspective, or a new aspect; in return, on the other hand.
Die einen lobten den Film, die anderen wiederum fanden ihn langweilig.
Some praised the film, while others, however, found it boring.
As an adverb, 'wiederum' (however/again) can stand in different positions in the sentence. If it is in position 1, the verb follows. Often, however, it also stands directly after the subject it refers to.
Wiederum zeigte sich, dass die Planung fehlerhaft war.
Again, it became clear that the planning was flawed.
In the meaning 'andererseits' (on the other hand) or 'im Gegenzug' (in return), 'wiederum' (however) belongs more to written or formal language. It helps clearly separate arguments or perspectives from one another.
Das Hotel war teuer, bot aber wiederum einen exzellenten Service.
The hotel was expensive, but on the other hand it offered excellent service.
The meaning 'erneut' (again) or 'noch einmal' (once more) is rare for 'wiederum' (again) today and is considered old-fashioned or very formal. It is hardly used in everyday language anymore.
Nachdem der König seine Rede beendet hatte, sprach wiederum der Kanzler.
After the king had finished his speech, the chancellor spoke again.
again / back
'wieder' and 'wiederum' can both express that something happens again. In this sense, however, 'wieder' is much more normal and everyday. 'wiederum' sounds more formal or old-fashioned here. More often, 'wiederum' is used when showing a contrast or another aspect, similar to 'andererseits' (on the other hand). That is why the words can be confused: they look similar, but 'wieder' is usually suitable for repetition, while 'wiederum' often introduces a new perspective or a contrast.
Wir wollten ans Meer fahren, die Kinder wiederum hatten mehr Lust auf die Berge.
We wanted to go to the sea, but the children, on the other hand, felt more like going to the mountains.
Nach der Pause arbeitet die Mitarbeiterin wieder am Bericht.
After the break, the employee is working on the report again.