einander

From Middle High German einander, from Old High German compound of ein (“one”) +‎ andar (“other”, modern German ander). Semantically limited by the ein (“one”) to singular contexts, similar to one another or each other.

The combined form einander does not inflect in modern German, and may be used unchanged in dative and accusative grammatical cases. Older forms could show inflection on the second ander portion, or on both the ein and ander portions. Any prepositions, which grammatically and semantically would apply to the second ander portion of the word, were already appearing in front of the compound term in Old High German, and in New High German (i.e., modern German), prepositions are compounded onto the front of the term.[1]

• IPA^(key): /aɪ̯ˈnandɐ/ (standard)
• IPA^(key): /aɪ̯nˈʔandɐ/ (uncommon and stiff, though etymologically correct)
• IPA^(key): /əˈnandɐ/ (colloquial)

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einander

1. each other

   Sie unterstützten einander, so gut sie konnten.

   They supported each other as much as they could.

• Prepositions are prefixed to the pronoun, thus “with each other” becomes miteinander. The spelling mit einander is not unfrequently seen, but is nonstandard nowadays.
• As opposed to English grammar, German einander is only obligatory after prepositions, or when the reciprocity of the action is not evident from the context:

Sie haben miteinander getanzt. ― They danced with each other.

Sie haben einander gewaschen. ― They washed each other.

• Otherwise it is common practice to use reflexive pronouns instead. Although not wrong, it would be quite formal—and sometimes even awkward—to use einander where it is not needed.

Sie gaben sich die Hand. ― They shook each other's hand. (literally, “They gave themselves the hand.”)

Wir trafen uns in der Kneipe. ― We met each other at the pub. (literally, “We met ourselves at the pub.”)

• In colloquial German, einander is generally used with prepositions only. Otherwise, the adverb gegenseitig (“mutually”) is added to accentuate reciprocity. This usage is also correct in standard German proper, but not quite as frequent as in the vernacular.

Sie haben sich gegenseitig gewaschen. ― They washed each other. (literally, “They have washed themselves mutually.”)

• “einander” in Duden online
• “einander” in Digitales Wörterbuch der deutschen Sprache