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I had a german teacher insist we write umlauts as two little dashes instead of dots, because "they're not trémas" (French for diaeresis) which are written as two little dots. The wikipedia article above seems to say they're the same. Was I lied to all these years ?

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To me, in Swedish, Ā and Ä and an A with two small dashes above would be the same letter, but with stylistic typeface differences. It doesn't change the letter itself. I have never heard of any German insisting their umlauts shouldn't be dots, so I think your German teacher was just a bit pedantic/insane.

Conceptually umlaut dots are different from trema dots, but who cares?

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