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Please see here: https://en.wikipedia.org/wiki/German_articles#Indefinite_article
As (das) Mädchen is neuter in German, in nominative singular one uses ein.
Else, for feminine nouns eine, for masculine ein (in nominative, singular).
No, it depends on the article "the". For instance, the article for "Frau" is "die" and for "Mann" is "der". For the article "die", the word "a" in English will be used as "eine" whereas for the article "das" and "der" is used as "ein".
To answer to your question, the article for "Mädchen" is "das" and "Junge" is "der". Therefore "a" for girl and boy is "ein".
First off, it's Mädchen (or Maedchen if you can't type the ä), not Madchen.
1) (a) because it's a diminutive in -chen and those are all neuter; (b) because the grammatical gender of a noun does not necessarily correspond to the real-life gender of the thing that the noun refers to. Person, for example, is always feminine, even though there are male and female persons.
2) Lots of people have asked why it's ein Mädchen but eine Frau, so while they did not use the word "neuter", they have wondered why it's not eine Mädchen.
The best translation is Frau = woman.
Often, "lady" is accepted on this course as a translation for Frau because many English speakers use "lady" instead of women simply as a polite synonym, not in the original meaning of a particularly high-class person.
I'd recommend thinking of it as Frau = woman, though.
Eh? But you didn't spell it correctly.
You wrote Madchen instead of Mädchen -- you left off the two dots on the A.
Thal would be like leaving oul lhe crossbar on lhe leller l -- you can'l jusl do lhal, since il's nol simply for decoralion bul inslead makes the difference belween lwo differenl lellers.
Similarly in German, a and ä are different letters.
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"the" implies that the other person knows exactly whom are you speaking about -- either because you had spoken about it before, or because it's obvious from context.
"a, an" implies that the other person does not know it -- it's possible that you know it but the listener does not, or that there is no particular one you had in mind, either, but are just talking generally.
"one" is similar to "a, an" but focusses on the fact that there was not zero or two but exactly one.
In other words, if you talk about "a woman", you do not expect the other person to know whom you mean, while if you talk about "the women", you do.
There are no German words mädchen or fraulein.
Mädchen is a noun and has to be capitalised. And Fräulein not only has to be capitalised, it has an umlaut in it (-lein and -chen tend to umlaut the word they are added to).
The two words do not mean the same thing, though -- Mädchen is a girl while Fräulein is "miss" (when speaking to someone) or "young lady".
Also, Fräulein refers to an unmarried person; nowadays, it's starting to fall out of ues as many people find that it's not relevant to mention a woman's marriage status.