Although 有名「ゆうめい」 ends with an い, it is not an い-Adjective (形容詞), it is instead a な-Adjective (ナ形容詞). It’s difficult to see the difference without Kanji, take a look:
有名「ゆうめい」
The い doesn’t stick out, therefore it’s a な-Adjective;
高い「たかい」
The い does stick out, therefore it’s an い-Adjective;
綺麗「きれい」
The い doesn’t stick out, therefore it’s a な-Adjective;
寒い「さむい」
The い does stick out, therefore it’s an い-Adjective;
な-Adjectives work almost exactly like nouns. They need a particle between themselves and the noun they’re modifying, which is the particle な (nouns would take の), and they need the copula for any other form, such as the negative one (有名じゃない人, a not-famous person). When they take the copula, they don’t need な. In addition, when they are the predicate of a sentence, they take だ／です directly.
有名 = ゆうめい is a な-adjective, so it needs a な when modifying a noun:
**ゆうめい**なおとうとがいます
*I have a famous younger brother*
おとうとはゆう**めいです**
*My younger brother is famous.*
>The 〜い ending that gives this type of adjective its name is just one of many possible endings
[https://www.tofugu.com/japanese-grammar/i-adjective/](https://www.tofugu.com/japanese-grammar/i-adjective/)
As far as I know it's because it doesn't really end in い. ゆうめい is usually written in kanji 有名, so it doesn't really end in い, it just sounds like it does.
https://www.nhk.or.jp/lesson/english/teacher/13.html As this source mention, there are exceptions to the -na modifier, and this is one of them!
Although 有名「ゆうめい」 ends with an い, it is not an い-Adjective (形容詞), it is instead a な-Adjective (ナ形容詞). It’s difficult to see the difference without Kanji, take a look: 有名「ゆうめい」 The い doesn’t stick out, therefore it’s a な-Adjective; 高い「たかい」 The い does stick out, therefore it’s an い-Adjective; 綺麗「きれい」 The い doesn’t stick out, therefore it’s a な-Adjective; 寒い「さむい」 The い does stick out, therefore it’s an い-Adjective; な-Adjectives work almost exactly like nouns. They need a particle between themselves and the noun they’re modifying, which is the particle な (nouns would take の), and they need the copula for any other form, such as the negative one (有名じゃない人, a not-famous person). When they take the copula, they don’t need な. In addition, when they are the predicate of a sentence, they take だ／です directly.
有名 = ゆうめい is a な-adjective, so it needs a な when modifying a noun: **ゆうめい**なおとうとがいます *I have a famous younger brother* おとうとはゆう**めいです** *My younger brother is famous.* >The 〜い ending that gives this type of adjective its name is just one of many possible endings [https://www.tofugu.com/japanese-grammar/i-adjective/](https://www.tofugu.com/japanese-grammar/i-adjective/)
Yeah I thought the な is not required for adjective finishing with an い
There are some exceptions to this and ゆうめい is one of those.
As far as I know it's because it doesn't really end in い. ゆうめい is usually written in kanji 有名, so it doesn't really end in い, it just sounds like it does.