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The full sentence is: 彼はクラスで一番頭がいいと勘違いしている。

The part I'm not understanding is: 一番頭がいい
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2016-11-03T16:56:39.420
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question
meaning
How do you translate: 一番頭がいい
(The comment section has sorted this out, but just to summarize the whole
thing,)

「頭がいい」 is an idiom (慣用句) that means bright/intelligent. 「一番」 is literally
number one, and from there it means best/top/most/first. 一番 is strictly
speaking a noun, so from that perspective this should be 一番に頭がいい by inserting
a particle, but this particle often gets dropped, and 一番 can work like an
adjective, just like "best."

So the whole sentence translates to "He wrongly thinks he is the brightest in
this class."
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「[一番]{いちばん}X」, with X an adjective, means "the most X" (literally, "the first
[in a ranked list] X").

This is a casual (though polite) way to say this, and the formal way is
「[最]{もっと}もX」. Besides, there are adjectives that are already superlatives,
that you might come across with:

> [最高]{さいこう}  
>  the highest  
>  [最新]{さいしん}  
>  the newest  
>  [最適]{さいてき}  
>  the best, the most appropriate

These adjective are actually rather frequent, so you might want to give it a
look.
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answer