「casual Japanese」と「formal Japanese」の日本語のことばは？
How would one reference "casual Japanese" and "formal Japanese"? As in requesting someone to speak using casual Japanese rather than using "です" and "ます” etc.

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「～だよ」、「～だな」、「～だよね」are casual japanese.

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「formal Japanese」：書き言葉.
「casual Japanese」：話し言葉

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There are so many ways in each way of speaking, both in the formal and informal ones. In the casual form, there are several level, for example, from the one sounds friendly until the one sounds rude, it's a bit complicated. However, in general, it is distinguished "using です/ます or not. I'd like to recommend the learners to learn the formal form first, then getting accustomed to the casual form is better in my opinion, not to speak rudely.