Say I was trying to write the number 3,476,521,893,421 in kanji. How would I go about this? My guess was 三万四千七百六十五億二千百八十九万三千四百二十一 but it looks really wrong to me because it feels too Western to be correct.
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The counting system for large numbers is almost the same as Western numbers, except that digits are delimited into blocks of four instead of three. You add "thousand, million, billion, trillion, ..." for each 3 digit, and in Japanese, we add "万, 億, 兆, 京【けい】, ..." for each 4 digit.
The "correct" format to write long numbers depends on the situation.
As spoken I think you're right on the money with
I would also give the possible option of writing it with an extra 「一」for visual clarity and to look more classical.