It seems that one month, one week, and one year in Korean are the following:
한 달 (one month)
한 주 (one week)
일 년 (one year)
Why do the first two use '한' while the last one uses '일'? And how can I classify them if I encounter a new word?
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It's complicated... sometimes you can even use both and it changes either the meaning or style of speech.
The book Using Korean devotes an entire chapter (about 9 pages) to this topic. Read the full details and numerous examples starting from page 169.
I'll try to summarize below (all quotes from Using Korean, examples have been greatly abridged):
In most cases, there is a strict division of labor in the use of native Korean numbers and Sino-Korean numbers. For example, native Korean numbers are used for counting (small numbers) and for o'clock (세시 "3 o'clock"), while Sino-Korean numbers are employed for minutes, dates, months, years, money, and so forth (16 절지 '8.5 by 11 size paper', 24 금 '24 K gold', and so on).
In some cases, both types of numbers can occur with the same counter, creating sharp contrasts in meaning.
Very rarely, either number can be used for the same concept, but this is correlated with a contrast in speech style.
Native numerals are used for counting smaller units. For larger quantities, Sino-Korean numerals are used, often in combination with native numbers. There is a tendency for Sino-Korean numbers to be used for multiples of 10, starting from 20, even when there is a native Korean counterpart.
Sino-Korean numbers are used for the metric system as well as some American units of measurement. Native numbers are employed only for a few traditional Korean units of measurement.
Both native and Sino-Korean numerals can be used for times, dates, and ages. A particularly notorious case involves telling time, which requires a native number for o'clock but a Sino-Korean number for minutes.
Sino-Korean numbers are used for arithmetical calculation, fractions (분수), decimals (소수), and multiplication tables (구구단). But native numbers are used for numerical comparison in general.
Sino-Korean numbers are used.
일 is not quantitative, in other words you should use 일 whenever you are not counting. So for example, telephone numbers, IDs, use 일 instead of 한.
Example: 일 항 -> paragraph one
한 is opposite, it is used only when you are counting.
Example: 한 마디 -> one sentence,
한 사람 -> one person
However, if there is no counter, use 하나 instead. Ex: 하나 있다 -> There is one
There are some exceptions, when 일 is used instead of 한(하나): 년, 초, 분, 주, 일, 개월.
There are still some others, but rarely will it happen, so just memorize them if you encounter one.
So in your example, 한 주 is wrong, but 일 주.
Correct me if I am wrong.