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I jump at every chance to practice Korean and the store clerk was speaking Korean to the previous customer so I tried my best.

After greeting with "안녕하세요" and then fielding questions on how I know Korean, haha. I said "12(십이) 주세요." Is that formal enough to order from an older lady?

Then, when specifying which ones I wanted, she asked me "둘개?", "세개?", or "네개?"

But, why did she use the other number system? And, did I use the correct number system originally?

  • you should use 하나, 둘, 셋, 넷, ... for counting items – user17915 Apr 13 '17 at 7:31
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Is that formal enough to order from an older lady?

Yes, it is and that's how you do it whenever you order something.

why did she use the other number system?

First of all, excuse me for not knowing the exact name for this system. But I know that this is the numbering system you use for counting.

12 pencils = 연필 열 두자루
12 trees = 나무 열 두그루
12 cats = 고양이 열 두마리
12 donuts = 도넛 열 두개

Frankly, if you mixed up the numbering system we would still understand but technically your usage was not correct.

The other numbering system is generally used when your doing math.

2 x 3 = 이 곱하기 삼
2x - 3 = 이 엑스 빼기 삼
-2 + 3 = 마이너스 이 더하기 삼
2/3 = 삼 분의 이
2.3 = 이 점 삼

... and, when your counting days.

I'll come back in three days = 삼 일 뒤에 돌아오겠습니다.

... or you can use a whole new numbering system for days!

1 = 하루
2 = 이틀
3 = 사흘
4 = 나흘*
5 = 닷새*
6 = 엿새*
7 = 이레*
8 = 여드레*
9 = 아흐레*
10 = 열흘
(marked with * means not used often)

But what happens after 10 days? You go back to the math numbering system.

십 일일, 이 십일, ...

... and weeks are a little weird too.

I'll see you in a week. = 일 주(일) 뒤에 봐요.
The task is likely going to take 6 weeks. = 그 일은 아마 육 주(일) 정도 걸릴거예요.

It's funny because if you use the counting system, you end up talking about stocks.

I just bought 7 shares from Apple. 나 방금 애플 주식을 일곱 주를 샀어.

But everything goes back to normal with months. Phew!

After 1 month = 한 달 뒤
Within 20 months = 스무 달 안에

But beware, years are here to mess with you just one more time.

After 1 year = 일 년 뒤
Within 20 years = 이십 년 안에

When we're talking time, remember that only the hours follow the counting scheme and everything else, the math scheme.

After an hour = 한 시간 뒤에
After two hours and a half = 두 시간 반 뒤에 Within two and a quarter hours = 두 시간 십 오분 안에

You can think of it as, anything from and below than "minutes" is considered as a unit of measurement like "nanoseconds", or "kilometers," but hours are treated as units of time-things.

But the measurements can get a little tricky, too. We use the counting scheme for traditional measures but the math scheme for western ones.

한 근, 두 척, 세 뼘, ...
일 피트, 이 킬로미터, 삼 뉴튼, ...

Finally, ages. When you're counting ages you use the counting scheme for "살" but the math scheme for "세".

한 살, 두 살, ...
만 한 살, 만 두 살...

Kids under 5 cannot enter here. = 만 오 세 이하 영•유아는 이 곳에 들어올 수 없습니다.
Louis XIV = 루이 십 사세

In writing, we of course prefer using Arabic numbers to writing then out.

Whelp, so yeah counting in Korean can get very messy... Good luck!

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And, did I use the correct number system originally?

No, you use native numbers with 개. 한 개, 두 개, 세 개, ...

You should have said 열두 개 주세요 instead of 십이 주세요.

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