25
As it turns out, there are Japanese numbers greater than 10!
Getting started, let's review the basics:
1 through 9:
[一つ]{ひとつ}、[二つ]{ふたつ}、[三つ]{みっつ}、[四つ]{よっつ}、[五つ]{いつつ}、[六つ]{むっつ}、[七つ]{ななつ}、[八つ]{やっつ}、[九つ]{ここのつ}
Going above 20, つ changes into そ. Here are the 10s through 90:
[十]{とお}、[二十]{はた}、[三十]{みそ}、[四十]{よそ}、[五十]{いそ}、[六十]{むそ}、[七十]{ななそ}、[八十]{やそ}、[九十]{...
23
つ is a generic inanimate counter. It might be confusing sometimes, but you can get away with using it for lots of inanimate objects if you don't know a more appropriate counter.
Don't use つ for animals or people.
You should use 匹 for kittens in your example, not つ. The basic counter for people is 人{にん}.
In general, animate objects (animals and people) ...
18
Its use is not limited to photos. もう一丁 is an idiomatic phrase which just means "one more try", "give it another shot".
丁 is also used as a counter for dishes (of food), and today もう一丁 is typically heard as a vigorous call in some restaurants (「牛丼、並一丁！」「ラーメン大盛り一丁！ もう一丁！」）. I think you can just memorize it as it is.
Photographers may say もう一丁 to their ...
16
I believe that the usual counter is 話{わ}, literally meaning stories, so you'd say 10話 for ten episodes, and 第10話 or 10話目 for the tenth episode.
Occasionally I've seen shows that used different counters for their own title cards. For example, ふしぎの海のナディア numbered all its episodes using 回, so for instance the tenth episode was 第10回. And 神秘の世界エルハザード numbered ...
16
You can read the time of day in 24-hour format using the pronunciation for the numbers 13–24 as for the numbers 1–10 followed by 時【じ】, e.g.
15:40 = 15時40分 = じゅうごじ よんじゅっぷん
19:20 = 19時20分 = じゅうくじ にじゅっぷん
In particular, exceptions to the usual readings are the same
4時 = よじ → 14時 = じゅうよじ, 24時 = にじゅうよじ
7時 = しちじ → 17時 = じゅうしちじ
9時 = くじ → 19時 = じゅうくじ
...
15
In old Chinese--where Japan borrowed the term--口 was a counter for people. More specifically, it referred to the number of people needing food. English has this construction too: the number of "mouths" (=people) to feed.
As for 個, it generally refers to individual non-animate items.
15
That is the 略字 for 枡, the square vessel used for sake and a measurement of volume.
EDIT: Punningly, also used as an abbreviation for the verb ending ーます during the Edo period:
また、「ます」と呼ぶことから丁寧の語尾（助動詞）の「ます」の置き換えとしても使用されることが多かった。（例：豆腐あり〼）この用例は江戸時代にはかなり多かったが現代になってからは使用頻度が少なくなった。
14
取り寄せるのに１日から２日は見ておいたほうがいいからね。
You know, (expect) it takes at least one day or two before the product arrives (to our office/department).
This 見る is "to expect", "to estimate", etc. See the tenth definition here.
1日 and 2日 refer to the time length between the order and the arrival. The ついたち/いちにち distinction is special and important, but ふつか, みっか and so ...
13
Googling the two terms in Japanese, there are a great deal of discussions among the Japanese about when to use which as clearly ambiguous to them as much as it is to you. One of such quoted the definitions from Dictionary of How to Count (『数え方の辞典』) written by Asako Iida (飯田朝子).
【匹】
・大型ではない生物全般。
・小型の哺乳類。
・小型の爬虫類、両棲類。
・魚類。
【頭】
・大型の哺乳類。
...
13
You can also use ２か国語, ３か国語 etc. to talk about how many languages one speaks, as in...
「私は、３か国語が話せます。」
「彼は、フランス語とドイツ語と英語の３か国語が話せます。」
You can also use 言語, as in...
「僕は、フランス語とドイツ語と英語の３つの言語が話せます。」
You could also use terms like バイリンガル, トリリンガル, マルチリンガル, as in:
「私は、フランス語と英語のバイリンガルです。」
「彼女は、フランス語と英語とドイツ語が話せるトリリンガルです。」
12
As for pizza, 枚 always stands for a whole round pizza. Each cut slice is counted with 切【き】れ (一【ひと】切れ, 二【ふた】切れ...).
(top: ピザ 一枚, bottom: ピザ 一切れ)
11
This is pretty common in restaurants etc, both by staff and by customers.
I think it's just to make counting easier.
For example, ビール is counted with 杯{はい} when seved in a glass, but 本{ほん} when served in a bottle. If rice is served in a chawan, it would be 杯{はい}, but when served on a plate, it would be 皿{さら}.
So ～つ is just being used as a generic "X ...
11
I think that the sequence beginning 一番・二番・三番 corresponds roughly to number one, number two, number three. Each word in this sequence is formed from the combination of a numeral (一・二・三) with a counter meaning number (番).
You can take this list and add the ordinal suffix 目 to each word, producing the list 一番目・二番目・三番目, corresponding roughly to first, second, ...
11
The ordinal prefix 第 is read だい. This is sense two in 大辞泉:
［接頭］数を表す語に付いて、ものの順序を表すのに用いる。「世界―一の都会」「―五巻」「―三レース」「―六感」
The counter 話 is read わ, and it attaches to Sino-Japanese numerals such as いち.
Put it all together and you get だいいちわ.
11
Yes, 人【にん】 is a counter "agrees" with person, but no, it cannot play the role of a noun. Counters only makes a number able to modify a noun, but grammar prohibits it from having noun meaning. Thus, if you want to tell "two hundred people" you always have to say 200人【にん】の人【ひと】. It however doesn't mean the noun is not omissible.
村には500人の人が住んでいたが、...
11
counters are based on size and shape of physical items
I'm not sure what you mean by this, but not really --- I'd say we choose counters based upon what we perceive the object to be.
So, for ebooks (as in Kindle), 冊 is used. When you see an object in immersive VR, corresponding counter for that object in real world is used. (On the other hand, developers ...
10
I believe the most common thing to do with sandwiches is use つ. （サンドイッチを３つもらえますか？） There probably is a correct counter for loaves of bread, but I don't know it, and again I think つ is more common.
10
"Counter" "counter word" is [助数詞]{じょすうし}, but when you ask and respond to "What is the counter word for XX?" you'd usually say like...
「傘は何で数えますか？」
(lit. Do you count umbrellas with what?)
--「（傘は）『本』で数えます。」
「傘はなんて/どうやって/どのように/どんなふうに/どういうふうに数えますか？」
(lit. How do you count umbrellas?)
--「（傘は）『一本、二本』と数えます。」
「傘の数え方は？」 (informal)
(lit. The ...
10
Both いちたーん and いったーん are valid. (As an aside, わんたーん is also very common, especially when it's part of a longer phrase like 1ターンキル.)
There may be exceptions I'm not aware of, but basically foreign counters/units that start with K, T or P can be read both いち○○ and いっ○○.
1カラット: いちからっと・いっからっと
1キロ: いちきろ・いっきろ
1テスラ: いちてすら・いってすら
1トラック: いちとらっく・いっとらっく
1トン: いちとん・いっとん
...
9
A good resource for this type of question is 『数え方の辞典』. It includes the following:
映画の作品数・上映数は「本」で数えます。細長い映画のフィルムが、巻かれた状態でひとつの作品として扱われることに由来します。
There is further information on various cases when it may be counted as 作, 作品, 巻き, 巻 (kan), 齣 (koma), カット, シーン, 場面 etc. For more details, I recommend consulting this at a library or getting your own copy.
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