# Maths sentences: 足す and 掛ける

I noticed a lack of case marking particles in mathematical sentences:

• １足す２は３である。
• ３掛ける４は１２である。

Compare this with the English version:

• 1 plus 2 is 3
• 3 times 4 is 12

After a cursory dictionary search, it is revealed that `plus` and `times` are not verbs, but prepositions. "plus 2" and "times 4" are prepositional phrases. It seems that the structure for English is 3(times 4). While the structure in Japanese appears to me as (3掛ける)4

If we parse 3 × 4 = 12 into English or Japanese, does the difference in grammatical syntax result in a different situation? Which situation comes to mind in English and in Japanese?:

a. 3 × 4 = OOOO + OOOO + OOOO
b. 3 × 4 = OOO + OOO + OOO + OOO

• If we have to overlay English parts of speech on `1 + 2 = 3`, why not call `+` a coordinator? But really, it seems like we're just pronouncing the symbols as English words, and `+` is really an infix operator, not a preposition.
– user1478
Jan 31, 2016 at 3:20
• @snailboat But it seems to make sense as plain language in English, where verbs stand in between arguments. It doesn't in Japanese unless you employ reverse Polish notation... Jan 31, 2016 at 12:52

They are not grammatical phrases. We just read the symbols verbatim like:

[⁠1]{いち} [+]{たす} [⁠2]{に} [=]{は} [⁠3]{さん}

It has nothing different than saying:

[⁠1]{いち} [+]{プラス} [⁠2]{に} [=]{イコール} [⁠3]{さん}

which is also commonly heard. Though we have both [+]{たす／プラス} and [−]{ひく／マイナス}, [×]{かける} and [÷]{わる} only have native pronunciations.

See this link for common readings of more advanced expressions including a [>]{だいなり} b and A [⊇]{ふくむ} B, which would be `aはbより大きい` and `AはBを含む` if read grammatically.

If we parse 3 × 4 = 12 into English or Japanese, does the difference in grammatical syntax result in a different situation? Which situation comes to mind in English and in Japanese?

It reminds me of "order of multiplication" controversy that was in Japan a while ago, but anyway, according to Japanese official teaching guidelines, multiplicand comes first and multiplier second, which results in:

b. 3 × 4 = OOO + OOO + OOO + OOO

But I'm not sure if this is grammatically elicited, because even when you say 3に4を掛ける or 3を4に掛ける in natural word order, neither of them has such implication, unless you read it 3を4回掛ける.