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12

R → ∞ is usually read R を限りなく大きくする[と・とき] R が限りなく大きくなる[と・とき] I don't think that 「R → ∞ のとき」 is supposed to have a fixed natural pronunciation. You can ignore the の and read it as above, or you could probably read it as [R]{アール} [→]{トゥ} [∞]{インフィニティ} のとき [R]{アール} [→]{ツー} [∞]{インフィニティ} のとき


9

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 ...


8

Two common ways of translating "if and only if" use the terms 必要十分条件 ("necessary and sufficient condition") and 同値 ("equivalence"). a > b は式 (15) である為の必要十分条件である。 Equation (15) holds if and only if a > b. 式 (15) と「a>b」とは同値である。 Equation (15) is equivalent to a > b.


7

In the context of mathematics, 「高々 (or たかだか) 一個」 is the standard expression. 高々 can also be used in non-technical context, but it's somewhat formal. In daily conversations, we'd say "多くて(も)一個" or "最大で一個", etc.


7

Not exactly (as several have commented). This is how you talk about fractions in Japanese: 7分の1 → 1/7 Literally, you can think about it as 'one part of seven'. It is not a ratio, i.e. 'one part to seven parts', as that equates to 1/8.


6

The sentence-final copula である ("be") is almost always omitted because it's obvious in definitions, leaving the sentences looking like ending with nouns. Both もの and こと are frequently used nominalizers translating "what do ~" and "doing ~" respectively. すなわち、1. (...) 2. (...) となるもの。 i.e. what satisfies 1. (...) and 2. (...). f:S→T が全射であるとは、f(S)=T ...


5

In Japanese, R is pronounced aaru (アール) → is pronounced yajirusi (矢印【やじるし】) ∞ is pronounced mugendai (無限大【むげんだい】) I think "n→∞" is often pronounced as follows in the differential and integral. エヌ矢印無限大 enu yajirusi mugendai エヌ無限大 enu mugendai where enu (エヌ) means the letter N. Therefore I guess that "R→∞" is pronounced in the same way. Although, ...


3

if and only if (= iff) a > b の時、(そして/かつ)その時に限り等式が成立する。 The equation is satisfied if and only if a > b. only ... if ~なければ~ない (colloquially ~なきゃ~ない or ~なけりゃ~ない) ≈ ~ないなら~ない アイスを買ってくれなきゃ行かない。 I'll only go if you buy me an ice. ~ないと~ない この植物は定期的に水をやらないと育たない。 The plant will only grow if it is watered regularly. The difference between ...


3

Yes, 「7分の1」 means same as 1:7 mathematically. 1:7 = (1/8):(7/8) = (1/8)/(7/8) = (1/8)x(8/7) = 8/56 = 1/7 However, in Japan, kids are taught that 1:7 is 「[比]{ひ}」 and 1/7 is 「[分数]{ぶんすう}」. I guess that 「比」 is translated as "ratios", and 「分数」 is translated as "fractions" generally. So, Japanese people tend to think that 1/7 is not a ratio, maybe. In ...



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