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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.


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 ...


証拠立てる is virtually never used in mathematical articles. I would translate it like so: すべてのxに対してP(x)が真であることを示せ。 すべてのxについてP(x)が成り立つことを証明せよ。


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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