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There are many rules to help us judge what the word "which" stands for in a sentence. But there always have so many exceptions. And the best way for judging seems to use these rules as well as make the meaning of a sentence logical. But do you meet the situations in which you don not have enough knowledge to see whether the sentence is logical, especially when you are a beginner in some field. For examples: enter image description here

In the above sentence, what does "which" stand for? The real variable or the symbol tan-1? If i have no idea about the argument of the complex variable of the logarithmic function, how can I make the judgement?

In these situations, as a native speaker, what will you do? Do you have a golden rules or some experiments? Or just search for the concepts until you know everything? Well, that makes sense, but sometimes will it be a great challenge?

Thanks in advance.

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    I'm not a native speaker, but the common approach we use (without being aware of it much, I believe, because reading anything in real-time happens so fast), we work backward, i.e., the first hypothesis would be the noun phrase right in front of it (on the left), if that fails (i.e., it doesn't make much sense), we'd try another hypothesis with the noun phrase further on the left, and keep going until we can make sense of the sentence. (Again, this usually happens in the blink of an eye.) That said, some sentences are really garden path-ish. – Damkerng T. Sep 14 '16 at 10:36
  • @Damkerng T Thank you for your answer~And what is the "-ish" means? A little? – Whisper of heart Sep 14 '16 at 16:22
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    You're welcome! I opted to use -ish in my comment because I was running out of space and wanted to shorten my last sentence a bit. It's a suffix that roughly means "having the nature or characteristic of". For more details, this question should be useful: ell.stackexchange.com/questions/19403/use-of-the-suffix-ish – Damkerng T. Sep 14 '16 at 16:32
  • For what it's worth, the word "argument" here refers to the concept of the argument of a complex number. – Tanner Swett Sep 14 '16 at 22:27
  • I believe it refers to "the inverse tangent function", rather than either of the two options you suggest. But I'd really need to see more of the context to be sure. I have a mathematics degree, and I don't believe it's in any way standard to use Arctan and tan^-1 to mean two different things in this way. – Dawood says reinstate Monica Sep 15 '16 at 2:45
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It's not you. It's them. I'm a mathematician and native English speaker, and I don't understand that sentence. I think that they are trying to say that they use the symbol tan⁻¹ for the 1-1 function on ℝ which is the inverse of the restriction of the tangent map to (-π/2, π/2), and use the symbol Arctan for an extension of it to ℂ (which either is thought of as multi-valued, or depends on the choice of branch of logarithm, since there's no single-valued analytic extension of this function).

As for what "which" means, well, it sounds like you already understand: it refers to the closest antecedent in the sentence that it could logically replace. As you say, if the reader is unfamiliar with a subject, it is difficult for the reader to determine what it could logically replace. (In this case, even with familiarity it seems difficult to tell).

Good technical writing often avoids long embedded clauses, for precisely this reason.

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    +1 I don't even know what characterizes means there. Represents? Stands for? Restricts? Qualifies? Gives shape to? – Tᴚoɯɐuo Sep 14 '16 at 11:39
  • Yes, I guess I understand it, but after about ten minutes. First, the two "of...of..."; then, "characterizes"; then, looked up two books which are translation edition. The first book just drop this sentence, which I hate this edition very much. The second, ah~ah~ah~ah~"the argument of the complex variable" is the phase angle..... then, every thing made sense. – Whisper of heart Sep 14 '16 at 16:10
  • @TRomano “describe” I guess, – Whisper of heart Sep 14 '16 at 16:18
  • @TRomano usually in math, "property P characterizes object O" means "Object O has property P and no other object does." It doesn't work in this context, though. – hunter Sep 16 '16 at 14:49
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I am a native speaker and quite unknowledgeable about maths, so I will try to answer.

I do not understand the sentence. However, I would assume "which" to refer to "the inverse tangent function of the real variable" or "the real variable". But I don't know which one is correct.

I would have to search the concepts individually and/or find a simpler resource.

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No, there is no golden rule. The sentence you quoted is an example of ambiguous pronoun reference. Ambiguous pronoun reference happens when there is more than one antecedent to which the pronoun may refer.

Really, it's up to the writer to make it clear. There are strategies that can be employed to avoid this problem.

Here's another example:

When Jon asked his father if he could borrow the car, he said he needed it to go to work.

In this example, he is ambiguous, and it's practically impossible to tell what the writer intended.

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Both arctan's take an argument (in one case it is a complex number, and in the other it is a real number). So the "which" allows you to choose which form of arctan you'd like to use.

The problem is that the same word is used for technically two different ideas, although, it is possible to draw a geometric diagram which indicates a strong parallel between the two styles of arctan!

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