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My question is about the place of to+infinitive in some sentences.

For example,

1- Before I went on holiday for a week, I was really eager to study at mathematics.

I know this is correct.

How about the following one I am just trying to give more emphasise to?

2- To study at mathematics, I was really eager before I went on holiday for a week.

I know 2 is not natural, however, I am just curious if it is grammatically correct and gives the right meaning still.

  • Related: english.stackexchange.com/a/177461/216106 – Davo Aug 23 '19 at 12:28
  • As a native US English speaker, I can't tell the intent of either sentence. I would remove "at" from both sentences (perhaps that's a British construct)? Are you eager to study math before holiday, or during holiday? That's what's not clear. Perhaps, "I was quite eager to study mathematics before going on holiday for a week." Or, "I was quite eager to study mathematics while on holiday for a week." Different meanings, though. – Edward Barnard Aug 27 '19 at 21:53
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Sentence 2 neither sounds natural nor is it grammatically correct.

(Ignoring to fact that you used "to study", the word "study" doesn't collocate with "at".)

You can sometimes treat the to-verb like a noun in these cases.

To play is to have fun.
To learn is to take in information.

You may notice that both the subject and object in my 2 examples above are to-verb. To be honest, it's hard for me to think of a sentence where the subject is a to-verb but the object isn't, so I think that's usually the correct form.

In your example #2, if you change it to:

Studying mathematics, I was really eager before I went on holiday for a week.

..., that would be grammatically correct but has a different meaning. It would mean:

I was really eager before I went on holiday for a week because I study mathematics.

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