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I see my friend always writing sentences with "Am" without the accompany of "I", like : "Am supposedly watching Veronica Mars." The sentence just seems grammatically incorrect to me. Can someone clear this up for me?

Also, "am" can be used once "I" has taken it's position of the pronoun in the sentence correct? Ex:

I like to write, surf the net, and am also a member of the International club.
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LRainDropslI see my friend always writing sentences with "Am" without the accompany of "I", like : "Am supposedly watching Veronica Mars." The sentence just seems grammatically incorrect to me. Can someone clear this up for me?

Also, "am" can be used once "I" has taken it's position of the pronoun in the sentence correct? Ex:

I like to write, surf the net, and am also a member of the International club.

It's just a short-cut.
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It's sometimes called "telegraphic style" - as if you were sending a telegram and paying for each word, you would leave out some words to make it shorter. You might write like this in a diary, or in a very informal letter, especially if almost every sentence would otherwidse begin with "I."

In the other example you give, "I like to write, surf the net, and am also a member of the International Club," you need the "am" because what follows it is not a construction parallel to "I like to write" and "(I like to) surf the net." It's not a great sentence either way - I wouldn't say it's incorrrect (with "am") but it seems to me that in a list of three things they should be parallel in construction. I would prefer either a more parallel construction : "I like to write, surf the net and attend the International Club" or else two more distinct clauses: "I like to write and surf the net, and I am also a member of the International Club." (without a comma after "write" it looks less like a list.) I'm not sure what the appropriate grammar rules here are, I'm just trying to explain why the original sentence sounds rather awkward to me.
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Comments  
It is acceptable in mean of formality?
 khoff's reply was promoted to an answer.
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Am coming

It's a casual way of writing quick notes.

But we don't speak this way.