What does 'even' mean in this context?

Even is one of those words with lots of meanings. It can mean divisible by two or flat/level.

But it can be used in a sentence such as this:

Are you even listening?

What time does the shop even close?

I realise how it makes a difference to the meaning of the sentence... but how? What does it actually mean something in this sentence? Does it even mean anything?

Solution 1:

Even in this context is an adverb that is

used to emphasize something surprising or extreme : they have never even heard of the U.S. | they wore fur hats, even in summer. [NOAD]

It is merely a kind of intensifier, usually used to express the outrageousness of a thing. Now let's get down to cases:

Are you even listening?

The speaker doubts you are listening and is rebuking you for your inattention.

What time does the shop even close?

This one's a little harder, but probably someone has suggested the speaker go to this shop and the speaker doubts that it's open, or else someone has suggested that shop may be closed but the speaker is emphasizing the fact that the shop in fact stays open quite late. It would depend on context.

Solution 2:

The excellent blog Language Log talked about this use of "even" a while ago: What does "even" even mean?