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Ni siquiera puedo dormir

Which is the translation for

I cannot even sleep.

So I was wondering why cannot it be

No siquiera puedo dormir.

How does 'No' became 'Ni' in the above sentence?

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neither and not even aren't equal in spanish language. Normally these mean tampoco and ni siquiera

I didn't even like these, neither those.

Here we say "ni siquiera me gustaron estos, tampoco esos." If we want to use ni siquiera, we need to put

I didn't even like these, not even those.

Which means "ni siquiera me gustaron estos, ni siquiera esos." However this sounds weird in spanish since we normally say the first sentence I put or

I didn't like these, not even those.

Which is "no me gustaron estos, ni siquiera esos."

The easiest way to differentiate ni from no in spanish is when we use no before a ni, this one comes immediately to state we don't want this stuff nor the other.

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Because Ni means Nor or Neither

Ni siquiera / Nisiquiera means neither or not even

An example of the difference with rough translation

No me gustan los tacos sucios a mi.

That is very straightforward... I don't like dirty tacos, but when you say this

Ni me gustan los tacos sucios a mi.

It implies that I don't like other things. Ni functions as a copulative conjunction meaning that it is used to aggregate, to add to the list of things.

A mi no me gustan los tacos sucios ni la basura mojada ni la carne perro ni el perder de dinero

It's also somewhat of an equivalent of the English me neither, although there is a word tampoco that does mean that already, I'm trying to make a point here.

A mi no me gustan los tacos sucios

Ni a mi

English phrases such as

I ordered an expensive beer and I didn't even get to drink it. :*)

can be translated to this

Ordené una cerveza cara y ni pude tomarla.

and

Not even George Washington liked dirty tacos.

translates to

Ni siquiera a George Washington le gustaban los tacos sucios

There are more descriptive definitions here

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