What is an Ellipsis?
Coined from the Greek word meaning "omission," an ellipsis (plural: ellipses) is used to show omitted words. Quotations are common examples of where you might use them. For example:
The professor said, "It is always better ... to fail honorably than to pass in a dishonest manner". 

In the example above, the omitted words are "for students." Omissions might be needed if some words are irrelevant or make understanding the text much harder. An ellipsis indicates that you've left something out. 
Another way to use an ellipsis is when showing that a sentence trails off or to make a pause. However, this should not be adopted in academic or formal writing as it signifies a casual tone. You'll want to limit this use to informal writing and fiction. Here's an example:
So, Tracy actually went to … sorry, I forgot the location. But, can we take a ride home? I'm very hungry, or will it be fine to go to the …?

How Many Dots?
An ellipsis is composed of 3 dots. However, in cases you are using an ellipsis after a grammatically complete sentence, it may appear to be four periods in a row since the sentence needs its own period. For example:
"Take the kid to his mother. Don't forget her message too. You may also take him to his uncle." might be shortened to:
"Take the kid to his mother …. You may also take him to his uncle."

There are style variations when it comes to spacing the dots. While The AP Stylebook allows no spaces between the dots but on either side of the ellipsis, The Chicago Manual of Style finds that spaces are needed between each ellipsis point. Hence, there's no one-size-fits-all approach, but you should be consistent with any style you chose to adopt.

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