simply put i believe when you have an the absolute value is how far the number is away from zero on the number line. So if you had the absolute value of -4, it would be four.
try this for further understanding, including the real mathmatical definition
It is well known that a vector space always admits an algebraic (Hamel) basis. This is a theorem that follows from Zorn's lemma based on the Axiom of Choice (AC).
Now consider any specific instance of vector space. Since the AC axiom may or may not be included in the underlying set theory, might there be examples of vector spaces in which an Hamel basis actually doesn't exist ?