This discusses Absolute Value as a geometric idea, in terms of lengths and distances.
A Quick History for Context
Historically numbers were used most commonly for two things; counting, and distances. As algebra and arithmetic developed however, negative numbers became a common tool and eventually with the advent of the Cartesian Plane, distances between points started getting a little weird. Intuitively we know that “the distance between and ” should be the same as “the distance between and ”, yet mathematically this was hard to write down uniformly. At first it was simple enough to say you “simply write down the larger number minus the smaller number” which would guarantee that the result was positive, but what if one of them was a variable? Then you don’t know which is larger, and regardless, having the notation somehow inherently depend on something qualitative (rather than quantitative) was exactly what mathematics was trying to move away from. This is where the absolute value was brought into play.
Notation and Usage
Absolute value is often referred to as the function that “makes the value positive”. We will discuss how this is done in the analytic viewpoint section, but as to why it is useful, the absolute value has two common usages.
The first usage is the historical one; ie to define a distance between two values. We notate the absolute value using two vertical bars, and so we can notate the phrase “the distance between and ” by . Most commonly we have some bound or equality for the distance. Most of the phrases are fairly self evident, but there is a table below with a sample of various words and their corresponding mathematical in(equalities) for reference. Keep in mind this table is by no means exhaustive, but should give a general idea of the process.
Key Word(s)  Math Symbol  Example Phrase  Corresponding Math 
“is”  The distance between and is 


“at most”  The distance between and is, at most, . 


“at least”  The distance between and is at least . 


“larger than”  The distance between and is larger than . 


“smaller than”  The distance between and is smaller than . 


“no more than”  The distance between and is no more than . 


“no less than”  The distance between and is no less than . 

