Properties of the absolute value
We all learn at school that a number is either positive or negative, with positive numbers not requiring a plus sign because it's the standard everywhere in the world. We always assume a number without a sign to be positive. Now textbooks that I had at school and the teachers that I had back then would always say that modulus "erases" the sign. I always felt that operation to be meaningless because if a number has no sign, does it mean that the number is neither positive nor negative? What is it?
I remember that I struggled a lot with signs and rules. There are many youtube videos with teachers discussing precisely the signs and rules with signs. Now I can't speak for others, but maybe many people out there make this mistake [math]\displaystyle{ |a - b| = a + b }[/math]. That is, if we are told that the modulus erases the sign, then does it mean that we erase the minus sign of every operation in between the two bars? I pretty sure that the way we are told to "erase" the minus sign from numbers can very well lead to that erroneous conclusion. I have no idea and have never witnessed it to be honest, but maybe some teacher out there in the world has witnessed this [math]\displaystyle{ |2 \pm 2| = 22 }[/math]. I know it's absurd, but what if someone understood the absolute value as literally removing the sign from existence?
The way I was told what the absolute value of a number is causes more harm than any good because sometimes I wondered if there was an operation that is beyond the four basic arithmetic operations that we all know. The definition of the absolute value is a function, but we don't know that because we learn it before learning about functions.
[math]\displaystyle{ |x| = \begin{cases} \ \ \ x, \ \text{if} \ x \gt 0 \\ \ \ \ 0, \ \text{if} \ x = 0 \\ -x, \ \text{if} \ x \lt 0 \end{cases} }[/math]
You may have asked about the middle case. Zero can be placed in the first or the second case, it's a matter of preference. Or we can use a third case. It doesn't change the definition in any way. A number can never be greater than or less than itself, it can only be equal to itself.
