What is $\operatorname{frac(x)}$ or $\{x\}$?

Question

I understand this is an opinion kind of a question...but still:

Well I know that $\operatorname{frac(x)}$ or $\{x\}$ stands for the fractional part of $x$ but how is it exactly defined?

Quoting Wikipedia exactly:

The fractional part of a non‐negative real number $x$ is the excess beyond that number integer part: $\operatorname{frac} (x)=x - \lfloor x \rfloor,\; x > 0$.

There are various conflicting ways to extend the fractional part function to negative numbers. It is either defined as $$\operatorname{frac} (x)=x-\lfloor x \rfloor\tag{1}$$as the part of the number to the right of the radix point, $$\operatorname{frac} (x)=|x|-\lfloor |x| \rfloor\tag{2}$$or as the odd function:

$$\operatorname{frac} (x)=\begin{cases} x - \lfloor x \rfloor & x \ge 0 \\ x - \lceil x \rceil & x < 0 \end{cases}\tag{3}$$

For example, the number −1.3 has a fractional part of 0.7 according to the first definition, 0.3 according to the second definition and −0.3 according to the third definition.

So it is giving 3 possible candidates...Which one is correct ... or better, which one is used more?

Also why does such type of confusion remain in mathematics?

(For me , I like the $3^{\text{rd}}$ one more!)

Answer 1

This is more of an opinion question, and cannot really be answered. Nevertheless, it should be noted that there isn't a defined definition for negative numbers... it is all contextual based on the author. All three are thus "correct", and you can't say one is correct and the other two are not. Personally, I prefer the second definition due to the fact that it breaks $-1.3$ into $-1$ and $0.3$. The number $-1$ is an integer (and is thus not a rational number $\frac{p}{q}$ in simplest form), meaning that only the part that cannot be expressed as a integer is represented by the frac$(x)$ function. I must reiterate though, this is entirely contextual and you must specify which definition you are using

Answer 2

To answer your second question first: I think there is no clear consensus in mathematics simply because the fractional part is not a concept mathematicians use very often. I've read a lot of mathematical papers, and $\lfloor x\rfloor$ comes up often, but I don't think I've ever seen $\operatorname{frac}(x)$ appear. What makes it harder to reach a consensus is that even if someone did use $\operatorname{frac}(x)$ it is quite likely that the context would force $x$ to be positive, and so they wouldn't bother defining it for negative $x$.

Personally I think the first is preferable, as it preserves the useful identity $\operatorname{frac}(x+y)=\operatorname{frac}\big(\operatorname{frac}(x)+\operatorname{frac}(y)\big)$. I can see the argument for the third as well, but the second makes no sense to me. If you split $-1.3$ into two parts, they would be $-1$ and $-0.3$; $-1$ and $0.3$ makes $-0.7$.