https://en.wikipedia.org/wiki/Fractional_part
I was reading the definition of fractional part, but I have not understood something and I want to make sure.
- Is the fractional part of a real number has to be greater than or equal to 0 but less than 1?
- What would happen if the fractional part of a number were greater or equal than 1?
My doubt arises because it mentions an interval, but I don't understand it well, I'm new
$\text{frac}(x)$ is defined to be $x-\lfloor x\rfloor$. Suppose we have a real number $x$. $x$ can be understood as an integer plus some more things that are too small to be represented as an integer.
Convention uses $\lfloor x\rfloor$ to represent this integer part, and $\text{frac}(x)$ to represent this fractional part. I think it is now clear where the formula $\text{frac}(x)=x-\lfloor x\rfloor$ come from.
Now to answer your questions.
Yes.
It can be equal to $0$ if $x$ is already an integer. For example, if we have $3$, It doesn't make sense to have a fractional part of $1$ and an integer part of $2$. The fractional part of $3$ is just $0$.
The fractional part can't be $1$ or greater than $1$ either. Remember that the fractional part is the "leftover bit that is too small to be represented as an integer". If we have $3.14$, it doesn't make sense to have a fractional part of $1.14$ and an integer part of $2$, since $1.14$ is big enough to be represented as $1$ and $0.14$. In this case, the integer part is $3$ and the fractional part is $0.14$. The fractional part must be smaller than 1.
As I've said above, this can't happen.
Now to understand why the inventors of Math defined so, suppose that the fractional part can be greater than $1$. Then, the fractional part of $3.14$ can be $0.14$, $1.14$, $2.14$ or just $3.14$. When a function has multiple values, it basically renders the function useless.