Real Number is Floor plus Difference

Theorem
Let $x \in \R$ be a real number.

Let $\floor x$ be the floor of $x$. Let $n$ be a integer.

Then:
 * $x = n + t: n \in \Z, t \in \hointr 0 1 \iff n = \floor x$

where $\floor x$ is the floor of $x$.

Sufficient Condition
Let $x = n + t$, where $t \in \hointr 0 1$.

We have that $1 - t > 0$.

Thus:
 * $0 \le x - n < 1$

Thus:
 * $n \le x < n + 1$

That is, $n$ is the floor of $x$.

Necessary Condition
Let $n = \floor x$.

Let $t = x - \floor x$.

Then $x = n + t$.

From Real Number minus Floor:
 * $t = x - \floor x \in \hointr 0 1$

Also see

 * Real Number is Ceiling minus Difference
 * Definition:Fractional Part