Sum of Floor and Floor of Negative
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Theorem
Let $x \in \R$. Then:
- $\floor x + \floor {-x} = \begin{cases} 0 & : x \in \Z \\ -1 & : x \notin \Z \end{cases}$
where $\floor x$ denotes the floor of $x$.
Proof
Let $x \in \Z$.
Then from Real Number is Integer iff equals Floor:
- $x = \floor x$
Now $x \in \Z \implies -x \in \Z$, so:
- $\floor {-x} = -x$
Thus:
- $\floor x + \floor {-x} = x + \paren {-x} = x - x = 0$
Now let $x \notin \Z$.
From Real Number is Floor plus Difference:
- $x = n + t$
where $n = \floor x$ and $t \in \hointr 0 1$.
Thus:
- $-x = -\paren {n + t} = -n - t = -n - 1 + \paren {1 - t}$
As $t \in \hointr 0 1$, we have:
- $1 - t \in \hointr 0 1$
Thus:
- $\floor {-x} = -n - 1$
So:
- $\floor x + \floor {-x} = n + \paren {-n - 1} = n - n - 1 = -1$
$\blacksquare$
Sources
- 1978: Thomas A. Whitelaw: An Introduction to Abstract Algebra ... (previous) ... (next): Chapter $2$: Some Properties of $\Z$: Exercise $2.1$