TAOCP 1.2.4 Exercise 1
By the definitions of floor and ceiling, \lfloor 1.
Section 1.2.4: Integer Functions and Elementary Number Theory
Exercise 1. [00] What are $\lfloor 1.1 \rfloor$, $\lceil -1.1 \rceil$, $\lfloor -1.1 \rfloor$, $\lfloor 0.99999 \rfloor$, and $\lfloor \lg 35 \rfloor$?
Verified: yes
Solve time: 36s
By the definitions of floor and ceiling,
$$ \lfloor 1.1 \rfloor = 1,\qquad \lceil -1.1 \rceil = -1,\qquad \lfloor -1.1 \rfloor = -2, $$
since $1 \le 1.1 < 2$, $-2 < -1.1 \le -1$, and $-2 \le -1.1 < -1$.
Also,
$$ \lfloor 0.99999 \rfloor = 0, $$
because $0 \le 0.99999 < 1$. Since $2^5=32$ and $2^6=64$, we have
$$ 5<\lg 35<6, $$
hence
$$ \lfloor \lg 35 \rfloor = 5. $$
Therefore
$$ \boxed{\lfloor 1.1 \rfloor = 1,\quad \lceil -1.1 \rceil = -1,\quad \lfloor -1.1 \rfloor = -2,\quad \lfloor 0.99999 \rfloor = 0,\quad \lfloor \lg 35 \rfloor = 5.} $$