convert ceil to floor Mathematically, why is this true?
$$\left\lceil\frac{a}{b}\right\rceil= \left\lfloor\frac{a+b-1}{b}\right\rfloor$$
Assume $a$ and $b$ are positive integers.
Is this also true if $a$ and $b$ are real numbers?

If $\frac{a}{b}$ is not an integer and $\frac{a}{b} > n + \frac{1}{b}$, where $n$ is an integer, then it is easy to see that $$\left\lceil\frac{a}{b}\right\rceil= \left\lfloor\frac{a}{b}+1-\frac{1}{b}\right\rfloor.$$
If $\frac{a}{b}$ is an integer then the relationship
$$\left\lceil\frac{a}{b}\right\rceil= \left\lfloor\frac{a}{b} + 1 - \frac{1}{b}\right\rfloor$$ holds.
The only situation which causes problem is when $\frac{a}{b} \in [n, n+\frac{1}{b})$, where $n$ is an integer.This is not going to happen since $a$ is a positive integer. Clearly, the above statement doesn't hold true if $a$ is real.
