Floors & Ceilings

Overview

The floor of $x$ is the greatest integer less than $x$. The ceiling of $x$ is the least integer greater than $x$. These values are denoted $\lfloor x\rfloor$ and $\lceil x\rceil$ respectively.

Identities

For integers $x$ and $y>0$, \begin{align*} \left\lfloor \frac{x}{y} \right\rfloor & = \left\lceil \frac{x}{y} - \frac{y - 1}{y} \right\rceil \\ \left\lceil \frac{x}{y} \right\rceil & = \left\lfloor \frac{x}{y} + \frac{y - 1}{y} \right\rfloor \end{align*}

Bibliography

• Bryant, Randal E., and David O’Hallaron. Computer Systems: A Programmer’s Perspective. Third edition, Global edition. Always Learning. Pearson, 2016.
• Ronald L. Graham, Donald Ervin Knuth, and Oren Patashnik, Concrete Mathematics: A Foundation for Computer Science, 2nd ed (Reading, Mass: Addison-Wesley, 1994).
• Thomas H. Cormen et al., Introduction to Algorithms, 3rd ed (Cambridge, Mass: MIT Press, 2009).