Greatest and Smallest Integer Functions

You can find the greatest integer less than or equal to a number by using the floor function, denoted $\left\lfloor\vphantom{ z}\right.$z$\left.\vphantom{ z}\right\rfloor$ .

$\blacktriangleright$ To put floor brackets around an expression

1.
Select the expression.

2.
Click itbpF0.2992in0.2992in0.0692infences.wmf.

3.
Click the left bracket itbpF0.1747in0.3157in0.0701inleftflor.wmf .

4.
Choose OK.

$\blacktriangleright$ To find a greatest integer value

1.
Place the insertion point in an expression enclosed between floor brackets.

2.
Choose Evaluate.

$\blacktriangleright$ Evaluate

$\left\lfloor\vphantom{ 5.6}\right.$5.6$\left.\vphantom{ 5.6}\right\rfloor$ = 5


$\left\lfloor\vphantom{ -11.3}\right.$ -11.3$\left.\vphantom{ -11.3}\right\rfloor$ = - 12


$\left\lfloor\vphantom{ \frac{43}{5}}\right.$${\frac{{43}}{{5}}}$$\left.\vphantom{ \frac{43}{5}}\right\rfloor$ = 8


$\left\lfloor\vphantom{ \pi +e}\right.$π + e$\left.\vphantom{ \pi +e}\right\rfloor$ = 5

You can find the smallest integer greater than or equal to a number by using the ceiling function, denoted $\left\lceil\vphantom{ z}\right.$z$\left.\vphantom{ z}\right\rceil$.

$\blacktriangleright$ To put ceiling brackets around an expression

1.
Select the expression.

2.
Click itbpF0.2992in0.2992in0.0692infences.wmf.

3.
Click the left ceiling bracket itbpF0.1686in0.3131in0.0701inFigure .

4.
Choose OK.

$\blacktriangleright$ To find a smallest greater integer value

1.
Place the insertion point in an expression enclosed between ceiling brackets.

2.
Choose Evaluate.

$\blacktriangleright$ Evaluate

$\left\lceil\vphantom{ 5.6}\right.$5.6$\left.\vphantom{ 5.6}\right\rceil$ = 6


$\left\lceil\vphantom{ -11.3}\right.$ -11.3$\left.\vphantom{ -11.3}\right\rceil$ = - 11


$\left\lceil\vphantom{ \frac{43}{5}}\right.$${\frac{{43}}{{5}}}$$\left.\vphantom{ \frac{43}{5}}\right\rceil$  = 9


$\left\lceil\vphantom{ \pi +e}\right.$π + e$\left.\vphantom{ \pi +e}\right\rceil$ = 6


The floor and ceiling brackets are also available in the panel under itbpF0.2992in0.2992in0.0692inFigure Special Delimiters, although these are not expanding brackets.