Floor function

From AoPSWiki

The greatest integer function, also known as the floor function, gives the greatest integer less than or equal to its argument. The floor of $x$ is usually denoted by $\lfloor x \rfloor$ or $[x]$ . The action of this function is the same as "rounding down." On a positive argument, this function is the same as "dropping everything after the decimal point," but this is not true for negative values.

Examples

$\lfloor 3.14 \rfloor = 3$

$\lfloor 5 \rfloor = 5$

$\lfloor -3.2 \rfloor = -4$

A useful way to use the floor function is to write $\lfloor x \rfloor=\lfloor y+k \rfloor$ , where y is an integer and k is the leftover stuff after the decimal point. This can greatly simplify many problems.

Alternate Definition

Another common definition of the floor function is

$\lfloor x \rfloor = x-\{x\}$

where $\{x\}$ is the fractional part of $x$ .

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Floor function

From AoPSWiki

Examples

Alternate Definition

See Also