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In Mathematical, the whole function left is the in the following way definite function  :

For all Real number X , the whole part, noted E ( X ), $\backslash \; left\; X\; \backslash \; right$ or $\backslash \; left\; \backslash \; lfloor\; X\; \backslash \; right\; \backslash \; rfloor$, is largest relative Entier lower or equal to X . It is about the single relative entirety such as is checked:

For example  : E (2,3) = 2, E (−2) = −2 and E (−2,3) = −3.

For entire relative K and for any real number X , there is

$\backslash \; left\; \backslash \; lfloor\; X\; +\; K\; \backslash \; right\; \backslash \; rfloor\; =\; \backslash \; left\; \backslash \; lfloor\; X\; \backslash \; right\; \backslash \; rfloor\; +\; k$

The round-off with the entirety nearest to a reality X can be expressed by E ( X + 0,5).

The whole left function is not continuous, but is on the right continuous. In fact it is constant on any interval of the form k+1 [and is not continuous in the relative entireties.

Another mathematical function of the same type is the function “  plafond  ” or left whole by excess or higher whole part ( ceiling in English), defined in the following way  :

For any real number X given, the upper whole part of X , noted $\backslash \; left\; \backslash \; lceil\; X\; \backslash \; right\; \backslash \; rceil$ou P ( X ), is the smallest entirety equal to or higher than X . It is the single relative entirety such as:

For example  : P (2,3) = 3, P (2) = 2 and P (−2,3) = −2.

The two whole parts, lower and higher are bound by:

$\backslash \; left\; \backslash \; lceil\; X\; \backslash \; right\; \backslash \; rceil\; =\; -\; \backslash \; left\; \backslash \; lfloor\; -\; X\; \backslash \; right\; \backslash \; rfloor$

For entire relative K , there is also the equality suivante :

$\backslash \; left\; \backslash \; lfloor\; K/2\; \backslash \; right\; \backslash \; rfloor\; +\; \backslash \; left\; \backslash \; lceil\; K/2\; \backslash \; right\; \backslash \; rceil\; =\; k$.

If m and N is whole natural Premiers between them then

$\backslash \; sum\_\; \{K\; =\; 1\}\; ^\; \{N\; -\; 1\}\; \backslash \; left\; \backslash \; lfloor\; \backslash \; frac\; \{K\; m\}\; \{N\}\; \backslash \; right\; \backslash \; rfloor\; =\; \backslash \; frac\; \{(M-1)\; (N\; -\; 1)\}\{2\}$

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