Is "foot of $X$" a common term in math? General questions about the foot?

Question

I moved and started taking classes at a new university. One term that has come up several times in class is the "foot of $X$" which has notation similar to a bracket but with the top bits cut off. (I'm not sure how to format this correctly.) I've never heard of the "foot" before and apparently it refers to the rounding down of a noninteger (ex: the foot of 5.384 is 5, 10.89 becomes 10, etc.).

Is the "foot" a common term that I just haven't encountered until now?

How do you format the notation in MathJax/Latex?

Is the opposite referred to as the "head"? How do you put in the symbols for that in Latex?

Answer 1

The usual terms in English are floor of $x$ for $\lfloor x\rfloor$ and ceiling of $x$ for $\lceil x\rceil$. You get them with \lfloor and \rfloor for $\lfloor$ and $\rfloor$, and with \lceil and \rceil for $\lceil$ and $\rceil$. Like parentheses, they can be automatically sized to fit the contents using \left and \right.

Answer 2

Ive never heard of the foot of a number, no. But dont let Brian's answer lead you astray. They may not be referring to the floor function at all. Suppose the "foot" of a number is everything to the left of the decimal place. In that case, the foot of a number and the floor of a number differ by 1 for all negative values. The floor means the same as "round down to the next lower integer", the ceiling is "round up to the next greater integer". Meanwhile the "foot" of a number COULD mean "everything to the left of the decimal point". These would be fundamentally different operations. You need to precisely define the foot function for the entire domain.