What can the floor of a square root be rewritten as?

Question

For example: Let $n$ be natural (positive) number. $\lfloor\sqrt{n+1}\rfloor$

Floor: round down to nearest integer.

What can I rewrite this as? Can I break it down or apart?

Answer 1

The expression $\lfloor \sqrt{n} \rfloor$ represents the largest integer $m$ such that $m^2 \le n$. There is no commonly known, specific symbol or notation to describe such a number, other than $\lfloor \sqrt{n} \rfloor$. That said, we could define such a number in advance; e.g., "For a positive integer $n$, let $f(n)$ be the largest integer such that $f(n)^2 \le n$." Then $n = f(n)^2 + r$, for some $r \in \{ 0, 1, 2, \ldots, \max(0,n-1) \}$. This specifies the decomposition of $n$ into the sum of its largest integer square and a "remainder" term $r$.