The existence of integers exceeding real numbers motivates the following definitions: * The greatest integer \(n\) less than or equal to \(x\) denoted by \(\lfloor x \rfloor\). * The least integer \(n\) greater than or equal to \(x\) denoted by \(\lceil x \rceil\).

Definition: Floor and Ceiling Functions

The floor $\lfloor \cdot \rfloor:\mathbb R\to\mathbb Z$ and the ceiling function $\lceil \cdot \rceil:\mathbb R\to\mathbb Z$ are functions from the set $\mathbb R$ of real numbers to the set $\mathbb Z$ of integers, defined by:

\[\begin{array}{rcl} \lfloor x \rfloor=n&\Longleftrightarrow& n\le x < n+1,\\ \lfloor x \rfloor=n&\Longleftrightarrow& x-1 < n \le x,\\ \lceil x \rceil=n&\Longleftrightarrow& x\le n < x+1,\\ \lceil x \rceil=n&\Longleftrightarrow& n-1 < x \le n.\\ \end{array}\]

  1. Proposition: Properties of Floors and Ceilings
  2. Proposition: Floor Function and Division with Quotient and Remainder
  3. Lemma: Sums of Floors
  4. Lemma: Reciprocity Law for Floor Functions

Definitions: 1
Lemmas: 2 3 4
Proofs: 5 6 7 8 9 10 11
Propositions: 12 13 14 15
Theorems: 16

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  1. Graham L. Ronald, Knuth E. Donald, Patashnik Oren: "Concrete Mathematics", Addison-Wesley, 1994, 2nd Edition