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1.2.3 Floor And Ceiling

Knowledge Base > Math

For any real number x, we denote the greatest integer less than or equal to x by ⌊x⌋, and the least integer greater than or equal to x by ⌈x⌉, that is:

x - 1 < ⌊x⌋ ≤ x ≤ ⌈x⌉ < x + 1 , ∀x ∈ R (1.2.1) ⌈n-⌉+ ⌊n⌋ = n , for an arbitrary integer n 2 2 ⌈⌈n2⌉⌉ = ⌈-n-⌉ , ∀n ∈ R,n ≥ 0 b a⋅b 2 ⋅⌊ n⌋ ≤ n (odd <, even = ) 2 ≥ (n- 1) (odd =, even >) lg(⌊n⌋) ≥ lg(n--1-) 2 2 = lg(n- 1)- lg2 = lg(n- 1)- 1 n ≤ lgn- 1 2 ⋅⌈ -⌉ ≥ n (odd >, even = ) 2 ≤ (n+ 1) (odd =, even <) lg(⌈n⌉) ≤ lg(n+-1-) 2 2 = lg(n+ 1)- lg2 = lg(n+ 1)- 1 ≥ lgn- 1

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