Greatest Integer Function

Greatest Integer Function

[x] denotes the greatest integer function and means greatest integer less than or equal to x.

i.e. [x] ≤ x.

e.g. [3.564] = 3 [.35] = 0 [4] = 4

[-3.564] = -4 and [-.03] = -1

In general if n is an integer and x is any real number between n and (n+1) that is
n ≤ x < n+1, then [x] = n.

Properties of Greatest Integer Function

1. If f(x) = [x+n], n ∈ I then f(x) = n + [x]
2. If x = [x] + {x}, then {x} denotes the fractional part of x.
3. x-1 < [x] ≤ x
4. [-x] = - [x], x ∈ I
5. [-x] = - [x] – 1 x ∉ 1
6. [x] – [-x] = 2n if x = n n ∈ I
7. [x] – [-x] = 2n+1 if x = n + {x} n ∈ I, {.} denotes fractional part