Mathematics Equation Involving Greatest Integer

The Greatest Integer function

`text(The Greatest Integer function)`
`[x]=`the largest integer that is less than or equal to `x.`

In mathematical notation we would write this as `[x]=max{m∈Z|m≤x}`
The notation `m∈Z` means `m` is an integer.

A couple of trivial facts about `[x]`
- `[x]` is the unique integer satisfying `x − 1 < [x] ≤ x.`
- `[x] = x` if and only if `x` is an integer.
- Any real number `x` can be written as `x = [x]+ θ`, where `0 ≤ θ < 1 .`

`text(Properties of Greatest Integer :)`

`(i) [x±n]= [x]±n,n in I`
`(ii) [-x]=- [x],x in I`
`(iii) [-x]=-1- [x],x in I`
`(iv) [x]- [- x] = 2n,` if `x= n, n in I`
`(v) [x]- [- x] = 2n + 1,` if `x = n + {x}, n in I` and `0< {x} < 1`
`(vi) [x] >= n => x >= n, n in I`
`(vii) [x]> n =>x >= n + 1, n in I`
`(viii) [x] <= n => x < n + 1, n in I`
`(ix) [x] < n => x < n, n in I`
`(x) n_2 <= [x] <= n_1 => n_2 <= x < n_1 + 1, n_1 , n_2 in I`
`(xi) [x+y] >= [x]+ [y]`
`(xii) [[x]/n] = [x/n] , n in N`
`(xiii) [(n+1)/2] + [(n+2)/4] + [(n+4)/8] + [(n+8)/16] + .... + n, n in N`
`(xiv) [x] + [x+ 1/n] + [x+ 2/n] + ......... + [x + (n-1)/n] = [nx] , n in N`

Least Integer

`(x)` or `ceilx` denotes the least integer greater than or equal to x i.e. `(x)>=x` or `ceilx >=x .` It is also known as ceilling of x.

e.g. `(3.578) = 4, \ \ \ \ (0.87) = 1, ( 4) = 4 `

`( -8.2391) =- 8, \ \ \ \ (-0.28) = 0`

In general, if n is an integer and xis any real number between `n` and `n + 1 `

i.e., `n < x < n + 1,` then `(x) = n + 1 `

`text(Relation between Greatest Integer and Least Integer :)`

`(x) = {tt (([x], x in I),([x]+1 , x notin I))`

i.e., If `x in I` then `x = [x] = (x)`

Remember :

If `(x) = n,` then `(n -1) < x < n .`

Fractional Part

{ x} denotes the fractional part of x, i.e. 0 s; { x} < 1.
Thus, {2 · 7} = 0. 7, { 5} = 0, {- 3. 72} = 0.28
If xis a real number, then
x=[x]+{x}
i.e., x = n + f, where `n in I` and `0 <= f < 1`
Properties of Fractional Part of x
(i) {x±n}={x},nei
(ii) If `0 <= x < 1,` then `{x}=x`

`text(Important Result)`

1. For proper fraction 0 < { x } < 1.
Domain and range of { x} are Rand [0, 1), respectively.
3. {- 5.238} = {- 5- 0.238} = {- 5-1 + 1- 0.238}
= {- 6+ 0.762}= (6.762}= 0.762