Let z = [y] and y = [x] − x, where [.] is the greatest integer function. If x is not an integer but positive, then what is the value of z ?

Concept:

• The greatest integer function of a number rounds off the number to the integer less than the number
• Every integer x can be witten as x = [x] + {x}, where [x] is the integer part of x and {x} is the fractional part of x
• 0 ≤ {x} < 1
• If x is an integer, then {x} = 0

Property of greatest integer function:

• [-x] = -[x] , if x ∈ Z
•  [-x] = -[x] - 1, if x ∉ Z

Calculation:

Given, z = [y] and y = [x] − x, where [.] is the greatest integer function

 If x is not an integer but positive, 

⇒ x > 0 and x ∉ Z

⇒ y = [x] − x =  [x] − ([x] + {x})

⇒ y = -{x} 

Now given z = [y] 

Putting the value of y,

⇒ z =[-{x}]

⇒ z = -[{x}] - 1   ({x} ∉ Z)

⇒ z = -0 - 1  (∵ 0 ≤ {x} < 1))

⇒ z = -1 

∴ The correct option is (1).