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Chapter 3
Elementary Number Theory and
Methods of Proof
3.5
Direct Proof and Counterexample 5
Floor & Ceiling
Floor & Ceiling
• Definition
– Floor
• Given any real number x, the floor of x, denoted ⎣x⎦, is
defined as: ⎣x⎦ = n ⇔ n ≤ x < n + 1.
– Ceiling
• Given any real number x, the ceiling of x, denoted ⎡x⎤, is
defined as: ⎡x⎤ = n ⇔ n-1 < x ≤ n.
Examples
• Compute ⎣x⎦ and ⎡x⎤ for the following:
– 25/4
• ⎣25/4⎦ = ⎣6+ 1/4⎦ = 6
• ⎡25/4⎤ = ⎡6+ 1/4⎤ = 7
– 0.999
• ⎣0.999⎦ = ⎣0 + 999/1000⎦ = 0
• ⎡0.999⎤ = ⎡0 + 999/1000⎤ = 1
Examples
• The 1,370 soldiers at a military base a re given
the opportunity to take buses into town for an
evening out. Each bus holds a maximum of 40
passengers
– What is the maximum number of buses the base
will send if only full buses are sent?
• ⎣1,370/40⎦ = ⎣34.25⎦ = 34
– How many buses will be needed if a partially full
bus is allowed?
• ⎡1,370/40⎤ = ⎡34.25⎤ = 35
Addition Property of Floor
• Does ⎣x + y⎦ = ⎣x⎦ + ⎣y⎦?
• Can you find a counterexample where the
case is not true. If so, then you can prove that
equality is false.
– How about x = ½ and y = ½ ?
• ⎣½ + ½⎦ = ⎣1⎦ = 1
• ⎣½⎦ + ⎣½⎦ = 0 + 0 = 0
• hence, the equality is false.
Proving Floor Property
• Prove that for all real numbers x and for all
integers m, ⎣x + m⎦ = ⎣x⎦ + m
– Suppose x is a particular but arbitrarily chosen real
number and m is particular but arbitrarily chosen
integer.
– Show: ⎣x + m⎦ = ⎣x⎦ + m
•
•
•
•
•
Let n = ⎣x⎦, n is integer n ≤ x < n+1
n + m ≤ x + m < n + m + 1 (add m to all sides)
⎣x + m⎦ = n + m (from previous)
since n = ⎣x⎦
Thus ⎣x + m⎦ = ⎣x⎦ + m
• Theorem 3.5.1
Floor of n/2
• Theorem 3.5.2 Floor of n/2
– For any n, ⎣n/2⎦ = n/2 (if n even) or (n-1)/2 (if n
odd)
• Examples
– Compute floor of n/2 for the following:
• n = 5: ⎣5/2⎦ = ⎣2 ½⎦ = 2 = (5-1)/2 = 2
• n = 8: ⎣8/2⎦ = ⎣4⎦ = 4 = (8)/2 = 4
Div / Mod and Floor
• There is a relationship between div and mod and
the floor function.
– n div d = ⎣n / d⎦
– n mod d = n – d⎣n/d⎦
• From the quotient-remainder theorem, n = dq + r
and 0≤r<d a relationship can be proven between
quotient and floor.
• Theorem 3.5.3
– If n is any integer and d is a positive integer, and if q =
⎣n/d⎦ and r = n – d⎣n/d⎦ then, n = dq + r and 0≤r<d