Chapter 3 Elementary Number Theory and Methods of Proof

3.5 Direct Proof and Counterexample 5 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 – = /1000 = = /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 – How many buses will be needed if a partially full bus is allowed? 1,370/40 = = 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 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 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 – dn/d From the quotient-remainder theorem, n = dq + r and 0≤r<d a relationship can be proven between quotient and floor. Theorem – If n is any integer and d is a positive integer, and if q = n/d and r = n – dn/d then, n = dq + r and 0≤r<d