Chapter 3 Elementary Number Theory and Methods of Proof PowerPoint Presentation, PPT - DocSlides

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.CeilingGiven 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⎤ = 70.999⎣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 = 0hence, the equality is false.

Proving Floor Property

Prove that for all real numbers

x

and for all integers

m

, ⎣x + m⎦ = ⎣x⎦ + mSuppose x is a particular but arbitrarily chosen real number and m is particular but arbitrarily chosen integer.Show: ⎣x + m⎦ = ⎣x⎦ + mLet n = ⎣x⎦, n is integer n ≤

x < n+1n + 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)ExamplesCompute floor of n/2 for the following:n = 5: ⎣5/2⎦ = ⎣2 ½⎦ = 2 = (5-1)/2 = 2n = 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 3.5.3If 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