Presentations text content in Chapter 3 Elementary Number Theory and Methods of Proof
Slide1
Chapter 3
Elementary Number Theory and Methods of Proof
Slide23.5
Direct Proof and Counterexample
5
Floor & Ceiling
Slide3Floor & 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.
Slide4Examples
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
Slide5Examples
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
Slide6Addition 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.
Slide7Proving 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
Slide8Floor 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
Slide9Div / 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
Chapter 3 Elementary Number Theory and Methods of Proof
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