There are limited applications. It's a great brain exercise. Uses simple objects (integers), yet there are many interesting results that can be proven.
Some topics:
divisibility
prime numbers
congruency relationships
There are some connections to abstract algebra
Assignment:
2.1 - 2, 6, 7, 10, 15
2.2 - 5, 10
Read the book before we cover the material. Read ahead a couple of sections.
Definitions:
square numbers - perfect squares. can be represented graphically with dots.
triangle numbers - same representation, but in a triangle: 1, 3, 6, 10, 15.
pythagorean triple - ppl try to develop a scheme that allows them to develop triples. some numbers cannot be in a triple. for instance, the hypotenuse must be the sum of two squares.
Chapter 2 is more formal. starts with definitions. pg 7 - divisibility.
a|b is read "a divides b".
a|b if there exists c in Z such that b=ac
there's also a symbol for "a does not divide b"
if a|b and x|y then ax|by
if a|b and b|c then a|c - transitivity
if a|b and b<>0 then |a|<=|b|
Follow the proofs that are given in the book and try to prove the others yourself.
Prime Numbers
Smallest prime number is 2. It's important that we do not consider 1 prime. Prime means that it is only divisible by itself and 1. I.e. the only divisors of p are 1 and p itself.
The opposite of prime is composite. Composite numbers can be written as a product of primes.
The division theorem: if we have two integers, b and a, there are two unique integers such that a=bq+r where r<|b|.
Q1: Divisors of 1 are... 1 and -1.
Positive integers are the natural numbers.
Whole numbers are the natural numbers plus 0.
2c. If a|b, then a|kb for all k in Z.
Proof:
Let a|b, then there is a c, such that b = ac.
Since b = ac, kb = kac or kb = a(kc)
Since k is in Z and c is in Z, then kc = n is in Z.
Therefore, kb = an where n is in Z.
Hence, a|kb.
Greatest integer function aka floor of x.
See definition on page 11.
The largest integer that does not exceed x.
[x] = n means
n<=x
n+1>x
n is in Z
Ceiling is the smallest integer that is greater than or equal to x.
If x is an integer, then x is equal to both its floor and its ceiling.
See lemma 2.1.13.
See also lemma 2.1.11 to find the number of positive multiples of a given integer within a given interval.
Number of multiples of d that are <=n is the floor of n/d.
Page 14, exercise 16:
Determine the number of integers between 200 and 400 that are divisible by 3.
[400/3]-[200/3] = 133-66 = 67
divisible by 7:
[400/7]-[200/3] = 57-28 = 29
by 3 or 7:
need to subtract out the multiples of 21. apply the union rule of the addition of sets:
n(AuB) = n(A)+n(B)-n(AintersectB)
[400/21]-[200-21] = 19-9 = 10
Therefore the answer is 67+29-10 = 86
exercise 8:
show that when n is odd, n2-1 is a multiple of 8
n is odd means n=2k+1
n2 = (2k+1)(2k+1) = 4k2 +4k+1
n2-1 = 4k2 +4k
n2-1 = 4k(k+1)
Case 1 - k is even
k=2L for some L in Z
n2-1 = 4(2L)(2L+1) = 8L(L+1)
Hence, 8|(n2-1)
Case 2 - k is odd
k+1=2L for some L in Z
n2-1 = 4k(2L) = 8kL
Hence, 8|(n2-1)
Exercise 9:
Show that a perfect square is never of the form 3k+2 for any k.
Assume that n2 = 3k+2.
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