In this lecture

1
In this lecture

• Number Theory
• ? Rational numbers
• ? Divisibility
• Proofs
• ? Direct proofs (cont.)
• ? Common mistakes in proofs
• ? Disproof by counterexample

2
Common mistakes in proofs

• Arguing from examples
• Using same letter
• to mean two different things
• Jumping to a conclusion
• (without adequate reasons)

3
Disproof by counterexample

• To disprove statement of the form
• ?x?D if P(x) then Q(x),
• find a value of x for which
• ? P(x) is true and
• ? Q(x) is false.
• Ex For any prime number a,
• a2-1 is even integer.
• Counterexample a2.

4
Rational Numbers

• Definition
• r is rational iff
• ? integers a and b such that
• ra/b and b?0.
• Examples
• 5/6, -178/123, 36, 0, 0.256256256
• Theorem Every integer
• is a rational number.

5
Properties of Rational Numbers

• Theorem The sum of two rational numbers
  is rational.
• Proof
• Suppose r and s are rational numbers.
• Then ra/b and sc/d for some integers
  a,b,c,d s.t. b?0, d?0. (by definition)
• So (by substitution)
• (by basic algebra)
• Let padbc and qbd.
• Then rsp/q where p,q ?Z and q?0.
• Thus, rs is rational by definition.
  

6
Types of Mathematical Statements

• Theorems Very important statements that
• have many and varied consequences.
• Propositions Less important and
  consequential.
• Corollaries The truth can be deduced
  almost immediately
• from other statements.
• Lemmas Dont have much intrinsic interest but
  help to prove other theorems.

7
Divisibility

• Definition For n,d ?Z and d?0 we say that n
  is divisible by d
• iff ndk for some k ?Z .
• Alternative ways to say
• n is a multiple of d , d is a factor of n ,
• d is a divisor of n , d divides n .
• Notation d n .
• Examples 648, 55, -48, 70, 19 .

8
Properties of Divisibility

• For ?x?Z, 1x .
• For ?x?Z s.t. x?0, x0 .
• An integer xgt1 is prime
• iff its only positive divisors are 1 and x .
• For ?a,b,c?Z, if ab and ac then a(bc) .
• Transitivity For ?a,b,c?Z,
• if ab and bc then ac .

9
Divisibility by a prime

• Theorem Any integer ngt1
• is divisible by a prime number.
• Sketch of proof
• Division into cases
• ? If n is prime then we are done (since n n).
• ? If n is composite
• then nr1s1 where r1,s1 ?Z and
  1ltr1ltn,1lts1ltn.
• (by definition of composite number)
• (Further) division into cases
• ?If r1 is prime then we are done (since r1
  n).
• ? If r1 is composite
• then r1r2s2 where r2,s2 ?Z and
  1ltr2ltr1,1lts2ltr1.

10
Divisibility by a prime

• Sketch of proof (cont.)
• Since r1n and r2r1 then r2 n (by
  transitivity).
• Continuing the division into cases,
• we will get a sequence of integers
• r1 , r2 , r3 ,, rk
• such that 1lt rklt rk-1ltlt r2lt r1ltn
• rp n for each p1,2,,k
• rk is prime.
• Thus, rk is a prime that divides n.

11
Unique Factorization Theorem

• Theorem
• For ? integer ngt1,
• ? positive integer k,
• distinct prime numbers ,
• positive integers
• s.t. ,
• and this factorization is unique.
• Example 72,000