Download Induction

Introduction to Number Theory
The Peano axioms were created to prove that the ZFC model of set theory implies the existence
of the natural numbers. For example, in one of the axioms, it states that the empty set is a set.

{}
{{}}
Another axiom in the ZFC model of set theory states;
If A is a set, then the set containing A is also a set.
With the two axioms above we can create a countably infinite in which the empty set
corresponds to zero, the set containing the empty set corresponds to 1, etc. etc.
Peano Axioms:
Let P(x) be a predicate statement, whose universe of discourse is the natural numbers.
Suppose the following are true statements.
1) Base Case: P(1) : the statement is true for n = 1
2) Induction Hypothesis: P(n) implies P(n+1): the statement being true for n implies the
statement is true for n +1
If the two statements above are true, then P(k) is true for all k 
Let’s prove that 1  2  3   n 
.
(n)(n  1)
using the Peano axioms. This type of proof is called
2
mathematical induction.
Proof: First we must show the statement is true for n = 1, P(1) is a true statement.
1
1(1  1) 2

2
2
Now we need to show that P(n) implies P(n+1).
Supposing P(n) is true, we would have that
(*)
1  2  3   n 
We need to show that
(n)(n  1)
2
(n  1)(n  2)
2
In order to do this, we will take the equation (*) and add n +1 to the left hand side.
1  2  3   n  (n  1) 
Then
1  2  3   n  (n  1)  (1  2  3  n)  (n  1)
By induction hypothesis,
(n)(n  1)

 (n  1)
2
(n)(n  1) 2(n  1)


2
2
(n)(n  1)  (2)(n  1)

2
(n  2)(n  1)

2
So then we have that P(n) implies P(n+1). Therefore, since, using the base case, we proved that P(1) is
true. By the induction hypothesis, since P(1) is true, then P(2) is true. Repeating this induction hypothesis,
we get that P(2) is true, implies P(3) is true. Following this to its inevitable conclusion, we find that the
statement is true for all natural numbers.
Examples in Class:
n
1.Show
 xk 
k 0
x n 1  1
is true for all n 
x 1
2.Show 12  22   n 2 
(n)(n  1)(2n  1)
for all n  .
6
Base Representation
Any natural number can be written base 10. We take that for granted because that is how we
learned the number system.
For example,
209  2(102 )  0(101 )  0(100 )
Or we can be more elegant and write it as
2
209   ai 10i where a0  9, a1  0, and a2  2
i 0
In fact, any natural number can be written as a finite series in base 10 whose coefficients are
between 0 and 9.
i.e., For all k 
, there exists
n
k   ai 10i such that ai 
i 1
and 0  ai  9
But why do we need to use 10 all the time. Is not a number by any other base, value just as
sweet?
For example, we can write 209 in base 2 as
209  1(27 )  1(26 )  0(25 )  1(24 )  0(23 )  0(22 )  0(21 )  1(20 )
209  1101001
In base 2 all coefficients of the powers of 2 must be between 0 and 1, inclusive.
Try writing the following numbers in base 2, or binary.
1. 14
2. 256
3. 117
However, we can represent a natural number, n, in any base, b. The representation would be of
the form,
n
k   ai bi , where ai 
i 1
such that 0  ai  b  1
Try writing the following number in the base indicated.
1. 163 base 3
2. 163 base 2
3. 163 base 5
Homework:
Prove the following statements are true for all natural numbers using mathematical induction.
(n)(n  1)(n  2)
3
3
3
3
3
2
2) 1  2  3   n  (1  2   n)
1) 1·2  2·3  3·4  (n)(n  1) 
3) 1  3  5    (2n  1)  n2
4) 24 divides n(n2  1)(3n  2)
n
1
n

5) 
n 1
i 1 (i )(i  1)
6) Define the Fibonacci sequence to be defined so that the next term in the sequence is the
sum of the previous two terms.
F1  1, F2  1, F3  2E, F4  3, ... Fk  Fk 1  Fk 2
n
Show that
F  F
i 1
i
n2
1
7) Write the following numbers in base 5
a. 25
b. 32
c. 56
8) Write the following numbers in base 2
a. 47
b. 68
c. 127
9) What is the least number of weights required to weigh any integral number of pounds of
up to 63 pounds if one is allowed to put weights in only one pan of a balance?
10) Prove that every natural number has a unique representation in each base.