Number Theory.pdf

NUMBER THEORY
Dr. Gabriel Obed Fosu
Department of Mathematics
Kwame Nkrumah University of Science and Technology
Google Scholar: https://scholar.google.com/citations?user=ZJfCMyQAAAAJ&hl=en&oi=ao
ResearchGate ID: https://www.researchgate.net/profile/Gabriel_Fosu2
Dr. Gabby (KNUST-Maths) Number Theory 1 / 27

Lecture Outline
1 Introduction to Numbers
Natural Numbers
Integers
Rational and Irrational Numbers
Real Number
2 Mathematical Induction

Introduction to Numbers
Outline of Presentation
Natural Numbers
Integers
Real Number

Introduction to Numbers Natural Numbers
Natural Numbers (N)
Definition
The set of natural numbers, denoted by N, admits a minimum number called one. It is
denoted by the symbol 1.
What happens if we add 1 to itself a certain number of times?
1+1 = 2,
1+1+1 = 3,
1+1+1+1 = 4, and
1+1+···+1
| {z }
n times
= n.
We obtain N (also called positive integers).

Natural Numbers
Some Properties
For any natural numbers a,b,c, the following are true
1 Addition is Commutative: a +b = b + a

Natural Numbers
Some Properties
2 Addition is Associative: a +(b +c) = (a +b)+c

Natural Numbers
Some Properties
3 Multiplication is commutative: ab = ba

Natural Numbers
Some Properties
4 Multiplication is Associative: a(bc) = (ab)c

Natural Numbers
Some Properties
5 Multiplication Distributes over Addition: a(b +c) = ab + ac = (b +c)a

Introduction to Numbers Integers
Integers (Z)
Conversely, by subtracting 1 i.e. adding −1,
1−1 = 0, (1)
0−1 = −1,
−1−1 = −2,
.
.
.
.
.
.
.
.
.
−1−1−···−1
| {z }
n times
= −n. (2)
Except 0, these numbers are called negative integers.

Integers (Z)
Conversely, by subtracting 1 i.e. adding −1,
1−1 = 0, (1)
0−1 = −1,
−1−1 = −2,
.
.
.
.
.
.
.
.
.
−1−1−···−1
| {z }
n times
= −n. (2)
Except 0, these numbers are called negative integers.
An integer (from the Latin integer meaning “whole") is colloquially/informally defined as a
number that can be written without a fractional component. For example, 21, 4, 0, and
-2048 are integers.

Integers
Definition
1 The set of integers
Z = {··· ,−2,−1,0,1,2,···} (3)

Integers
Definition
Z = {··· ,−2,−1,0,1,2,···} (3)
2 The set of non-negative integers
Z+ = {0,1,2,···} (4)

Integers
Definition
Z = {··· ,−2,−1,0,1,2,···} (3)
2 The set of non-negative integers
Z+ = {0,1,2,···} (4)
3 The set of non-positive integers
Z− = {··· ,−2,−1,0} (5)

Integers
Some properties (a,b,c ∈ Z)

Integers

Integers
5 Multiplication Distributes over Addition: a(b +c) = ab + ac

Integers
6 Identity Element of Integer Addition is Zero: a +0 = a = 0+ a

Integers
7 Identity Element of Integer Multiplication is One: a ·1 = a = 1· a

Integers
8 Transitivity: a > b and b > c then a > c.

Integers
9 Cancellation law: If a ·c = b ·c and c ̸= 0 then a = b

Rational Number (Q)
Definition
A rational number is a number that can be in the form p/q where p and q are integers and
q is not equal to zero.

Rational Number (Q)
Definition
A rational number is a number that can be in the form p/q where p and q are integers and
q is not equal to zero.
1 Also, we can say that any fraction fits under the category of rational numbers, where
the denominator and numerator are integers and the denominator is not equal to zero.
2 When the rational number (i.e., fraction) is divided, the result could be an integer or a
decimal. The decimal may either be terminating or repeating.
3 Example are 1/1 = 1, 1/2 = 0.5, 20/7 = 2.85714285714285714

Introduction to Numbers Rational and Irrational Numbers
Rational Numbers Q
Example: Rational numbers are solutions of the equation
nr −m = 0.
2
3 is a rational number and solution of the equation −3r +2 = 0.

Rational Numbers Q
Example: Rational numbers are solutions of the equation
nr −m = 0.
2
3 is a rational number and solution of the equation −3r +2 = 0.
Some Properties
1 Integers are rational numbers i.e. Z ⊂ Q.
2 Multiplicative inverse of nonzero elements of Q belong to Q (i.e.
∀q ∈ Q,q ̸= 0 =⇒ 1
q ∈ Q.)

Irrational Numbers
Definition (Irrational Numbers)
These are numbers that cannot be represented as a simple fraction. It cannot be
expressed in the form of a ratio, such as p/q, where p and q are integers, q ̸= 0.
Again, the decimal expansion of an irrational number is neither terminating nor
recurring.

Irrational Numbers
Definition (Irrational Numbers)
These are numbers that cannot be represented as a simple fraction. It cannot be
expressed in the form of a ratio, such as p/q, where p and q are integers, q ̸= 0.
Again, the decimal expansion of an irrational number is neither terminating nor
recurring.
Example
Some examples are
p
2,
p
3, π, and the Euler number e

Introduction to Numbers Real Number
Real Numbers (R)
Definition (Real Numbers)
Real numbers can be defined as the union of both the rational and irrational numbers.
They can be both positive or negative, decimals or fractions.

Real Numbers (R)
Definition (Real Numbers)
Real numbers can be defined as the union of both the rational and irrational numbers.
They can be both positive or negative, decimals or fractions.
Real numbers are often represented along a graduated line with extreme values −∞ and
+∞.
Figure 1: Real numbers

Real Numbers

Real Numbers
Some properties (a,b,c ∈ R)
6 Identity Element of Addition is Zero: a +0 = a = 0+ a
7 Identity Element of Multiplication is One: a ·1 = a = 1· a
9 Cancellation law: If a ·c = b ·c and c ̸= 0 then a = b
10 R is closed under addition (a +b ∈ R), and multiplication (ab ∈ R)

Theorem (Q is dense in R)
For any two real numbers x and y, if x < y then there exists a rational number q such that
x < q < y (∀x, y ∈ R;x < y =⇒ ∃q ∈ Q;x < q < y).

x < q < y (∀x, y ∈ R;x < y =⇒ ∃q ∈ Q;x < q < y).
Archimedean Property
If x and y are real numbers, and x > 0, then there exist a positive integer n such that
nx > y (6)
It also means that the set of natural numbers is not bounded above.

x < q < y (∀x, y ∈ R;x < y =⇒ ∃q ∈ Q;x < q < y).
Archimedean Property
If x and y are real numbers, and x > 0, then there exist a positive integer n such that
nx > y (6)
It also means that the set of natural numbers is not bounded above.
Trichotomy
Every real number is negative, zero, or positive.
The law is sometimes stated as for any x and y, then exactly one of these applies
x < y, x = y, x > y (7)

Mathematical Induction
Outline of Presentation
Natural Numbers
Integers
Real Number

1 The principle of mathematical induction is an important property of positive integers.
2 It is especially useful in proving statements involving all positive integers when it is
known for example that the statements are valid for n = 1,2,3 but it is suspected or
conjectured that they hold for all positive integers.
3 The method of proof consists of the following steps:

1 Prove the statement for
n = 1 (8)
This is the Basis Step, that is P(1) is true

n = 1 (8)
2 Assume the statement true for
n = k (9)
where k is any positive integer. The premise P(n) in the inductive step is called
Induction Hypothesis.

n = 1 (8)
2 Assume the statement true for
n = k (9)
where k is any positive integer. The premise P(n) in the inductive step is called
Induction Hypothesis.
3 From the assumption in (2) prove that the statement must be true for
n = k +1 (10)
That is if P(n) is true, then P(n +1) is true.

a. The basis step states that P(1) is true.
b. Then the inductive step implies that P(2) is also true.
c. By the inductive step again we see that P(3) is true, and so on.
d. Consequently, the property is true for all positive integers.

a. The basis step states that P(1) is true.
b. Then the inductive step implies that P(2) is also true.
c. By the inductive step again we see that P(3) is true, and so on.
d. Consequently, the property is true for all positive integers.
In the basis step we may replace 1 with some other integer m. Then the conclusion is that
the property is true for every integer n greater than or equal to m.

Example
Prove that the sum of the n first odd positive integers is n2
, that is
1+3+5+···+(2n −1) = n2
.

Example
, that is
1+3+5+···+(2n −1) = n2
.
Let S(n) = 1+3+5+···+(2n −1). We want to prove by induction that for every positive
integer n, S(n) = n2
.
1 Basis Step: If n = 1 we have S(1) = 1 = 12
, so the property is true for 1

Example
, that is
1+3+5+···+(2n −1) = n2
.
.
2 Inductive Step: Assume (Induction Hypothesis) that the property is true for some
positive integer n, ie S(n) = n2
.

Example
, that is
1+3+5+···+(2n −1) = n2
.
.
2 Inductive Step: Assume (Induction Hypothesis) that the property is true for some
positive integer n, ie S(n) = n2
.
3 We must prove that it is also true for n +1, i.e.,
S(n +1) = (n +1)2
(11)
that is the Expected results.

n +1 case
1 Picking up from the n = k case, we have
S(k) = 1+3+5+···+(2k −1) (12)

n +1 case
S(k) = 1+3+5+···+(2k −1) (12)
2 then the n = k +1
S(k +1) = 1+3+5+···+(2k −1)+(2k +1) (13)
= S(k)+(2k +1) (14)

n +1 case
S(k) = 1+3+5+···+(2k −1) (12)
2 then the n = k +1
S(k +1) = 1+3+5+···+(2k −1)+(2k +1) (13)
= S(k)+(2k +1) (14)
But by induction hypothesis (12), S(k) = k2
, hence equation (14) reduces to
S(k +1) = k2
+2k +1

n +1 case
S(k) = 1+3+5+···+(2k −1) (12)
2 then the n = k +1
S(k +1) = 1+3+5+···+(2k −1)+(2k +1) (13)
= S(k)+(2k +1) (14)
S(k +1) = k2
+2k +1 = (k +1)2
(15)

n +1 case
S(k) = 1+3+5+···+(2k −1) (12)
2 then the n = k +1
S(k +1) = 1+3+5+···+(2k −1)+(2k +1) (13)
= S(k)+(2k +1) (14)
S(k +1) = k2
+2k +1 = (k +1)2
(15)
This completes the induction, and shows that the property is true for all positive
integers.

Example
Prove that 2n +1 ≤ 2n
for n ≥ 3

Example
for n ≥ 3
Solution
This is an example in which the property is not true for all positive integers but only for
integers greater than or equal to 3.
1 Basis Step: If n = 3 we have 2n +1 = 2(3)+1 = 7

Example
for n ≥ 3
Solution
1 Basis Step: If n = 3 we have 2n +1 = 2(3)+1 = 7 and 2n
= 23
= 8, so the property is true
in this case.

Example
for n ≥ 3
Solution
= 23
in this case.
2 Inductive Step : Assume (Induction Hypothesis) that the property is true for some
positive integer n, that is
2n +1 ≤ 2n
(16)

Example
for n ≥ 3
Solution
= 23
in this case.
2 Inductive Step : Assume (Induction Hypothesis) that the property is true for some
positive integer n, that is
2n +1 ≤ 2n
(16)
3 We must prove that it is also true for n +1, that is ,
2(n +1)+1 ≤ 2n+1
expected result (17)

1 By the induction hypothesis we know that for n = k we have
2k +1 ≤ 2k
(18)

2k +1 ≤ 2k
(18)
2 Multiplying both sides by 2
2(2k +1) ≤ 2×2k
(19)
4k +2 ≤ 2k+1
(20)

2k +1 ≤ 2k
(18)
2(2k +1) ≤ 2×2k
(19)
4k +2 ≤ 2k+1
(20)
3 for k > 3 then 4k +2 > 2k +3,

2k +1 ≤ 2k
(18)
2(2k +1) ≤ 2×2k
(19)
4k +2 ≤ 2k+1
(20)
3 for k > 3 then 4k +2 > 2k +3, thence
2k +3 ≤ 2k+1
(21)

2k +1 ≤ 2k
(18)
2(2k +1) ≤ 2×2k
(19)
4k +2 ≤ 2k+1
(20)
3 for k > 3 then 4k +2 > 2k +3, thence
2k +3 ≤ 2k+1
(21)
This completes the induction, and shows that the property is true for all n ≥ 3.

Example
Prove that 1+2+···+n =
n(n +1)
2

Example
n(n +1)
2
Solution
This formula is easily verified for small numbers such as n = 1,2,3, or 4, but it is impossible
to verify for all natural numbers on a case-by-case basis. To prove the formula true in
general, a more generic method is required.
We need to show that
n0 = 1, since 1 =
1(1+1)
1
(22)

Example
n(n +1)
2
Solution
This formula is easily verified for small numbers such as n = 1,2,3, or 4, but it is impossible
to verify for all natural numbers on a case-by-case basis. To prove the formula true in
general, a more generic method is required.
We need to show that
n0 = 1, since 1 =
1(1+1)
1
(22)
Expected results
1+2+···+n +(n +1) =
(n +1)(n +2)
2
(23)

So for n = k
1+2+···+k =
k(k +1)
2
(24)

So for n = k
1+2+···+k =
k(k +1)
2
(24)
then for next time step
1+2+···+k +(k +1) =
k(k +1)
2
+k +1 (25)

So for n = k
1+2+···+k =
k(k +1)
2
(24)
1+2+···+k +(k +1) =
k(k +1)
2
+k +1 (25)
=
k2
+3k +2
2
(26)

So for n = k
1+2+···+k =
k(k +1)
2
(24)
1+2+···+k +(k +1) =
k(k +1)
2
+k +1 (25)
=
k2
+3k +2
2
(26)
=
(k +1)(k +2)
2
(27)
This is exactly the formula for the (n +1)th case.

Exercise
1 Find a condition on n ∈ N such that 2n2
−7n −4 ∈ N.
2 Prove that for all integers n,
1 if n ≥ 3 then 2n
> n +4.
2 1+ x + x2
+...+ xn
= 1−xn+1
1−x ,n ∈ {0,1,...} where x ∈ R−{1}.
3 12
+22
+...+n2
= 1
6 n(n +1)(2n +1),n ≥ 1.
4 if n ≥ 1, then (1+ x)n
≥ 1+nx for x ∈ R+ (i.e. x is a non-negative real number).

END OF LECTURE
THANK YOU

Number Theory.pdf

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