This document provides an outline and introduction to the topic of number theory. It begins with definitions and properties of various number sets, including natural numbers, integers, rational numbers, irrational numbers, and real numbers. It discusses how rational numbers can be represented as fractions and irrational numbers cannot. The document also states that the set of rational numbers is dense in the set of real numbers and presents the Archimedean property. The overall summary is an introduction to number sets and basic concepts in number theory.
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Number Theory.pdf
1. 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
2. Lecture Outline
1 Introduction to Numbers
Natural Numbers
Integers
Rational and Irrational Numbers
Real Number
2 Mathematical Induction
Dr. Gabby (KNUST-Maths) Number Theory 2 / 27
3. Introduction to Numbers
Outline of Presentation
1 Introduction to Numbers
Natural Numbers
Integers
Rational and Irrational Numbers
Real Number
2 Mathematical Induction
Dr. Gabby (KNUST-Maths) Number Theory 3 / 27
4. 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+ยทยทยท+1
| {z }
n times
= n.
We obtain N (also called positive integers).
Dr. Gabby (KNUST-Maths) Number Theory 4 / 27
5. Introduction to Numbers Natural Numbers
Natural Numbers
Some Properties
For any natural numbers a,b,c, the following are true
1 Addition is Commutative: a +b = b + a
Dr. Gabby (KNUST-Maths) Number Theory 5 / 27
6. Introduction to Numbers Natural Numbers
Natural Numbers
Some Properties
For any natural numbers a,b,c, the following are true
1 Addition is Commutative: a +b = b + a
2 Addition is Associative: a +(b +c) = (a +b)+c
Dr. Gabby (KNUST-Maths) Number Theory 5 / 27
7. Introduction to Numbers Natural Numbers
Natural Numbers
Some Properties
For any natural numbers a,b,c, the following are true
1 Addition is Commutative: a +b = b + a
2 Addition is Associative: a +(b +c) = (a +b)+c
3 Multiplication is commutative: ab = ba
Dr. Gabby (KNUST-Maths) Number Theory 5 / 27
8. Introduction to Numbers Natural Numbers
Natural Numbers
Some Properties
For any natural numbers a,b,c, the following are true
1 Addition is Commutative: a +b = b + a
2 Addition is Associative: a +(b +c) = (a +b)+c
3 Multiplication is commutative: ab = ba
4 Multiplication is Associative: a(bc) = (ab)c
Dr. Gabby (KNUST-Maths) Number Theory 5 / 27
9. Introduction to Numbers Natural Numbers
Natural Numbers
Some Properties
For any natural numbers a,b,c, the following are true
1 Addition is Commutative: a +b = b + a
2 Addition is Associative: a +(b +c) = (a +b)+c
3 Multiplication is commutative: ab = ba
4 Multiplication is Associative: a(bc) = (ab)c
5 Multiplication Distributes over Addition: a(b +c) = ab + ac = (b +c)a
Dr. Gabby (KNUST-Maths) Number Theory 5 / 27
10. Introduction to Numbers Integers
Integers (Z)
Conversely, by subtracting 1 i.e. adding โ1,
1โ1 = 0, (1)
0โ1 = โ1,
โ1โ1 = โ2,
.
.
.
.
.
.
.
.
.
โ1โ1โยทยทยทโ1
| {z }
n times
= โn. (2)
Except 0, these numbers are called negative integers.
Dr. Gabby (KNUST-Maths) Number Theory 6 / 27
11. Introduction to Numbers Integers
Integers (Z)
Conversely, by subtracting 1 i.e. adding โ1,
1โ1 = 0, (1)
0โ1 = โ1,
โ1โ1 = โ2,
.
.
.
.
.
.
.
.
.
โ1โ1โยทยทยทโ1
| {z }
n times
= โn. (2)
Except 0, these numbers are called negative integers.
An integer (from the Latin integer meaning โwhole") is colloquially/informally defined as a
number that can be written without a fractional component. For example, 21, 4, 0, and
-2048 are integers.
Dr. Gabby (KNUST-Maths) Number Theory 6 / 27
12. Introduction to Numbers Integers
Integers
Definition
1 The set of integers
Z = {ยทยทยท ,โ2,โ1,0,1,2,ยทยทยท} (3)
Dr. Gabby (KNUST-Maths) Number Theory 7 / 27
13. Introduction to Numbers Integers
Integers
Definition
1 The set of integers
Z = {ยทยทยท ,โ2,โ1,0,1,2,ยทยทยท} (3)
2 The set of non-negative integers
Z+ = {0,1,2,ยทยทยท} (4)
Dr. Gabby (KNUST-Maths) Number Theory 7 / 27
14. Introduction to Numbers Integers
Integers
Definition
1 The set of integers
Z = {ยทยทยท ,โ2,โ1,0,1,2,ยทยทยท} (3)
2 The set of non-negative integers
Z+ = {0,1,2,ยทยทยท} (4)
3 The set of non-positive integers
Zโ = {ยทยทยท ,โ2,โ1,0} (5)
Dr. Gabby (KNUST-Maths) Number Theory 7 / 27
15. Introduction to Numbers Integers
Integers
Some properties (a,b,c โ Z)
1 Addition is Commutative: a +b = b + a
Dr. Gabby (KNUST-Maths) Number Theory 8 / 27
16. Introduction to Numbers Integers
Integers
Some properties (a,b,c โ Z)
1 Addition is Commutative: a +b = b + a
2 Addition is Associative: a +(b +c) = (a +b)+c
Dr. Gabby (KNUST-Maths) Number Theory 8 / 27
17. Introduction to Numbers Integers
Integers
Some properties (a,b,c โ Z)
1 Addition is Commutative: a +b = b + a
2 Addition is Associative: a +(b +c) = (a +b)+c
3 Multiplication is commutative: ab = ba
Dr. Gabby (KNUST-Maths) Number Theory 8 / 27
18. Introduction to Numbers Integers
Integers
Some properties (a,b,c โ Z)
1 Addition is Commutative: a +b = b + a
2 Addition is Associative: a +(b +c) = (a +b)+c
3 Multiplication is commutative: ab = ba
4 Multiplication is Associative: a(bc) = (ab)c
Dr. Gabby (KNUST-Maths) Number Theory 8 / 27
19. Introduction to Numbers Integers
Integers
Some properties (a,b,c โ Z)
1 Addition is Commutative: a +b = b + a
2 Addition is Associative: a +(b +c) = (a +b)+c
3 Multiplication is commutative: ab = ba
4 Multiplication is Associative: a(bc) = (ab)c
5 Multiplication Distributes over Addition: a(b +c) = ab + ac
Dr. Gabby (KNUST-Maths) Number Theory 8 / 27
20. Introduction to Numbers Integers
Integers
Some properties (a,b,c โ Z)
1 Addition is Commutative: a +b = b + a
2 Addition is Associative: a +(b +c) = (a +b)+c
3 Multiplication is commutative: ab = ba
4 Multiplication is Associative: a(bc) = (ab)c
5 Multiplication Distributes over Addition: a(b +c) = ab + ac
6 Identity Element of Integer Addition is Zero: a +0 = a = 0+ a
Dr. Gabby (KNUST-Maths) Number Theory 8 / 27
21. Introduction to Numbers Integers
Integers
Some properties (a,b,c โ Z)
1 Addition is Commutative: a +b = b + a
2 Addition is Associative: a +(b +c) = (a +b)+c
3 Multiplication is commutative: ab = ba
4 Multiplication is Associative: a(bc) = (ab)c
5 Multiplication Distributes over Addition: a(b +c) = ab + ac
6 Identity Element of Integer Addition is Zero: a +0 = a = 0+ a
7 Identity Element of Integer Multiplication is One: a ยท1 = a = 1ยท a
Dr. Gabby (KNUST-Maths) Number Theory 8 / 27
22. Introduction to Numbers Integers
Integers
Some properties (a,b,c โ Z)
1 Addition is Commutative: a +b = b + a
2 Addition is Associative: a +(b +c) = (a +b)+c
3 Multiplication is commutative: ab = ba
4 Multiplication is Associative: a(bc) = (ab)c
5 Multiplication Distributes over Addition: a(b +c) = ab + ac
6 Identity Element of Integer Addition is Zero: a +0 = a = 0+ a
7 Identity Element of Integer Multiplication is One: a ยท1 = a = 1ยท a
8 Transitivity: a > b and b > c then a > c.
Dr. Gabby (KNUST-Maths) Number Theory 8 / 27
23. Introduction to Numbers Integers
Integers
Some properties (a,b,c โ Z)
1 Addition is Commutative: a +b = b + a
2 Addition is Associative: a +(b +c) = (a +b)+c
3 Multiplication is commutative: ab = ba
4 Multiplication is Associative: a(bc) = (ab)c
5 Multiplication Distributes over Addition: a(b +c) = ab + ac
6 Identity Element of Integer Addition is Zero: a +0 = a = 0+ a
7 Identity Element of Integer Multiplication is One: a ยท1 = a = 1ยท a
8 Transitivity: a > b and b > c then a > c.
9 Cancellation law: If a ยทc = b ยทc and c ฬธ= 0 then a = b
Dr. Gabby (KNUST-Maths) Number Theory 8 / 27
24. Introduction to Numbers Integers
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.
Dr. Gabby (KNUST-Maths) Number Theory 9 / 27
25. Introduction to Numbers Integers
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
Dr. Gabby (KNUST-Maths) Number Theory 9 / 27
26. Introduction to Numbers Rational and Irrational Numbers
Rational Numbers Q
Example: Rational numbers are solutions of the equation
nr โm = 0.
2
3 is a rational number and solution of the equation โ3r +2 = 0.
Dr. Gabby (KNUST-Maths) Number Theory 10 / 27
27. Introduction to Numbers Rational and Irrational Numbers
Rational Numbers Q
Example: Rational numbers are solutions of the equation
nr โm = 0.
2
3 is a rational number and solution of the equation โ3r +2 = 0.
Some Properties
1 Integers are rational numbers i.e. Z โ Q.
2 Multiplicative inverse of nonzero elements of Q belong to Q (i.e.
โq โ Q,q ฬธ= 0 =โ 1
q โ Q.)
Dr. Gabby (KNUST-Maths) Number Theory 10 / 27
28. Introduction to Numbers Rational and Irrational Numbers
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 ฬธ= 0.
Again, the decimal expansion of an irrational number is neither terminating nor
recurring.
Dr. Gabby (KNUST-Maths) Number Theory 11 / 27
29. Introduction to Numbers Rational and Irrational Numbers
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 ฬธ= 0.
Again, the decimal expansion of an irrational number is neither terminating nor
recurring.
Example
Some examples are
p
2,
p
3, ฯ, and the Euler number e
Dr. Gabby (KNUST-Maths) Number Theory 11 / 27
30. 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.
Dr. Gabby (KNUST-Maths) Number Theory 12 / 27
31. 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 are often represented along a graduated line with extreme values โโ and
+โ.
Figure 1: Real numbers
Dr. Gabby (KNUST-Maths) Number Theory 12 / 27
32. Introduction to Numbers Real Number
Real Numbers
Dr. Gabby (KNUST-Maths) Number Theory 13 / 27
33. Introduction to Numbers Real Number
Real Numbers
Some properties (a,b,c โ R)
1 Addition is Commutative: a +b = b + a
2 Addition is Associative: a +(b +c) = (a +b)+c
3 Multiplication is commutative: ab = ba
4 Multiplication is Associative: a(bc) = (ab)c
5 Multiplication Distributes over Addition: a(b +c) = ab + ac
6 Identity Element of Addition is Zero: a +0 = a = 0+ a
7 Identity Element of Multiplication is One: a ยท1 = a = 1ยท a
8 Transitivity: a > b and b > c then a > c.
9 Cancellation law: If a ยทc = b ยทc and c ฬธ= 0 then a = b
10 R is closed under addition (a +b โ R), and multiplication (ab โ R)
Dr. Gabby (KNUST-Maths) Number Theory 14 / 27
34. Introduction to Numbers Real Number
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 (โx, y โ R;x < y =โ โq โ Q;x < q < y).
Dr. Gabby (KNUST-Maths) Number Theory 15 / 27
35. Introduction to Numbers Real Number
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 (โx, y โ R;x < y =โ โq โ 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.
Dr. Gabby (KNUST-Maths) Number Theory 15 / 27
36. Introduction to Numbers Real Number
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 (โx, y โ R;x < y =โ โq โ 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)
Dr. Gabby (KNUST-Maths) Number Theory 15 / 27
37. Mathematical Induction
Outline of Presentation
1 Introduction to Numbers
Natural Numbers
Integers
Rational and Irrational Numbers
Real Number
2 Mathematical Induction
Dr. Gabby (KNUST-Maths) Number Theory 16 / 27
38. Mathematical Induction
Mathematical Induction
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:
Dr. Gabby (KNUST-Maths) Number Theory 17 / 27
40. Mathematical Induction
Mathematical Induction
1 Prove the statement for
n = 1 (8)
This is the Basis Step, that is P(1) is true
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.
Dr. Gabby (KNUST-Maths) Number Theory 18 / 27
41. Mathematical Induction
Mathematical Induction
1 Prove the statement for
n = 1 (8)
This is the Basis Step, that is P(1) is true
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.
Dr. Gabby (KNUST-Maths) Number Theory 18 / 27
42. Mathematical Induction
Mathematical Induction
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.
Dr. Gabby (KNUST-Maths) Number Theory 19 / 27
43. Mathematical Induction
Mathematical Induction
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.
Dr. Gabby (KNUST-Maths) Number Theory 19 / 27
44. Mathematical Induction
Example
Prove that the sum of the n first odd positive integers is n2
, that is
1+3+5+ยทยทยท+(2n โ1) = n2
.
Dr. Gabby (KNUST-Maths) Number Theory 20 / 27
45. Mathematical Induction
Example
Prove that the sum of the n first odd positive integers is n2
, that is
1+3+5+ยทยทยท+(2n โ1) = n2
.
Let S(n) = 1+3+5+ยทยทยท+(2n โ1). 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
Dr. Gabby (KNUST-Maths) Number Theory 20 / 27
46. Mathematical Induction
Example
Prove that the sum of the n first odd positive integers is n2
, that is
1+3+5+ยทยทยท+(2n โ1) = n2
.
Let S(n) = 1+3+5+ยทยทยท+(2n โ1). 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
2 Inductive Step: Assume (Induction Hypothesis) that the property is true for some
positive integer n, ie S(n) = n2
.
Dr. Gabby (KNUST-Maths) Number Theory 20 / 27
47. Mathematical Induction
Example
Prove that the sum of the n first odd positive integers is n2
, that is
1+3+5+ยทยทยท+(2n โ1) = n2
.
Let S(n) = 1+3+5+ยทยทยท+(2n โ1). 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
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.
Dr. Gabby (KNUST-Maths) Number Theory 20 / 27
48. Mathematical Induction
n +1 case
1 Picking up from the n = k case, we have
S(k) = 1+3+5+ยทยทยท+(2k โ1) (12)
Dr. Gabby (KNUST-Maths) Number Theory 21 / 27
49. Mathematical Induction
n +1 case
1 Picking up from the n = k case, we have
S(k) = 1+3+5+ยทยทยท+(2k โ1) (12)
2 then the n = k +1
S(k +1) = 1+3+5+ยทยทยท+(2k โ1)+(2k +1) (13)
= S(k)+(2k +1) (14)
Dr. Gabby (KNUST-Maths) Number Theory 21 / 27
50. Mathematical Induction
n +1 case
1 Picking up from the n = k case, we have
S(k) = 1+3+5+ยทยทยท+(2k โ1) (12)
2 then the n = k +1
S(k +1) = 1+3+5+ยทยทยท+(2k โ1)+(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
Dr. Gabby (KNUST-Maths) Number Theory 21 / 27
51. Mathematical Induction
n +1 case
1 Picking up from the n = k case, we have
S(k) = 1+3+5+ยทยทยท+(2k โ1) (12)
2 then the n = k +1
S(k +1) = 1+3+5+ยทยทยท+(2k โ1)+(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 = (k +1)2
(15)
Dr. Gabby (KNUST-Maths) Number Theory 21 / 27
52. Mathematical Induction
n +1 case
1 Picking up from the n = k case, we have
S(k) = 1+3+5+ยทยทยท+(2k โ1) (12)
2 then the n = k +1
S(k +1) = 1+3+5+ยทยทยท+(2k โ1)+(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 = (k +1)2
(15)
This completes the induction, and shows that the property is true for all positive
integers.
Dr. Gabby (KNUST-Maths) Number Theory 21 / 27
54. Mathematical Induction
Example
Prove that 2n +1 โค 2n
for n โฅ 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
Dr. Gabby (KNUST-Maths) Number Theory 22 / 27
55. Mathematical Induction
Example
Prove that 2n +1 โค 2n
for n โฅ 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 and 2n
= 23
= 8, so the property is true
in this case.
Dr. Gabby (KNUST-Maths) Number Theory 22 / 27
56. Mathematical Induction
Example
Prove that 2n +1 โค 2n
for n โฅ 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 and 2n
= 23
= 8, so the property is true
in this case.
2 Inductive Step : Assume (Induction Hypothesis) that the property is true for some
positive integer n, that is
2n +1 โค 2n
(16)
Dr. Gabby (KNUST-Maths) Number Theory 22 / 27
57. Mathematical Induction
Example
Prove that 2n +1 โค 2n
for n โฅ 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 and 2n
= 23
= 8, so the property is true
in this case.
2 Inductive Step : Assume (Induction Hypothesis) that the property is true for some
positive integer n, that is
2n +1 โค 2n
(16)
3 We must prove that it is also true for n +1, that is ,
2(n +1)+1 โค 2n+1
expected result (17)
Dr. Gabby (KNUST-Maths) Number Theory 22 / 27
58. Mathematical Induction
1 By the induction hypothesis we know that for n = k we have
2k +1 โค 2k
(18)
Dr. Gabby (KNUST-Maths) Number Theory 23 / 27
59. Mathematical Induction
1 By the induction hypothesis we know that for n = k we have
2k +1 โค 2k
(18)
2 Multiplying both sides by 2
2(2k +1) โค 2ร2k
(19)
4k +2 โค 2k+1
(20)
Dr. Gabby (KNUST-Maths) Number Theory 23 / 27
60. Mathematical Induction
1 By the induction hypothesis we know that for n = k we have
2k +1 โค 2k
(18)
2 Multiplying both sides by 2
2(2k +1) โค 2ร2k
(19)
4k +2 โค 2k+1
(20)
3 for k > 3 then 4k +2 > 2k +3,
Dr. Gabby (KNUST-Maths) Number Theory 23 / 27
61. Mathematical Induction
1 By the induction hypothesis we know that for n = k we have
2k +1 โค 2k
(18)
2 Multiplying both sides by 2
2(2k +1) โค 2ร2k
(19)
4k +2 โค 2k+1
(20)
3 for k > 3 then 4k +2 > 2k +3,
Dr. Gabby (KNUST-Maths) Number Theory 23 / 27
62. Mathematical Induction
1 By the induction hypothesis we know that for n = k we have
2k +1 โค 2k
(18)
2 Multiplying both sides by 2
2(2k +1) โค 2ร2k
(19)
4k +2 โค 2k+1
(20)
3 for k > 3 then 4k +2 > 2k +3, thence
2k +3 โค 2k+1
(21)
Dr. Gabby (KNUST-Maths) Number Theory 23 / 27
63. Mathematical Induction
1 By the induction hypothesis we know that for n = k we have
2k +1 โค 2k
(18)
2 Multiplying both sides by 2
2(2k +1) โค 2ร2k
(19)
4k +2 โค 2k+1
(20)
3 for k > 3 then 4k +2 > 2k +3, thence
2k +3 โค 2k+1
(21)
This completes the induction, and shows that the property is true for all n โฅ 3.
Dr. Gabby (KNUST-Maths) Number Theory 23 / 27
65. Mathematical Induction
Example
Prove that 1+2+ยทยทยท+n =
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)
Dr. Gabby (KNUST-Maths) Number Theory 24 / 27
66. Mathematical Induction
Example
Prove that 1+2+ยทยทยท+n =
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+ยทยทยท+n +(n +1) =
(n +1)(n +2)
2
(23)
Dr. Gabby (KNUST-Maths) Number Theory 24 / 27
68. Mathematical Induction
So for n = k
1+2+ยทยทยท+k =
k(k +1)
2
(24)
then for next time step
1+2+ยทยทยท+k +(k +1) =
k(k +1)
2
+k +1 (25)
Dr. Gabby (KNUST-Maths) Number Theory 25 / 27
69. Mathematical Induction
So for n = k
1+2+ยทยทยท+k =
k(k +1)
2
(24)
then for next time step
1+2+ยทยทยท+k +(k +1) =
k(k +1)
2
+k +1 (25)
=
k2
+3k +2
2
(26)
Dr. Gabby (KNUST-Maths) Number Theory 25 / 27
70. Mathematical Induction
So for n = k
1+2+ยทยทยท+k =
k(k +1)
2
(24)
then for next time step
1+2+ยทยทยท+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.
Dr. Gabby (KNUST-Maths) Number Theory 25 / 27
71. Mathematical Induction
Exercise
1 Find a condition on n โ N such that 2n2
โ7n โ4 โ N.
2 Prove that for all integers n,
1 if n โฅ 3 then 2n
> n +4.
2 1+ x + x2
+...+ xn
= 1โxn+1
1โx ,n โ {0,1,...} where x โ Rโ{1}.
3 12
+22
+...+n2
= 1
6 n(n +1)(2n +1),n โฅ 1.
4 if n โฅ 1, then (1+ x)n
โฅ 1+nx for x โ R+ (i.e. x is a non-negative real number).
Dr. Gabby (KNUST-Maths) Number Theory 26 / 27