www.covenantuniversity.edu.ng
Raising a new Generation of Leaders
MAT 211
REAL ANALYSIS I
Course Lecturer:
Dr. G.O. Alao
Course Outlines for Module I
Part I:
• The Real Number System
• Euler Number
• Absolute Value Function
Part II:
• Mathematical Induction
2
Course Outlines for Module I (Cont.)
Part III:
• Sequences in R
• Convergence of Sequences in R
• Properties of Convergence of Sequences in R
3
Working Guidelines
 Punctuality to classes,
 75% lecture attendance for eligibility to write
semester examination,
 No eating nor drinking in the class,
 The use of distractive electronic devices is not
allowed.
4
PART I
5
Number Systems
Recall the following number systems:
-
Set of Natural Numbers
-
Set of Integer Numbers
- Set of Rational Numbers
C
- Set of Irrational Numbers
- Set of Real Numbers
6
Set of Natural Numbers
Set of Integers
 1, 2,3, 4,5, 6, 7,8,9,10
=  , 5, 4, 3, 2, 1, 0,1, 2,3, 4,5,


Set of Rational Numbers - Numbers which can be written
p
, where
q
p, q  , q  0

 0, 1, 2, 3, 

1 1 1
 ,  ,  , ,
4 3 2
in the form:



7
Set of Irrational numbers - Numbers which CANNOT be written
in the form:

p
, where
q
p, q  , q  0
   2,  3,  5, 

Set of Real numbers - The combination of all the sets discussed
above makes the set of real numbers.
8
The Real Number System
Rational numbers
• Can be represented as a fraction of 2 integers
Integers
.… -3, -2, -1, 0, 1, 2, 3, ….
Whole numbers
0, 1, 2, 3, ….
Natural numbers
1, 2, 3, ….
Irrational
numbers
• Cannot be
represented
as a fraction
of 2 integers
QUIZ
Write the categories the following numbers belong:
7
0.25
 9.1732
1.237
21
0
36
3
11

13
3
10
Real Number System
The real number system is a set on which the operations
addition    and multiplication   are defined and has
the following properties for all a, b, c  R :
1) a  b  b  a ,
ab  ba
Commutative
2)  a  b   c  a   b  c 
Associative
3) a  b  c   a  b  a  c
Distributive
11
4) There exist distinct real numbers, 0 each 1
such that:
a  0  a,
a 1  a
5) For each a  R there exist,  a and 1
such that:
a    a   0,
a
a  1 1
a
12
Euler Number (e)
An irrational number
The base of the natural logarithm  log e 
The exponential function  e 
x
Used in natural experiment function, compound interest
n
 1
 It is defined as follows: e  1   as n  
 n
13
Calculating Euler Number (e)
Value of n
Formula
Value
1
1
2
4
 1
1  
 1
2
 1
1  
 2
2
 1
1  
 4
4
The value of e up to 12 decimal places is:
2.25
2.718281828459
2.44140625
12
12
1

1  
 12 
2.61303529
14
Calculating
x
e
n
Recall:
 1
e  lim 1   . Hence
n 
 n
x
nx


 1
 1
x
e  lim 1     lim 1  
n 
 n   n   n 
Applying Binomial theorem
n
nx
x x
x
x
x
 1
e  lim 1    1      
n 
1! 2! 3! 4! 5!
 n
2
3
4
5
x
15
Differentiating
d e 
x
dx
x
e
x
x
x
d  x x
 1      
dx  1! 2! 3! 4! 5!
2
3
4
2
3
4
5



x
x
x x
x
 0 1      e
1! 2! 3! 4!
x
d e 
x
e
Thus
dx
e is the only number that posses this property.
16
Absolute Value Function
The absolute value of a real number, x denoted by x
is defined by:
x

x  0
 x

Example: if
if
x0
if
x0
if
x0
x  5, then x    5   5
17
Properties
For all x, y  R
1) xy  x y
2) x  x
2
2
3)  x  x  x
4) x  y  x  y
5) x  y  x  y

 Triangle Inequalities

18
Triangle Inequality
If x, y  R then 1 x  y  x  y
 2 x  y  x  y
Proof: From property 3 we have that  x  x  x and
 y  y  y . On adding these two inequalities we get
 x  y  x y  x  y

x y  x  y
1
In (1) replace x with x  y


 2
x y  x  y
x y y  x y  y
19