This document provides an overview of the real number system and real line. It defines the real line as the set of all real numbers represented by R or IR. Real numbers can be represented on a number line with positive and negative integers shown in equal intervals extending indefinitely in both directions. The set of real numbers is the union of rational numbers Q and irrational numbers. Bounded and unbounded sets are also defined, where a bounded set has an upper or lower bound and an unbounded set does not.
2. Real Number System
In this unit, we shall study the following:
• Field of Real Numbers and
• Order Structure of real numbers
• Countable sets and uncountable sets
• Real line
• Bounded and unbounded sets-Supremum and
Infimum
• Order Completeness property
• Archimedean Property of real Numbers
• Intervals: Open sets, Closed sets and Countable sets.
3. Real line
I hope we are well acquainted with Set, Subset,
equality of sets, union and intersection of sets,
and union and intersection of an arbitrary family of
sets, Universal sets, functions, equivalent sets and
composition of structure, that is, addition and
multiplication
Now, we define Real line
4. Real
Real line
The real line or real number line is the line whose
points are real numbers. ie, the real line is the set R
of all real numbers, real line can be show below ;
The real line is usually denoted by R (or IR)
5. Real line
Representation of real numbers on number line :-
Real numbers can be represented on a number line,
which is a straight line that represents the i ntegers in
equal internals. Both positive and negative integer,
can be represented on a number line in a sequence
as shown below this line extended indefinitely at both
ends.
6. Real line
• Real numbers definitions:
• The set of real numbers is the union of the set of
rational numbers Q and the set of irrational number
Q. Therefore, all the number such as natural
numbers, whole numbers, integers, rational numbers
and irrational numbers are subset of the set of real
numbers.
• The set of real number is represented by R.
7. Bounded sets
Bounded sets :- Bounded and unbounded subset of
real numbers
Set bounded above definition: Let S be a subset of real
numbers. We say that S is bounded above if there
exists a real number b, not necessarily a member of
S, such that
..(I)
The number b is called an upper bound of S. If there
exists no real number be satisfying
(1) , then the set S is said to be not bounded or
unbounded above.
x s x b
x s
8. Set bounded below
Set bounded below
Definition :- Let S be a subset of real numbers we say
that S is bounded below if there exists a real number
a, not necessarily a member of S, such that
……..(2) The number a
is called a lower bound of S. If there exists no real
number a Satisfying (2), then the set S is said to be
not bounded or unbounded below Thus, a set is
unbounded below if however, small a real number a
may be chosen, there exists at least one such
that x<S
x s x a
x s
x s
10. References
1) G K Ranganath College Mathematics
(S Chand& co).
2)G.K Ranganath College Mathematics Vol .1 (part 1)
(S Chand& co).
3) Dr.Syeda Rasheeda Parveen and Dr.Niyaz Bugum
Algebra II and Calculus II ( S.S.Bhavikatti Prakashana).
