EXTENDED REAL NUMBER SYSTEM & THE SYMBOL " INFINITY (∞) # EXTENDED #REAL #NUMBER #INFINITY

March 15, 2025

https://youtu.be/-Z3T5vcfaSE?si=OJxFNMZd7G5RdMGB

contents :

1. Extended real number system

2. The symbol infinity

3. Properties of infinity

THE EXTENDED REAL NUMBER SYSTEM :

Basically the real number system not contain + or - ∞, it contains all the real numbers lies between for -∞ and +∞ i.e. the real number system R=(-∞.∞).

The extended real number system is a union of the real number set R and the symbols -∞ and +∞ i.e. Extended real system = R ⋃ {-∞,∞} = [-∞,∞]

In extended real number system, ∞ is an upper bound and -∞ is a lower bound of every subset.

Here we see an example to tell the main difference between real and extended real number systems.

Consider an open interval (2,∞). The interval (2,∞) is not bounded above in R, as in R, ∞ is not an upper bound but the same interval (2,∞) is bounded above in extended real number system as ∞ is an upper bound of every subset.

If you observe thoroughly in the above introduction of extended number system, I do not mention ∞ as a number. Then what is the symbol ∞ & what is its priority in mathematics?

The symbol " ∞ " :

First of all ∞ is not a number. ∞ is a symbolic representation of non- terminating occurrence of numbers or values etc.

In mathematics, in so many incidents or circumstances we have to express some terms or definitions or values of a number are never ended. It is impossible to express every time. To rectify that problem the symbol ∞ is introduced.

For example if we consider the set of natural numbers N={ 1,2,3,...∞} means the numbers in N are never ended.

We have 1 , 1², 1³, …, 1ⁿ= 1 for n Ꜫ N.

We do not conclude 1^∞ as 1, since the process of multiplication 1 to itself ∞ times is not terminated to tell the value as 1.

Not only 1^∞ but also the values of 0^∞,∞^∞, a^∞ etc

i.e. any number with ∞ as integral power is not known . Hence they are called indeterminate forms.

Properties of ∞ :

For a finite real number a ,