Infinity: The numbers without boundaries.


The Infinity can simply be defined as a number without boundary or we say a boundless or endless number that is larger than any real or natural number. 
Denoted by the symbol $\infty$, the symbol was introduced by John Wallis. 
In real analysis, the Infinity is used to denote an unbounded limit and that is why we say the notion $x\rightarrow{\infty}$ means $x$ increases without bound. 
And when $x\rightarrow{-\infty}$, it means $x$ decreases without bound. 

In integral calculus, if $f(t)$ is a function and $f(t)\geq{0}$ then:
1. $\int^b_a{f(t)}dt=\infty$ means that $f(t)$ does not bound the finite area from $a$ to $b$.
2. $\int^\infty_{-\infty}f(t)dt=\infty$ means that the area under $f(t)$ is infinite.
3. $\int^\infty_{-\infty}f(t)dt=a$ means that the total area under $f(t)$ is finite and is equal to $a$.
Infinity can also be used to define infinite series.
1. $\sum^\infty_{i=0}f(i)=a$ means that the sum of the infinite series converges to $a$.
2. $\sum^\infty_{i=0}f(i)=\infty$ means that the sum of the infinite series diverges to $\infty$.
In the extended real number system, we can extend infinity to represent values. For example in topological space, the points $+\infty$ and $-\infty$ are used to define compactness of the real numbers.

In complex analysis, $\infty$ is used to define an unsigned infinite limit. When we say $x\rightarrow{\infty}$ it means that the magnitude of $x$, $|x|$, grows beyond any assigned value i.e it grows without bound.
The Riemann sphere can be obtained when $\infty$ is added to the complex plane as a topological space thereby resulting in a one point compact complex plane.

In set theory, we can imagine the $\infty$ as a system of transfinite number known as the ordinal and cardinal infinities, it was first developed by George cantor.
The ordinal numbers characterizes a well ordered set(countable set) while the cardinal numbers define the size of a set meaning how many members they contain, and can be standardized by choosing the first ordinal number of a certain size to represent the cardinal number of that size.
The smallest ordinal infinity is that of the positive integers, and any set which has the cardinality of the integers is countably infinite.
And if the set is too large to be represented on a one-to-one mapping with positive integers then we say the set is uncountable.