Gaurav Tiwari

A Microblog by Gaurav Tiwari

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What is Real Analysis?

Real analysis is the branch of Mathematics in which we study the development on the set of real numbers. We reach on real numbers through a series of successive extensions and generalizations starting from the natural numbers. In fact, starting from the set of natural numbers we pass on successively to the set of integers, the set of rational numbers and that of real numbers.
Let $ \mathbf {Q}$ be the set of rational numbers. It is well known that $\mathbf {Q}$ is an ordered field, i.e., $ \mathbf {Q}$ is an Algebraic Structure on which the operations of addition, subtraction, multiplication and division by a non-zero number can be carried out. Also the set $\mathbf {Q}$ is equipped with a relation called “less than” which is an order relation. Between two rational numbers there exist infinite number of elements of $\mathbf {Q}$. Thus the system of rational numbers seems to be dense and so apparently complete. But it is quite easy to show that there exist some numbers which are not rational. Such numbers are called irrational numbers. The set/collection of rational and irrational numbers combined is called the set of real numbers. Study of these numbers is Real Analysis.