“Enigmatic exhibition of each real number on several sets”

Abstract

In the most primitive times our ancestors employed the principle of one-one correspondence for counting purposes. In those days when the shepherd came back in the evening from the grazing fields he used to count his sheep by pebbles (or sticks) in his bag and he was satisfied if he had as many sheep as pebbles in his bag. Later on different vocal sounds were developed as word tally against the number of obstacles and later on symbols were evolved to stand for these numbers. In this way it took many thousands of years to come to 1, 2, 3, 4, 5,…… which are called natural numbers ℕ. The properties of natural numbers were developed from a few of simpler properties known as Peano's Axioms after the Italian Mathematician Guiseppe Peano(1858-1932) who gave these axioms in 1899.We thus find that the operations of addition and multiplication are fully defined in the set of natural numbers and then union , intersection in different sets. After a long struggle we have been able to form a well-decorated set ℝ, the set of real numbers. This ℝ contains all kind of numbers namely natural numbers, negative integers, zero, rational numbers and irrational numbers. There are infinitely many real numbers in ℝ. Every real number has a specific name. In my research work, I would like to convey the concerned readers that every real number takes a new name when we consider that real number on different sets.