Counting numbers or natural numbers are probably the first (and the most important) mathematical objects to be invented. You have encountered them from early childhood and learnt to work with them - adding, multiplying, subtracting, dividing, comparing them etc. You were then introduced to 0 (zero) and this along with the natural numbers gave you the set of whole numbers. The next step, was to introduce you to integers ('negatives' of the natural numbers were the extra 'numbers' that you got). In the last two years you have been told about fractions and rational numbers. At each stage you worked with them - adding, multiplying, comparing them etc. You either proved or observed various properties of these sets of numbers. One of the basic properties that you learnt was that each successive step gave you a larger set of 'numbers'. Using long - division or other wise, you learnt about decimal expansions of rational numbers. At this stage, you may have noticed that certain decimal expansions cannot be of rational numbers. So there must exist a larger set of 'numbers' which includes other 'numbers' besides the rational numbers. 

This set is called the set of real numbers and the members of this set which are not rational numbers are called irrational numbers.

The set of real numbers is unique in the world of mathematics. Any other collection of mathematical objects with properties similar to those of real numbers is just the set of real numbers under a different name. These rather imprecise statements will become clear to you in higher classes. We leave this part of the discussion with just one more rather imprecise statement, which again will be understandable only at a much later stage - there exist sets very dissimilar to the other sets of 'numbers' (natural numbers, integers, rational numbers) but having very similar properties. 

The set of real numbers, though unique in nature, is not the end-point in the development of numbers. In the next two - three years you will be introduced to complex numbers and if you continue with mathematics, you will come across algebraic numbers, transcendental numbers, cardinal numbers, ordinal numbers etc.

But all said and done, real numbers are the most 'used' numbers. A decent understanding of mathematics is impossible without grasping the basic properties of real numbers.

The object of this lesson is to view the set of real numbers in a few different ways, learn some of its properties  which can be used to understand other mathematical objects. Quite a few properties like Euclid's Division Lemma, Euclid's division algorithm, Fundamental Theorem of  Arithmetic are stated without proofs - trust us!- you will be able to comprehend their importance and proofs in later classes.