 Hello and welcome to the session. In this session we are going to discuss real numbers and its properties. First of all let us discuss what are real numbers. Rational numbers, irrational numbers combine to form a set called the set of real numbers. Now the real numbers can be positive, negative or zero. Now any numbers you can think of is a real number. Numbers like 2, 14.23, minus 0.23, 2 by 3, root 3 and 1090 are all real numbers. Now the numbers which are not real are square root of minus 1, imaginary number and infinity is not a real number. Now here the real numbers are shown diagrammatically and here we have not real numbers which are the imaginary numbers and infinity. Now here n denotes the set of natural numbers which start from 1 so it contains 1, 2, 3, 0 to the set of natural numbers and we get the set of whole numbers which contains elements 0, 1, 2, 3, 4 and so on. Now the set of natural numbers is a subset of, now the whole numbers are positive the negative numbers in this set that is minus 1, minus 2, minus 3 and the set of integers which is denoted by i. Now if to the set of integers if we add non-integral values like 1 by 2 with the set of rational numbers which is denoted by q to the set of rational numbers root 2, root 3 of real numbers which is denoted by the diagram that the set of natural numbers is a subset of the set of whole numbers then the set of whole numbers is the subset of set of integers and the set of integers is the subset of rational numbers of the set of real numbers. Now let us discuss the properties of real numbers, real numbers for the operations of addition and multiplication the closure property, the closure property for addition is if we add then the result which is p plus q also a real number, so for multiplication if we multiply any two real numbers that is p and q then the result which is p into q is also a real number. Now the second property is the commutative property, if addition is r to real numbers then plus q is equal to q plus p and for multiplication it is p into q is equal to q into p. Now the third property with real numbers then plus q the whole plus r is equal to p plus q plus r in case of multiplication it is into q the whole into r is equal to p into q into r the whole and the next is in case of addition real number p plus 0 is equal to 0 plus p is equal to p that is 0 and in case of multiplication for each p that is for each real number p into 1 is equal to 1 into p is equal to p that is operation inverse in case of addition for each real number p you may minus p of the x of minus p is equal to minus p real number p which is not equal to 0 now p the multiplication inverse v into 1 by p is equal to 1 by p into p which is equal to 1. Now next property which is common to both addition and multiplication now according to this if p, q and r are any three real numbers then into q the whole plus is called the left distributive q, q and r of the whole into p is equal to q into p the whole plus into p the whole and this is called hope you all have enjoyed the session.