 Hello and welcome to the session. In this session we will discuss rational numbers. We already know that natural numbers, whole numbers and integers, they all make a connection of rational numbers. Now let's define a rational number, a number which can be written in the form p upon q where p and q are integers and q is not equal to 0 is called a rational number. Consider the numbers minus 1 upon 3, 6 upon 7, 3 upon 8, these are all rational numbers. Now we shall discuss different properties of rational numbers under different operations. First we discuss the closure property. We have the rational numbers are closed under addition that is for rational numbers a and b their sum that is a plus b is also a rational number. Then we have the rational numbers are closed under subtraction also that is for rational numbers a and b, a minus b is also a rational number. Rational numbers are closed under multiplication also, for rational numbers a and b a multiplied by b that is the product of the two rational is also a rational number. But we have the rational numbers are not closed under division because we find that for any rational number A, A divided by 0 is not defined. So we say the rational numbers are not closed under division. However if we exclude the number 0 from the collection of the rational numbers then we can say that the rational numbers are closed under addition but when we include the number 0 we find that rational numbers are not closed under division. Next property that we discuss is commutativity. The rational numbers can be added in any order and we have that addition is commutative for rational numbers. Like if you consider the rational numbers 1 upon 2 and 1 upon 4, 1 upon 2 plus 1 upon 4 is equal to 3 upon 4. Now if you change the order of the rational numbers while adding them that is if you have 1 upon 4 plus 1 upon 2 this would also give us 3 upon 4. So we say that for any two rational numbers A and B, A plus B is equal to B plus A that is addition is commutative for rational numbers. Then we have subtraction is not commutative for rational numbers considering the rational numbers 1 upon 2 and 1 upon 4, 1 upon 2 minus 1 upon 4 is equal to 1 upon 4. Now let's change the order of the rational numbers while subtracting them that is we have 1 upon 4 minus 1 upon 2 and this would give us minus 1 upon 4 and you find that these two are not equal so we say that for rational numbers A and B, A minus B is not equal to B minus A hence the traction is not commutative for rational numbers. Multiplication is commutative for rational numbers. Consider the rational numbers 1 upon 2 and 1 upon 4 let's multiply them we get 1 upon 8. Now when we change the order and multiply that is we have 1 upon 4 multiplied by 1 upon 2 this would also give us 1 upon 8. So we say that for rational numbers A and B, A into B is equal to B into A. The vision is not commutative for rational numbers 1 upon 2 when divided by 1 upon 4 gives us 2 then 1 upon 4 when divided by 1 upon 2 gives us 1 upon 2. Now as you can see both these results are not equal so we say that division is not commutative for rational numbers that is we have when A divided by B then we have this is not equal to B divided by A where A and B are the two rational numbers. Next property that we discuss is associativity. Addition is associative for rational numbers. Let's consider three rational numbers 1 upon 2, 1 upon 4 and 1 upon 3. Now let's add them in this way. This is equal to 1 upon 2 plus 7 upon 12 and this is equal to 13 upon 12. Next we have 1 upon 2 plus 1 upon 4 plus 1 upon 3. Now this would be equal to 3 upon 4 plus 1 upon 3 that is this is equal to 13 upon 12. So from this we get that for any three rational numbers A B and C we have A plus B plus C is equal to A plus B plus C that is addition is associated for rational numbers. Next we have subtraction is not associated for rational numbers. Again let's consider the rational numbers 1 upon 2, 1 upon 4 and 1 upon 3. Now let's subtract them in this way. So this would be equal to 1 upon 2 minus minus 1 upon 12 which comes out to be equal to 7 upon 12. Now let's subtract them in this way 1 upon 2 minus 1 upon 4 minus 1 upon 3. Now this would be equal to 1 upon 4 minus 1 upon 3 and this would be equal to minus 1 upon 12 and thus from this result we say that for any two rational numbers A B and C we have A minus B minus C is not equal to A minus B minus C. Then we have multiplication is associated for rational numbers. Consider the rational numbers 1 upon 2, 1 upon 4, 1 upon 3. Let's multiply them in this way. So this would be equal to 1 upon 2 multiplied by 1 upon 12 which is equal to 1 upon 24. Now let's multiply them in this way that is 1 upon 2 multiplied by 1 upon 4, the whole multiplied by 1 upon 3. This is equal to 1 upon 8 multiplied by 1 upon 3 which is equal to 1 upon 24. Thus we say for three rational numbers A B and C A into B into C is equal to A into B into C. Then we have division is not associated for rational numbers. Considering the rational numbers 1 upon 2, 1 upon 4 and 1 upon 3, let's divide them in this way. So this is equal to 1 upon 2 divided by 3 upon 4 which is equal to 2 upon 3. Now let's divide them in this way that is 1 upon 2 divided by 1 upon 4, the whole divided by 1 upon 3. Now this is equal to 2 divided by 1 upon 3 that is equal to 6. Now these two results are not equal so we say that for three rational numbers A, B and C, A divided by B divided by C is not equal to A divided by B, the whole divided by C. So we have now discussed the properties of rational numbers under different operations. Next we shall discuss role of zero. The rational number zero is the additive identity for rational numbers that is when we add zero to a rational number like for a rational number A if we add zero to this we get the rational number itself. Next we discuss role of one. The rational number one is the multiplicative identity for rational numbers like if you consider the rational number one and you multiply this rational number by one you get the rational number itself. Next is negative of a number for a rational number A upon B we have A upon B plus minus A upon B and this is equal to minus A upon B plus A upon B and this is equal to zero. So we say that minus A upon B is the additive inverse of A upon B and A upon B is the additive inverse of minus A upon B. Like for a rational number minus 6 upon 11 its additive inverse is given by 6 upon 11 since minus 6 upon 11 plus 6 upon 11 would be equal to zero. Next we discuss reciprocal. This rational number C upon D is the reciprocal or the multiplicative inverse of another rational number A upon B if we have that A upon B when multiplied by C upon D is equal to one. So for the rational number minus 6 upon 11 its reciprocal or you can say the multiplicative inverse is given by minus 11 upon 6 since minus 6 upon 11 into minus 11 upon 6 would give us one. Now we have distributivity of multiplication over addition and subtraction for rational numbers. For all rational numbers A B and C we have A into B plus C is equal to A B plus AC and A into B minus C is equal to A B minus AC. Consider the rational numbers one upon two, one upon four and one upon three. Now one upon two into one upon four plus one upon three is equal to one upon two into seven upon twelve that is equal to seven upon twenty four. Now let's consider one upon two into one upon four plus one upon two into one upon three this is equal to one upon eight plus one upon six and this is equal to seven upon twenty four. So you see that one upon two multiplied by one upon four plus one upon three is equal to one upon two into one upon four plus one upon two into one upon three. This is how we show distributivity of multiplication over addition. Now we discuss distributivity of multiplication over subtraction. So we have one upon two multiplied by one upon four minus one upon three this would be equal to one upon two multiplied by minus one upon twelve and this is equal to minus one upon twenty four. Now let's consider one upon two multiplied by one upon four minus one upon two multiplied by one upon three this is equal to one upon eight minus one upon six and this is also equal to minus one upon twenty four. So we have one upon two multiplied by one upon four minus one upon three is equal to one upon two multiplied by one upon four minus one upon two multiplied by one upon three. So this completes the session hope you have understood the concept of rational numbers and different properties of rational numbers under different operations multiplicative inverse, additive inverse, rule of zero, rule of one and distributivity of multiplication over addition and subtraction.