 Hello and welcome to the session. In this session we will learn how to represent addition and subtraction of rational numbers on a horizontal and vertical number line. Now we know that on a number line we have positive numbers on the right side of zero and negative numbers on the left side of zero. So a number line consists of positive numbers, negative numbers and zero. Now to show addition and subtraction on a number line we always start from zero. Now if a and v are ever two numbers then we have a type of expression. First is a plus b, second is a plus of minus b, third is minus a plus b and fourth one is minus a plus of minus b. Now we are going to show each one of them on a number line. First of all let us show the expression a plus b on a number line. For this example in which we have to find two plus three on a number line. Now we can also write it as plus two the whole plus of plus three. Now here plus two and plus three both are positive numbers and we have to add two positive numbers. First of all let us plot the point plus two on a number line. Now we know that on the right side of zero we have positive numbers. So we will start from zero and for locating the point plus two we will move to the right of zero and we will reach the point plus two. And then we will have plus three which is again a positive number. So again to the right of plus two and we will reach plus five therefore plus two plus of plus three is equal to plus five. So this is equal to plus five or we can write two plus three is equal to. Now let us see one more example in which we have to find one point four plus one point five using a number line. Now again one point four and one point five both are positive numbers. So we will start from zero since one point four is positive so move one point four units to the right of zero and we will reach to this point that is the point one point four. Then we have to add one point five to one point four, one point five units to the right of one point four and moving one point five units to the right of one point four we will reach at this point which is the point two point nine. So one point four plus one point five is equal to two point nine. Now let us show the second expression on a number line. What is considered an example in which we have to find two point three plus or minus five by three on a number line. We can also write it as plus two point three plus or minus five by three. Now here two by three is positive and five by three is negative and we have to add these two fractions. Now these two are like fractions that is the fractions with same denominators. Now here the denominator is three so we have drawn a number line in which we have divided each unit in. Now here is of length one by three we will plot the fraction plus two by three on a number line. Now two by three is positive so starting from zero we will move two by three units of zero and we will reach at this point. Now we have to add minus five by three to the fraction two by three. Now minus five by three is a negative fraction this means we have to move five by three now here is of length one by three. So to move five by three units to the left of two by three we will move to the left of two by three that is one, two, three, four and five. And we have reached at this point so two by three minus five by three is equal to minus one. Also we can verify this now two upon three plus of minus five upon three is equal to two plus of minus five four upon three which is equal to two minus five four upon three which is equal to minus three upon three which is equal to minus one. Hence it is verified. So we can find four plus of minus zero point eight using a number line. Now here we will start from zero and since one point four is positive so we will move one point four units to the right of zero and we will reach at this point. So we have reached at this point now we have to add minus zero point eight to one point four now minus zero point eight is negative this means we have to move zero point eight units to the left of one point four. So we will reach at this point when we move zero point eight units to the left of one point four as here each part is of length zero point two. So one point four of minus zero point eight is zero point six. Now let us show the third type of expression on number line. For this consider the example in which we have to find minus three by eight plus seven by eight using a number line. For this we have drawn a number line in which we have divided each unit into eight parts and here the length of each part is one by eight. Now here we have minus three by eight which is a negative fraction. So we move to the left of zero and we will reach to this point which is minus three by eight. We have to add seven by eight to minus three by eight. Now seven by eight is a positive fraction. So we will move seven by eight units to the right of minus three by eight and we will reach at this point which is four by eight. So minus three by eight plus seven by eight is equal to four by eight. You can verify this now minus three by eight plus seven by eight is equal to minus three plus seven four upon eight which is equal to four upon eight. Now let us show the fourth type of expression on a number line. For this consider the example in which we have to find minus two by three plus of minus five by three by using a number line. Now these are like fractions with denominator three. So we have drawn a number line in which we have divided each unit into three parts. This two by three is a negative fraction. So we will start from zero and we will move two by three units to the left of zero and we will reach at this point and then minus five by three is again negative and we have to add minus five by three to minus two by three. So we will move three units to the left of minus two by three and we have reached at the point minus seven by three. So minus two by three plus of minus five by three is equal to minus seven by three. On solving we get the sum equal to minus seven by three. Now consider one more example. Now here first of all we have to plot one by two. So we will move one by two units to the right of zero and we will reach at this point. Now we have to add minus one by two to one by two. Now minus one by two is negative. So we will move one by two units to the left of one by two and we will reach at this point which is the point zero. So the answer of this is zero. Now using this vertical number line let us solve the given example. In this it is given two stones are thrown vertically upwards. One reaches the height one point two feet and the other stone reaches the height zero point six feet and then the previous stone what is the height of certain stone. Now here you can see that the numbers above zero are positive and numbers below zero are negative. Now it is given that the first stone reaches the height of one point two feet that is from zero. It will reach at this point. The second stone reaches the height zero point six feet then the previous stone will move zero point six units upwards. Each at this point we have reached at this point which is the point one point eight. It means the second stone reached the height of one point eight feet. So in this way we have solved the given example using the vertical number line. In the same manner we can subtract two rational numbers using a vertical number line. So you have learnt how to represent a lesion and subtraction of rational numbers on a horizontal and vertical number line. And this completes our session. Hope you all have enjoyed the session.