 Hello and welcome to this session. In this session we will discuss number systems. We know that N that is the set of natural numbers consist of the numbers 1, 2, 3, 4 and so on. W that is the set of whole numbers consist of the numbers 0, 1, 2, 3, 4 and so on. Then we have Z which is the set of integers which consist of the negative numbers like minus 3, minus 2, minus 1, then 0 and then we have the positive numbers. Now first we will discuss what are rational numbers. A number R is called a rational number if it can be written in the form p upon q where we have p and q are the integers and q is not equal to 0. Consider the numbers 1 upon 3, 4 upon 5. These are all rational numbers. Rational numbers do not have a unique representation in the form p upon q where the p and q are integers and q is not equal to 0. Let's consider a rational number 4 upon 7. Now when we multiply the numerator and denominator of this rational number by 2 we get 8 upon 14. In the same way let's multiply the numerator and denominator of rational number 4 upon 7 by 3. This would give us 12 upon 21. So in this way we can multiply the numerator and denominator of a rational number by any number so as to get a rational number and the rational number that we obtain by doing this are called equivalent rational numbers and we say that this 4 upon 7 is equal to 8 upon 14 is equal to 12 upon 21. Then we also have that there are infinitely many rational numbers between any two given rational numbers. Consider the numbers 3 and 4. Let's find out two rational numbers between 3 and 4. Now we have two methods to do this. First let's see which is the first method to do this. In this we consider let R equal to the number 3 and S be equal to the given number 4. So we have first rational number between 3 and 4 is given by 3 plus 4 upon 2 which is equal to 7 upon 2. Then the second rational number between 3 and 7 upon 2 is given by 3 plus 7 upon 2 whole upon 2. So this gives us 13 upon 4. So we have 3, then 13 upon 4, then 7 upon 2 and then 4. So these are the two rational numbers between the given numbers 3 and 4. Now let's see what is the second method to do so. In this method we can find both the rational numbers in one step only for this we write the given numbers 3 and 4 as rational numbers with denominator equal to 2 plus 1. We have taken 2 since we need to find two rational numbers between the given numbers 3 and 4. That is we write the given numbers 3 and 4 as rational numbers with denominator as 3. We can write 3 as 9 upon 3 that is we have got the denominator as 3 and then we write 4 as 12 upon 3 here also we have got the denominator as 3. So now we need to find two rational numbers between the rational numbers 9 upon 3 and 12 upon 3. So we get 9 upon 3 that is 3 then we have 10 upon 3 then 11 upon 3 and then finally 12 upon 3 which is the given number 4. So these are the two rational numbers between the given numbers 3 and 4. Now let's see what are the rational numbers. A number s is called irrational if it cannot be written in the form p upon q where p and q are the integers and we have q is not equal to 0 like cube root 3 then root 2 pi these are all irrational numbers. A collection of real numbers consist of the rational numbers and the irrational numbers. So we can say that a real number is either rational or irrational number. Every real number is represented by a unique point on the number line also every point on the number line represents a unique real number. That's why we call the number line the real number line. Now let's see how we can locate an irrational number on a number line. Let's try and locate square root 5 on the number line. Let's consider a unit square oabc that is each side is of one unit length. So the diagonal o b of the square oabc would be of length square root 2. Now we transfer the square oabc on this number line so that this point o of the square coincides with the 0 of the number line. So we get this and this oab is of length square root 2. Now to mark the point square root 2 on the number line what we do is we take oabc center and radius oab that is square root 2 and we mark that point on the number line. So this point we name this point p. This point p represents square root 2 on this number line. Now we shall mark square root 3 on this number line for this what we do is we make a perpendicular bd of length 1 unit at point b. So this bd is of 1 unit length and now when we apply Pythagoras theorem to this right triangle dob we get od equal to square root 3. Now we take ost center and radius od and then we mark a point on this number line. So this point let this be q. This point represents square root 3 on the number line. Now let's move on to represent square root 4 on the number line. For this again at b we will draw perpendicular of unit length. So here we have taken de of 1 unit in length and by applying Pythagoras theorem to this right triangle od we get e o equal to square root 4. Now to mark this point square root 4 or you can say 2 on the number line we take ost center and radius as oe and we mark a point. So we get this point let this be point r represents square root 4 or you can simply say 2. Now finally we mark the point square root 5 on the number line. So this again at point e we will draw a perpendicular of unit length. So we have taken ef of unit length that is we have drawn a perpendicular of unit length at the point e. By Pythagoras theorem applied to the right triangle f oe we get of equal to square root 5. Now to mark this point square root 5 on the number line we take ost center and radius of f and we mark a point on the number line. This point let this be point s this is square root 5 on this number line. So this is how we can represent any irrational number on number line. This completes the session hopefully you understood the concept of rational numbers and irrational numbers.