 Hi and welcome to the session. Today we will learn about rational numbers. A rational number is defined as a number that can be expressed in the form p by q where p and q are integers and q is not equal to 0. For example, 5 by 6 minus 2 by 8 5 are all rational numbers. Rational numbers include integers, fractions. That means all integers and all fractions are rational numbers. In the rational number p by q, p is the numerator and q is the denominator where q is not equal to 0. Now let's see equivalent rational numbers if numerator and denominator of a rational number are multiplied or divided by a non-zero integer. We get a rational number which is said to be equivalent to the given rational number. For example, suppose we have given a rational number minus 1 by 3. Now let's multiply the numerator and denominator of the given rational number by the non-zero integer 2. So we get minus 1 into 2 upon 3 into 2 which is equal to minus 2 upon 6. So here the rational number minus 2 upon 6 is equivalent to the given rational number minus 1 upon 3. So that means minus 1 upon 3 and minus 2 upon 6 are equivalent rational numbers. Next we have positive and negative rational numbers. Rational numbers are classified as positive rational numbers and negative rational numbers. Rational numbers in which both numerator and denominator are positive integers are called positive rational numbers. And the rational numbers in which either numerator or denominator is a negative integer on negative rational numbers. For example, 3 by 7 is a positive rational number and minus 2 by 5 and 3 by minus 4 are negative rational numbers. Now let's see the rational number say minus 1 upon minus 3. Now let us multiply the numerator and denominator by minus 1. So this will be equal to minus 1 into minus 1 upon minus 3 into minus 1 which will be equal to 1 by 3. Now 1 by 3 is a positive rational number that means minus 1 by minus 3 is also a positive rational number. That means 3 by 7 and minus 1 by minus 3 are the examples of positive rational numbers. The number 0 is neither positive nor a negative rational number. Now let's see how to represent a rational number on a number line. Suppose we want to represent minus 1 by 3 on the number line. So first of all let us draw the number line. The points to the right of 0 are denoted by positive sign and a positive integer. And the points to the left of 0 are denoted by negative sign and are negative integers. Now the positive rational numbers will be marked on the right of 0 and the negative rational numbers will be marked on the left of 0. Here minus 1 by 3 is a negative rational number that means it will be marked on the left of 0. As we know that it is tensed between 0 and 1 and 0 and minus 1 is equal. And that means the distance between 0 and 1 by 3 and 0 and minus 1 by 3 will also be equal. We also know that 1 by 3 lies between 0 and 1. So that means here minus 1 by 3 will lie between 0 and minus 1. Here the denominator of the given rational number is 3. So let us divide the distance between 0 and minus 1 in 3 equal parts like this. So here first point we got is minus 1 by 3. Second point is minus 2 by 3. And third point is minus 3 by 3 which is equal to minus 1 itself. Now we want it to mark minus 1 by 3. So here this point is minus 1 by 3. Thus in this way we can represent rational numbers on number 9. Our next topic is rational numbers in standard form. A rational number is said to be in standard form if its denominator is a positive integer, its numerator and denominator have no common factor other than 1. For example the rational number minus 2 upon 5 is in the standard form. The denominator 5 is a positive integer and the numerator minus 2 and denominator 5 have no common factor other than 1. If a rational number is not in a standard form then we can reduce it to the standard form. So to reduce a rational number to its standard form, divide the numerator and denominator by del hc ignoring the negative sign. Let us take an example for this to reduce the rational number 12 upon minus 18 to its standard form. Now we know that the hcf of 12 and 18 is equal to 6. So let us divide the numerator and denominator of the given rational number 12 upon minus 18 by the hcf that is 6. So this will be equal to 2 upon minus 3. The numerator and denominator have no common factor other than 1 but the denominator is a negative integer. So we need to change it to a positive integer. So let us multiply the numerator and denominator by minus 1 to make it positive. So we have 2 into minus 1 upon minus 3 into minus 1. This will be equal to minus 2 upon 3. So the standard form of 12 upon minus 18 is minus 2 upon 3. With this we finished this session. Hope you must have understood all the concepts. Goodbye and take care and have a nice day.