introduction what are you doing Amit hey welcome you know today our maths teacher taught us a new topic rational numbers I know about natural numbers whole numbers and integers but what are the rational numbers these numbers are in the form numerator by denominator where numerator is either 0 or positive integer and the denominator a positive integer I did not understand come I will explain you objectives at the end of this lesson you'll be able to define rational numbers right equivalent rational numbers compare rational numbers reduce the rational number to its standard form add subtract multiply and divide rational numbers the word rational arises from the term ratio you mean that a ratio like 3 is to 2 which can also be written as 3 by 2 is a rational number yes similarly the ratio of two integers P and Q in which Q is not equal to zero can be written in the form P by Q this is the form in which rational numbers are expressed 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 zero well now you tell is four by five a rational number yes four by five is a rational number because it is in the form P by Q where Q is not equal to 0 and is minus three by four also a rational number yes because P is equal to minus three and Q is equal to four and both are integers do you know what is numerator and what is denominator in rational number P by Q P is a numerator and Q is a denominator for example in minus 3 by 7 the numerator is minus 3 and the denominator is 7 omit are integers also rational numbers yes for example the integer minus 5 is a rational number because you can write it as minus 5 by 1 similarly the integers 0 is a rational number because 0 can also be written as 0 by 2 or 0 by 7 etc I will now tell you about equivalent rational numbers look at this rational number now look at this rational number these both rational numbers are equivalent but how when we will multiply the numerator and denominator of a rational number by the same nonzero integer we obtain another rational number which is equivalent to that number similarly minus 2 by 3 is equivalent to minus 10 by 15 because when we multiply the numerator and denominator by 5 we get minus 10 by 15 do you know what do we call a rational number which has positive numerator and positive denominator that rational number is called a positive rational number I guess and what are the negative rational numbers when the numerator is a negative integer and the denominator is a positive integer or for example minus four by seven how do we represent a rational number on a number line look at this number line points on right side of zero are called positive integers and points on the left side of zero uphold negative integers let's mark one by two on this line one by two will lie in between 1 and 0 and minus one by two will lie in between 0 and minus 1 now read this statement a rational number is said to be in the standard form if its denominator is a positive integer and the numerator and denominator have no common factor other than one that is these all rational numbers are in the standard form yes because their numerator and denominator have no common factor other than 1 and if the numbers are not in the standard form then it is reduced in a standard form for example minus 45 by 30 is not in a standard form we will reduce it by dividing its numerator and denominator by their HCF here the HCF of 45 and 30 is 15 yes so we will divide the numerator and denominator by 15 and we will get the rational number minus 3×2 which is in a standard form if we have given two rational numbers then can you compare them no I will tell you how to compare rational numbers we know that the negative number is always smaller than the positive number we can also compare the rational numbers using number line the more the number is in the right the greater it is for example two by three lies here and one by three lies here so which one is in the right two by three correct therefore two by three is greater than one by three similarly minus 1 by 3 is greater than minus 2 by 3 because minus 1 by 3 comes right to the minus 2 by 3 can we act the rational numbers yes if the denominators of the rational numbers are same then we just add the numerators for example minus 11 by 5 plus 7 by 5 is equal to minus 11 plus 7 by 5 which is equal to minus 4 by 5 and if the denominators of the rational numbers are not same in that case firstly we find the LCM of the denominators here the LCM is 15 then we find the equivalent rational numbers of the given rational numbers with this LCM as the denominator here minus 7 by 5 is equivalent to minus 21 by 15 and minus 2 by 3 is equivalent to minus 10 by 15 then add the two rational numbers like addition when the denominators are same then we just subtract the numerators and keep the denominators same and if the denominators are not same we find the LCM of the denominators then we find the equivalent rational numbers of the given rational numbers with this LCM as the denominator then we find the equivalent rational numbers of the given rational numbers with this LCM as the denominator then subtract the two rational numbers can you guess how can we multiply rational numbers while multiplying a rational number by a positive integer we multiply the numerator by that integer keeping the denominator unchanged when we multiply the rational numbers we multiply the numerators then we multiply the denominators whoo well done you have learned now before starting about division I will tell you about reciprocals what are reciprocals let's take an example the reciprocal of 2 by 7 is 7 by 2 the reciprocal of 7 by 9 is 9 by 7 okay okay I got it the saprykin of -5 by 4 is 4 by -5 correct so when we divide one rational number by another rational number we multiply the rational number with the reciprocal of the other number summary let us summarize what we have learned 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 is called a rational number if the numerator and denominator of a rational number are multiplied or divided by a nonzero integer we get a rational number which is said to be equivalent to the given rational number when the numerator and denominator both are positive integers it is a positive rational number when either the numerator or the denominator is a negative integer it is a negative rational number a rational number is set to be in the standard form if its denominator is a positive integer and the numerator and denominator have no common factor other than one two rational numbers with the same denominator can be added by adding their numerators keeping the denominator same two rational numbers with different denominators are added by first taking the LCM of the two denominators and then converting both the rational numbers to their equivalent forms having the LCM as the denominator

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