**Understanding Rational Numbers**

Rational numbers are a fundamental concept in mathematics, and they have several properties that make them useful for solving various mathematical problems. In this article, we will explore some of these properties, including how to perform operations with rational numbers.

**Closure Property**

One of the key properties of rational numbers is the closure property. This property states that when two rational numbers are subtracted, the result is always a rational number. For example, consider the two rational numbers 3/7 and 1/7. When we subtract these two numbers, the answer is (3-1)/7 = 2/7. Again, 2/7 is a rational number, demonstrating that the closure property holds true for subtraction of rational numbers.

**Commutative Property**

The commutative property states that the order of subtraction cannot be changed. In other words, we cannot subtract one rational number from another in a different order. For example, if we subtract 3/7 and 1/7, the answer is still 2/7. Similarly, if we subtract 1/7 and 3/7, the answer is also 2/7. This property highlights that the order of subtraction does not affect the result.

**Associative Property**

The associative property states that the order in which we perform subtractions can be changed without affecting the result. In other words, we can subtract three rational numbers and then subtract another one, or we can subtract two pairs of rational numbers simultaneously. For example, consider the rational numbers 9/14, 3/14, and 1/14. When we subtract these three numbers in a different order, we get a different result: (9-3-1)/14 = 5/14 vs (9-1) - (3-1)/14 = 8/14. As we can see, the associative property allows us to change the order of subtractions without affecting the final answer.

**Subtractive Identity**

The subtractive identity is a concept that does not exist in rational numbers. This means that there is no number that, when subtracted from another rational number, leaves the original number unchanged. For example, if we subtract 0 from 3/4, the result is still 3/4. However, if we subtract 0 - (3/4), the answer is -3/4, which is not equal to the original number. This highlights that the concept of subtractive identity does not apply to rational numbers.

**Multiplication of Rational Numbers**

When multiplying two rational numbers, we multiply the numerators and denominators separately. For example, consider the two rational numbers 3/7 and 5/6. When we multiply these numbers, we get (3*5)/(7*6) = 15/42. This demonstrates that when multiplying rational numbers, we can combine the products of the numerators and denominators to obtain a new rational number.

By understanding these properties and how to perform operations with rational numbers, we can build a solid foundation for working with these fundamental mathematical concepts.

"WEBVTTKind: captionsLanguage: ennext topic is subtraction of rational numbers that how can we subtract two rational numbers let us take an example subtract 2 pi 5 from 3 PI 4 that we have to subtract 2 by 5 from 3 PI 4 first of all we write 3 PI 4 minus 2 by 4 for this firstly we find the LCM of 4 & 5 the LCM of 4 & 5 is 20 now we find the it will add fraction of 3 PI 4 & 2 by 5 so equivalent fraction of 3 PI 4 is 15 by 20 remember that if we multiply 5 with 4 then we also have to multiply 5 with 3 that is equal to 15 by 20 and 2 by 5 multiplied by 4 by 4 that is equal to 8 by 20 here note that denominators and LCM are same after that we subtract these two numbers 15 by 20 minus 8 y 20 that is equal to 7 by 20 next question is what should be subtracted from 4 by 7 to get 3 PI for that which number we subtract from 4 by 7 that answer is 3 by 4 for example if I say which number I subtract around 5 to get answer for so answer is 1 because 5 minus 1 is equal to 4 similarly in this question that what should be we have to subtract from 4 by 7 to get 3 by 4 for this let x be rational number we take X be a rational number that they can be subtracted from 4 by 7 to get 3 by 4 so according to our question 4 by 7 minus X is equal to 3 by 4 so now take X on right hand side here this is X of negative and on that side it becomes positive so 3 by 4 plus X now we take 3 by 4 on left hand side and 3 by 4 becomes negative because here 3 by 4 is positive now we take the LCM of 7 & 4 that is 28 and 16-21 that is equal to X and answer is equal to X is equal to minus 5 upon 28 so the number is minus 5 upon 28 that should be subtracted from 4 by 7 and you get the answer 3 by 4 you can check your answer also by subtracting 4 this number from this if your answer is 3 by 4 it means value of x is correct next example is sum of two rational number is minus 5 by 3 if one number one of the number is minus 4 fine the other rational number in this question given is minus 5 by 3 is the sum of two rational number and one number is given that is minus 4 we have to find the other number for example sum of two number is not 1 number is 5 then find the second number 4 as you know 5 plus 4 is 9 so similarly in this way we have to find the other rational number so solution is let the other rational number B X so we take X be the other rational number so according to question minus four plus three plus X is equal to minus five by three so take minus four own right hand side that become positive because here for is of negative and on right-hand side it becomes positive and now we solve it take the album of three and one as you know as sum of three and one is three and minus 4 plus twelve do we find the equivalent fractions and the answer is 7 by 3 so 7 by 3 is the another number that should be added to minus 4 and answer becomes minus 5 by 3 next is properties of subtraction of rational number first one is closer closure property in this property the difference of two rational number is always a rational number let us stay to a rational number for example 3 by 7 and 1 by 7 these two are rational number when we subtract these two rational numbers the answer is 2 by 7 again 2 by 7 is also a rational number so the difference of two rational numbers of this rational number next is commutative property in subtraction the order of numbers cannot be changed means we can't subtract like 3 by 7 and 1 by 7 when we subtract these two number its answer is 2 by 7 but when we subtract 1 by 7 minus 3 by 7 the answer is - 2x7 so in this property order of numbers cannot be changed third one is associative property in this property the order of 3 - 3 rational numbers cannot be changed in subtraction we can't change the order of the rational numbers let us take the example the 3 rational numbers be 9 by 14 3 5 14 and 1 by 14 9 by 14 minus 3 by 14 minus 1 by 14 so firstly if we subtract these two numbers 9 by 14 - when we subtracted 3 by 14 minus 1 by 14 answer is 2 by 14 and when we subtract 9 by 14 MA and two by 14 answer is 7 by 40 now 9 by 14 minus 3 by 14 minus 1 by 14 in this firstly we subtracted these two numbers when we subtract these two number answer is 6 by 14 minus 1 by 14 the answer is 5 by 14 so as you see this answer is not equal to this answer so in this in associative property we can't change the order of the rational numbers so P upon k - r upon s minus t upon you this is not equal to P upon Q minus R upon s minus T upon you next is subtractive identity this identity does not exist in rational numbers for example if 3 by 4 is a rational number and we subtract a 0 from it answer is 3 by 4 but when we subtract 0 - 3x4 the answer is minus 3 by 4 so both are not equal so that's why this identity does not exist in rational numbers next we will discuss about multiplication of rational number let us take an example 3 by 7 and 5 by 6 we have to multiply these two rational numbers so in this the product of nominated upon product of denominator so answer is 3/5 say 15 6 AMSA 42 so product of two rational number is equal to product of nominator upon product of denominator remember that we can't multiply 3 & 6 and we can't multiply 5 with 7 so product of nominator upon product of denominatornext topic is subtraction of rational numbers that how can we subtract two rational numbers let us take an example subtract 2 pi 5 from 3 PI 4 that we have to subtract 2 by 5 from 3 PI 4 first of all we write 3 PI 4 minus 2 by 4 for this firstly we find the LCM of 4 & 5 the LCM of 4 & 5 is 20 now we find the it will add fraction of 3 PI 4 & 2 by 5 so equivalent fraction of 3 PI 4 is 15 by 20 remember that if we multiply 5 with 4 then we also have to multiply 5 with 3 that is equal to 15 by 20 and 2 by 5 multiplied by 4 by 4 that is equal to 8 by 20 here note that denominators and LCM are same after that we subtract these two numbers 15 by 20 minus 8 y 20 that is equal to 7 by 20 next question is what should be subtracted from 4 by 7 to get 3 PI for that which number we subtract from 4 by 7 that answer is 3 by 4 for example if I say which number I subtract around 5 to get answer for so answer is 1 because 5 minus 1 is equal to 4 similarly in this question that what should be we have to subtract from 4 by 7 to get 3 by 4 for this let x be rational number we take X be a rational number that they can be subtracted from 4 by 7 to get 3 by 4 so according to our question 4 by 7 minus X is equal to 3 by 4 so now take X on right hand side here this is X of negative and on that side it becomes positive so 3 by 4 plus X now we take 3 by 4 on left hand side and 3 by 4 becomes negative because here 3 by 4 is positive now we take the LCM of 7 & 4 that is 28 and 16-21 that is equal to X and answer is equal to X is equal to minus 5 upon 28 so the number is minus 5 upon 28 that should be subtracted from 4 by 7 and you get the answer 3 by 4 you can check your answer also by subtracting 4 this number from this if your answer is 3 by 4 it means value of x is correct next example is sum of two rational number is minus 5 by 3 if one number one of the number is minus 4 fine the other rational number in this question given is minus 5 by 3 is the sum of two rational number and one number is given that is minus 4 we have to find the other number for example sum of two number is not 1 number is 5 then find the second number 4 as you know 5 plus 4 is 9 so similarly in this way we have to find the other rational number so solution is let the other rational number B X so we take X be the other rational number so according to question minus four plus three plus X is equal to minus five by three so take minus four own right hand side that become positive because here for is of negative and on right-hand side it becomes positive and now we solve it take the album of three and one as you know as sum of three and one is three and minus 4 plus twelve do we find the equivalent fractions and the answer is 7 by 3 so 7 by 3 is the another number that should be added to minus 4 and answer becomes minus 5 by 3 next is properties of subtraction of rational number first one is closer closure property in this property the difference of two rational number is always a rational number let us stay to a rational number for example 3 by 7 and 1 by 7 these two are rational number when we subtract these two rational numbers the answer is 2 by 7 again 2 by 7 is also a rational number so the difference of two rational numbers of this rational number next is commutative property in subtraction the order of numbers cannot be changed means we can't subtract like 3 by 7 and 1 by 7 when we subtract these two number its answer is 2 by 7 but when we subtract 1 by 7 minus 3 by 7 the answer is - 2x7 so in this property order of numbers cannot be changed third one is associative property in this property the order of 3 - 3 rational numbers cannot be changed in subtraction we can't change the order of the rational numbers let us take the example the 3 rational numbers be 9 by 14 3 5 14 and 1 by 14 9 by 14 minus 3 by 14 minus 1 by 14 so firstly if we subtract these two numbers 9 by 14 - when we subtracted 3 by 14 minus 1 by 14 answer is 2 by 14 and when we subtract 9 by 14 MA and two by 14 answer is 7 by 40 now 9 by 14 minus 3 by 14 minus 1 by 14 in this firstly we subtracted these two numbers when we subtract these two number answer is 6 by 14 minus 1 by 14 the answer is 5 by 14 so as you see this answer is not equal to this answer so in this in associative property we can't change the order of the rational numbers so P upon k - r upon s minus t upon you this is not equal to P upon Q minus R upon s minus T upon you next is subtractive identity this identity does not exist in rational numbers for example if 3 by 4 is a rational number and we subtract a 0 from it answer is 3 by 4 but when we subtract 0 - 3x4 the answer is minus 3 by 4 so both are not equal so that's why this identity does not exist in rational numbers next we will discuss about multiplication of rational number let us take an example 3 by 7 and 5 by 6 we have to multiply these two rational numbers so in this the product of nominated upon product of denominator so answer is 3/5 say 15 6 AMSA 42 so product of two rational number is equal to product of nominator upon product of denominator remember that we can't multiply 3 & 6 and we can't multiply 5 with 7 so product of nominator upon product of denominator\n"