Properties of Integers: A Comprehensive Guide

Properties of integers are mathematical concepts that describe the behavior and relationships between integers under various operations such as addition, subtraction, multiplication, and division. In this article, we will explore some of these properties in detail.

Distributive Property Over Addition

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The distributive property over addition states that for any two integers a and b, and c being an integer, a + (b x c) = (a + b) x (a + c). This means that the distribution of multiplication over addition holds true. For example, let's consider the equation -5 into 6 plus -3. We can solve this equation in two ways:

Firstly, we multiply -5 by 6 and then add -3 to the result. So, (-5) x 6 = -30, and now adding -3 to -30 gives us a final answer of -15.

Alternatively, we can first add -5 and -3 to get -8, and then multiply -8 by 6. However, since -8 is not an integer, this method does not give us the correct result. Therefore, we observe that by solving this equation in two different ways, the answer remains the same.

Additive Inverse Property

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The additive inverse property states that for any integer a, there exists an integer -a such that a + (-a) = 0. This means that when you add a number to its negative, the result is always zero. For example, let's consider the equation -5 into 6 plus -3. We can also represent -3 as 3, since -3 and 3 are additive inverses.

Distributive Property Over Multiplication

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The distributive property over multiplication states that for any two integers a and b, and c being an integer, (a x b) + c = (a + c) x (b + c). This means that the distribution of addition over multiplication holds true. For example, let's consider the equation -5 into 6 plus minus 3. We can solve this equation in two ways:

Firstly, we multiply -5 by 6 and then add -3 to the result. So, (-5) x 6 = -30, and now adding -3 to -30 gives us a final answer of -15.

Alternatively, we can first multiply -5 by minus 3, which gives us a result of 15. Now, multiplying this result by 6 gives us a final answer of -90, not -15. Therefore, we observe that by solving this equation in two different ways, the answer remains the same.

Closure Property

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The closure property states that for any integer a, there exists an integer b such that (a x b) is also an integer. However, this property does not hold true in all cases. For example, when we divide 8 by 9, the result is not an integer. Therefore, integers are not always closed under division.

Commutative Property of Division

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The commutative property of division states that for any two integers a and b, (a ÷ b) = (b ÷ a). However, this property does not hold true in all cases. For example, when we divide -12 by 1, the result is -12, but when we divide 1 by -12, the result is -1/12, not -12.

Associative Property of Division

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The associative property states that for any three integers a, b, and c, (a ÷ (b ÷ c)) = ((a ÷ b) ÷ c). However, this property does not hold true in all cases. For example, when we divide 64 by 8 divided by 2, the result is different from when we group it differently.

Division by 1

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When any integer is divided by 1, the quotient is always the same number itself. For example, if we divide 275 by 275, the result is 1. Therefore, a ÷ 1 is always equal to a.

Division by Itself

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If we divide any integer by itself, the quotient is always 1. For example, when we divide 275 by 275, the result is 1. Therefore, a ÷ a is always equal to 1.

Division by Zero

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When any integer is divided by zero, it is undefined and meaningless. This is because division by zero does not have a meaningful result. For example, if we try to divide 35 by 0, the result is undefined. Therefore, dividing by zero is never allowed in mathematics.

Division of 0 by Any Number

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When 0 is divided by any integer, the quotient is always 0. For example, when we divide 0 by 7, the result is 0. This property holds true for all integers other than 0 itself.

"WEBVTTKind: captionsLanguage: engood morning students in the second part of the video we will discuss properties of operations on integer so let us start we have the properties closure property commutative property associative property distributive property additive inverse and identity so first of all let us discuss closure property so if a and b are the two integers then their sum is also an integer so here we have example as you see 14 plus 30 14 is an integer and 13 30 is also an integer so when we add its answer is 44 so 44 is also an integer so when we add two integers its answer is also integer next is 14 plus minus 5 so as you see 14 and minus 5 both are integers so when we add its answer is 9 so again we get an integer so remember that in closure property if a and b are the two integers then their answer will always be an integer so integers are closed under addition okay so next is commutative property this property states that changing the order of the addends does not affect the sum means in this property let us take an example as you see 6 and 4 when we add 6 and 4 its answer is always 10 okay and if we add 4 and 6 again its answer is 10 so what will we observe we observe that in both the cases the answer remain same so we observe that if a plus b is always equal to b plus a okay so this property states that changing the order of the attendance does not affect the sum whether you can write b here or here answer remain same in the both cases so addition of integers is commutative okay next we have one more example minus 6 plus 4 when we add its answer is minus 2 and 4 plus minus 6 its answer is again minus 2 so in the both cases answer remains same so addition of integers is commutative next is associative property this property states that changing the grouping of numbers being added thus does not change its value for example we have 6 plus 2 plus 1 which is equal to 6 plus 2 plus 1 so first of all we can add 6 plus 2 so when we add 6 plus 2 its answer is 8 next is plus 1 so 8 plus 1 9 now in this case firstly we add 2 plus 1 6 plus 2 plus 1 answer is 3 so 6 plus 3 answer is 9 so what will you observe in the both the cases the answer remain same so whether you can add firstly two numbers or you can add last two numbers in the both cases answer remain same so this is known as the associative property okay so next we have one more example 6 plus minus 2 plus 3 so first of all we can add these two numbers when we add its answer is 4 plus 3 its answer is 7 okay now we solve this firstly we add these 2 minus 2 plus 3 its answer is 1 so 6 plus 1 answer is 7 so what will we observe we will observe that in the both cases answer remain same so while adding integers when the grouping is changed the result does not change so this property is called associative property of addition next is distributive property this property states that when two numbers have been added or subtracted and then multiplied by a factor the result will be same when each number is multiplied the factor and the product are then added or subtracted so what does it means like we have one example here is minus 2 into 4 plus 3 so firstly add them 4 plus 3 its answer is 7 minus 2 into 4 7 answer is minus 14 now solve this side firstly we multiply minus 2 into 4 answer is minus 8 plus minus 2 into 3 answer is minus 6 now minus 8 in plus minus 6 what is its answer its answer is minus 14 so what will we observe we observe that when we add two numbers and then multiply a factor the result will be same when each number is multiplied by the factor and the products are then added okay next is identity property for addition this property states that the sum of any numbers and 0 is the number itself okay so we have one we have an example 5 plus 0 we add 0 to number 5 so its answer is 5 and when we add 0 to minus 4 its answer is again minus 4 so always remember that the sum of any number and 0 the answer remain the number itself so adding 0 to any integer does not change the value or identity of the integer so a plus 0 is equal to a next is inverse property for addition so this property states that the sum of any number and its additive inverse is 0 so like minus 6 plus 6 when we add its answer is 0 and 10 plus minus 10 its answer is 0 so remember that for every integer there exists another integer such that its sum is 0 which is the additive identity okay so as you see the example in this example minus 6 is called the opposite or additive inverse of 6 okay and similarly six is called the opposite or additive inverse of minus six so remember that there exists another integer for every integer a that is minus a such that if we add them its answer is 0 okay which is the additive identity so minus a is called the opposite or additive inverse of a and vice versa next we have some properties of subtraction let us discuss the first property of subtraction that is closure property so when we subtract two integers we will get an integer let us take an example so minus five and two as we see that minus 5 and 2 both are integers so we subtract these two integers the answer is minus 7 so as you see 7 minus 7 is also an integer so always remember that when we subtract two integers we will always get an integer so integers are closed under subtraction the next property is commutative property so in subtraction commutative property not is not true okay while subtracting two integers the order of subtraction alters the result for example six minus minus three as you see six and minus 3 both are integers when we subtract the integer answer is 9 okay and minus 3 minus 6 when we subtract these two integers answer is minus sign so what will you observe you will observe that both the answers are not same so a minus b is not equal to b minus a so subtraction is not commutative for integers next is associative property so in associative property first of all we will discuss an example so firstly here subtract 8 and 3 when we subtract 3 from 8 answer is 5 minus 2 so 5 minus 2 answer is 3 okay now firstly subtract 3 and 2 when we subtract 2 from 3 its answer is 1 so 8 minus 1 answer is 7 so what will you observe you will observe that in both the cases answers are different so what it means it means subtraction is not associated so in subtraction a change in order changes the answer so integers are not associative under addition so a minus b minus c is not equal to a minus b minus c clear next is property of 0 for subtraction when we subtract 0 from any integer we get the same number for example 3 minus 0 its answer is 3 minus 3 minus 0 its answer is again minus 3 so what will you observe in the both cases when we subtract 0 we get the same answer okay so a minus 0 is equal to a so always remember that property of 0 is when 0 is subtracted from an integer we get the same integer we have an example we subtract 0 from 7 its answer remains 7. next is we have some properties of multiplication first property is closure property so let us take an example minus 3 multiplied by minus 2 when we multiply these two integer answer is 6 that is again an integer so if a and b are the two integers then a into b is also an integer so this is called closure property of multiplication of integers that when we multiply two integers its answer is also an integer next is property 2 firstly let us take an example 8 x 4 when we multiply 8 ma into minus 4 its answer is minus 32 and when we multiply minus 4 into it its answer is again minus 32 so if a and b are any two integers then a into b is equal to b into a so this is called the commutative property of multiplication of integers that grouping in multiplication does not matter so multiplication is associative for integers next property number third so let us take an example first of all multiply minus 10 and 9 when we multiply minus 10 with 9 answer is minus 90 now multiplying minus 90 with minus 8 we get 720 okay now firstly we multiply minus 8 and minus 10 then we multiply the answer with 9 so when we multiply minus 8 by 10 minus 10 its answer is 80 and now multiply 80 by 9 its answer is 720 so what will you observe in the both cases you will observe that in the both cases answer remain same it means we can say that the grouping in multiplication does not matter matter whether you can multiply first two numbers then multiply the answer with the third number or firstly you can multiply second and third number and the answer that we get we can multiply that answer with first number in the both cases answer remain same so if a b and c are the three integers then a into b into c is always equal to a into b into c so associative property of multiplication of integers hold true next is property 4 as you see here is an example multiply 1 with minus 5 so when we multiply 1 by minus 5 what will we get we get the same answer that is minus 5 so if any integer a is multiplied by 1 the product is the same integer a that is a into 1 is equal to 1 into a is always equal to a so integers do not lose their identity on being multiplied by one so one is called the multiplicative identity of integers next is property 5 when we multiply -4 by 0 its answer is 0 so when an integer is multiplied by 0 the product is always 0 if a is any integer then a into 0 is equal to 0 into a is equal to zero next is property number six when an integer is multiplied by minus one we get the additive inverse of that integer for example 5 into minus 1 its answer is minus 5 so which is the additive inverse of 5 so if a is an integer then a multiplied by minus 1 is equal to minus 1 multiplied by a is equal to minus a where minus a is the additive inverse of a next is property 7 that is distributive property over addition so here we have two methods to solve these this equation that is minus 5 into 6 plus minus 3 we can solve this question in two ways first one is firstly multiply firstly add these add these two numbers 6 plus minus 3 its answer is 3 now add 3 with minus 5 its answer is minus 15 the second way to solve this question firstly we can multiply minus 5 with 6 its answer is minus 30 now multiply minus 5 into minus 3 its answer is 15 now add minus 30 and 15 its answer is minus 15 so what will you observe we observe by solving this question in this in the two different ways the answer remain same so if a b and c are any three integers then a into b plus c is always equal to a into b plus a into c this is called the distributive property of multiplication of integers over addition next we have some properties of division of integer so first one is closure property so division is closed for any integer for example if we divide the first integer by the second integer we get again our answer an integer but in sometimes in the some cases this property does not hold true for example let us take an example 8 divided by 9 so as you see that 8 is an integer and 9 is also an integer when we divide its answer is 8 by 9 so as you see 8 by 9 is not an integer so always remember that integers are not always closed under division so one more example as you see 25 divided by 5 when we divide 25 by 5 its answer is minus 5 so 25 and 5 both are integers when we divide its answer is minus 5 that is also an integer but in this case 8 by nine is not an integer so integers are not closed under division next is commutative property in case of subtraction the order in which division is carry out is important a change in the order results in a change in the answer for example when we divide minus 12 by 1 its answer is minus 12 and when we divide minus 4 by 1 its answer is minus 4 so as you see when you divide minus 12 by 1 its answer is minus 12 and when we divide 1 by minus 12 its answer is minus 1 by 12 so in both the cases what will you observe that both the answers are not same so if a and b are the two integers then a divided by b is not equal to b divided by a so always remember that integers do not possess the commutative property for division next is associative property so order is important in division always remember that associative property does not apply to the division of the integers so in division of integers the result changes when grouping is changed so this property does not hold true for division let us take an example 64 divided by 8 divided by 2 so firstly divide 64 by 8 its answer is 8 now divide it by 2 its answer is 4 now 64 divided by 8 divided by 2 so firstly divide 8 by 2 its answer is 4 now divide 64 by 4 its answer is 16 so what will you observe you will observe in the both cases the answers does not seem so always remember that the result changes when the grouping is changed in region so a divided by b divided by c is not equal to a divided by b divided by c next is we have some another properties of division division by 1 so if we divide a number by 1 the quotient is the number itself for example we divide 1 7542 by 1. its answer remain same 7542 so when any an integer is divided by one it always give the same integer so a divided by 1 is always equal to a next is division by itself if we divide a number by the number itself the quotient is 1 for example if we divide 275 by 275 its answer is 1 so a divided by a is always equal to 1 next is division any number by 0 so division of any number by 0 is meaningless if we divide any number by 0 it is meaningless so here we have 35 divided by 0 which is equal to no meaning so it is a meaningless okay so next is division of 0 by any number property so 0 divided by a number gives a 0 as a quotient so when we divide 0 by any integer its answer is always 0 for example 0 divided by 7 again its answer is 0 so if a is any integer other than 0 then 0 divided by a answer remains 0 so thank yougood morning students in the second part of the video we will discuss properties of operations on integer so let us start we have the properties closure property commutative property associative property distributive property additive inverse and identity so first of all let us discuss closure property so if a and b are the two integers then their sum is also an integer so here we have example as you see 14 plus 30 14 is an integer and 13 30 is also an integer so when we add its answer is 44 so 44 is also an integer so when we add two integers its answer is also integer next is 14 plus minus 5 so as you see 14 and minus 5 both are integers so when we add its answer is 9 so again we get an integer so remember that in closure property if a and b are the two integers then their answer will always be an integer so integers are closed under addition okay so next is commutative property this property states that changing the order of the addends does not affect the sum means in this property let us take an example as you see 6 and 4 when we add 6 and 4 its answer is always 10 okay and if we add 4 and 6 again its answer is 10 so what will we observe we observe that in both the cases the answer remain same so we observe that if a plus b is always equal to b plus a okay so this property states that changing the order of the attendance does not affect the sum whether you can write b here or here answer remain same in the both cases so addition of integers is commutative okay next we have one more example minus 6 plus 4 when we add its answer is minus 2 and 4 plus minus 6 its answer is again minus 2 so in the both cases answer remains same so addition of integers is commutative next is associative property this property states that changing the grouping of numbers being added thus does not change its value for example we have 6 plus 2 plus 1 which is equal to 6 plus 2 plus 1 so first of all we can add 6 plus 2 so when we add 6 plus 2 its answer is 8 next is plus 1 so 8 plus 1 9 now in this case firstly we add 2 plus 1 6 plus 2 plus 1 answer is 3 so 6 plus 3 answer is 9 so what will you observe in the both the cases the answer remain same so whether you can add firstly two numbers or you can add last two numbers in the both cases answer remain same so this is known as the associative property okay so next we have one more example 6 plus minus 2 plus 3 so first of all we can add these two numbers when we add its answer is 4 plus 3 its answer is 7 okay now we solve this firstly we add these 2 minus 2 plus 3 its answer is 1 so 6 plus 1 answer is 7 so what will we observe we will observe that in the both cases answer remain same so while adding integers when the grouping is changed the result does not change so this property is called associative property of addition next is distributive property this property states that when two numbers have been added or subtracted and then multiplied by a factor the result will be same when each number is multiplied the factor and the product are then added or subtracted so what does it means like we have one example here is minus 2 into 4 plus 3 so firstly add them 4 plus 3 its answer is 7 minus 2 into 4 7 answer is minus 14 now solve this side firstly we multiply minus 2 into 4 answer is minus 8 plus minus 2 into 3 answer is minus 6 now minus 8 in plus minus 6 what is its answer its answer is minus 14 so what will we observe we observe that when we add two numbers and then multiply a factor the result will be same when each number is multiplied by the factor and the products are then added okay next is identity property for addition this property states that the sum of any numbers and 0 is the number itself okay so we have one we have an example 5 plus 0 we add 0 to number 5 so its answer is 5 and when we add 0 to minus 4 its answer is again minus 4 so always remember that the sum of any number and 0 the answer remain the number itself so adding 0 to any integer does not change the value or identity of the integer so a plus 0 is equal to a next is inverse property for addition so this property states that the sum of any number and its additive inverse is 0 so like minus 6 plus 6 when we add its answer is 0 and 10 plus minus 10 its answer is 0 so remember that for every integer there exists another integer such that its sum is 0 which is the additive identity okay so as you see the example in this example minus 6 is called the opposite or additive inverse of 6 okay and similarly six is called the opposite or additive inverse of minus six so remember that there exists another integer for every integer a that is minus a such that if we add them its answer is 0 okay which is the additive identity so minus a is called the opposite or additive inverse of a and vice versa next we have some properties of subtraction let us discuss the first property of subtraction that is closure property so when we subtract two integers we will get an integer let us take an example so minus five and two as we see that minus 5 and 2 both are integers so we subtract these two integers the answer is minus 7 so as you see 7 minus 7 is also an integer so always remember that when we subtract two integers we will always get an integer so integers are closed under subtraction the next property is commutative property so in subtraction commutative property not is not true okay while subtracting two integers the order of subtraction alters the result for example six minus minus three as you see six and minus 3 both are integers when we subtract the integer answer is 9 okay and minus 3 minus 6 when we subtract these two integers answer is minus sign so what will you observe you will observe that both the answers are not same so a minus b is not equal to b minus a so subtraction is not commutative for integers next is associative property so in associative property first of all we will discuss an example so firstly here subtract 8 and 3 when we subtract 3 from 8 answer is 5 minus 2 so 5 minus 2 answer is 3 okay now firstly subtract 3 and 2 when we subtract 2 from 3 its answer is 1 so 8 minus 1 answer is 7 so what will you observe you will observe that in both the cases answers are different so what it means it means subtraction is not associated so in subtraction a change in order changes the answer so integers are not associative under addition so a minus b minus c is not equal to a minus b minus c clear next is property of 0 for subtraction when we subtract 0 from any integer we get the same number for example 3 minus 0 its answer is 3 minus 3 minus 0 its answer is again minus 3 so what will you observe in the both cases when we subtract 0 we get the same answer okay so a minus 0 is equal to a so always remember that property of 0 is when 0 is subtracted from an integer we get the same integer we have an example we subtract 0 from 7 its answer remains 7. next is we have some properties of multiplication first property is closure property so let us take an example minus 3 multiplied by minus 2 when we multiply these two integer answer is 6 that is again an integer so if a and b are the two integers then a into b is also an integer so this is called closure property of multiplication of integers that when we multiply two integers its answer is also an integer next is property 2 firstly let us take an example 8 x 4 when we multiply 8 ma into minus 4 its answer is minus 32 and when we multiply minus 4 into it its answer is again minus 32 so if a and b are any two integers then a into b is equal to b into a so this is called the commutative property of multiplication of integers that grouping in multiplication does not matter so multiplication is associative for integers next property number third so let us take an example first of all multiply minus 10 and 9 when we multiply minus 10 with 9 answer is minus 90 now multiplying minus 90 with minus 8 we get 720 okay now firstly we multiply minus 8 and minus 10 then we multiply the answer with 9 so when we multiply minus 8 by 10 minus 10 its answer is 80 and now multiply 80 by 9 its answer is 720 so what will you observe in the both cases you will observe that in the both cases answer remain same it means we can say that the grouping in multiplication does not matter matter whether you can multiply first two numbers then multiply the answer with the third number or firstly you can multiply second and third number and the answer that we get we can multiply that answer with first number in the both cases answer remain same so if a b and c are the three integers then a into b into c is always equal to a into b into c so associative property of multiplication of integers hold true next is property 4 as you see here is an example multiply 1 with minus 5 so when we multiply 1 by minus 5 what will we get we get the same answer that is minus 5 so if any integer a is multiplied by 1 the product is the same integer a that is a into 1 is equal to 1 into a is always equal to a so integers do not lose their identity on being multiplied by one so one is called the multiplicative identity of integers next is property 5 when we multiply -4 by 0 its answer is 0 so when an integer is multiplied by 0 the product is always 0 if a is any integer then a into 0 is equal to 0 into a is equal to zero next is property number six when an integer is multiplied by minus one we get the additive inverse of that integer for example 5 into minus 1 its answer is minus 5 so which is the additive inverse of 5 so if a is an integer then a multiplied by minus 1 is equal to minus 1 multiplied by a is equal to minus a where minus a is the additive inverse of a next is property 7 that is distributive property over addition so here we have two methods to solve these this equation that is minus 5 into 6 plus minus 3 we can solve this question in two ways first one is firstly multiply firstly add these add these two numbers 6 plus minus 3 its answer is 3 now add 3 with minus 5 its answer is minus 15 the second way to solve this question firstly we can multiply minus 5 with 6 its answer is minus 30 now multiply minus 5 into minus 3 its answer is 15 now add minus 30 and 15 its answer is minus 15 so what will you observe we observe by solving this question in this in the two different ways the answer remain same so if a b and c are any three integers then a into b plus c is always equal to a into b plus a into c this is called the distributive property of multiplication of integers over addition next we have some properties of division of integer so first one is closure property so division is closed for any integer for example if we divide the first integer by the second integer we get again our answer an integer but in sometimes in the some cases this property does not hold true for example let us take an example 8 divided by 9 so as you see that 8 is an integer and 9 is also an integer when we divide its answer is 8 by 9 so as you see 8 by 9 is not an integer so always remember that integers are not always closed under division so one more example as you see 25 divided by 5 when we divide 25 by 5 its answer is minus 5 so 25 and 5 both are integers when we divide its answer is minus 5 that is also an integer but in this case 8 by nine is not an integer so integers are not closed under division next is commutative property in case of subtraction the order in which division is carry out is important a change in the order results in a change in the answer for example when we divide minus 12 by 1 its answer is minus 12 and when we divide minus 4 by 1 its answer is minus 4 so as you see when you divide minus 12 by 1 its answer is minus 12 and when we divide 1 by minus 12 its answer is minus 1 by 12 so in both the cases what will you observe that both the answers are not same so if a and b are the two integers then a divided by b is not equal to b divided by a so always remember that integers do not possess the commutative property for division next is associative property so order is important in division always remember that associative property does not apply to the division of the integers so in division of integers the result changes when grouping is changed so this property does not hold true for division let us take an example 64 divided by 8 divided by 2 so firstly divide 64 by 8 its answer is 8 now divide it by 2 its answer is 4 now 64 divided by 8 divided by 2 so firstly divide 8 by 2 its answer is 4 now divide 64 by 4 its answer is 16 so what will you observe you will observe in the both cases the answers does not seem so always remember that the result changes when the grouping is changed in region so a divided by b divided by c is not equal to a divided by b divided by c next is we have some another properties of division division by 1 so if we divide a number by 1 the quotient is the number itself for example we divide 1 7542 by 1. its answer remain same 7542 so when any an integer is divided by one it always give the same integer so a divided by 1 is always equal to a next is division by itself if we divide a number by the number itself the quotient is 1 for example if we divide 275 by 275 its answer is 1 so a divided by a is always equal to 1 next is division any number by 0 so division of any number by 0 is meaningless if we divide any number by 0 it is meaningless so here we have 35 divided by 0 which is equal to no meaning so it is a meaningless okay so next is division of 0 by any number property so 0 divided by a number gives a 0 as a quotient so when we divide 0 by any integer its answer is always 0 for example 0 divided by 7 again its answer is 0 so if a is any integer other than 0 then 0 divided by a answer remains 0 so thank you\n"