introduction students look at these numbers these are the whole numbers now look at these numbers these are the negative numbers but what do we call a bigger collection of numbers which contains whole numbers and negative numbers those numbers are called integers come let's learn more about integers objectives at the end of this lesson you will be able to define integers write the properties of integers and write the operations on integers students what do we get if we add two integers let us see Jean has six balls his friend Solon has five balls so how many balls are there in total yes eleven balls in total from this example we can say that addition of two integers is also an integer we can see that addition of integers gives integers so integers are closed under addition look at Rena she has five chocolates if she gives three chocolates to her brother then how many chocolates are left with Rena yes you're right she is left with two chocolates in this example we see that difference of two integers is also an integer hence integers are closed under subtraction in general for any two integers a and B a plus B is an integer similarly if a and B are two integers then a minus B is also an integer let's now move to its another property that is commutative property this is a 10 rupee note if we add one coin of 2 rupees in it we will get 12 rupees now take a coin of 2 rupee and add one note of 10 rupee in it again we will get 12 rupees or we can write it as 10 plus 2 is equal to 12 and 2 plus 10 equals 12 in both the cases we get the same answer this means the integers can be added in any manner as addition is commutative for integers in general for any two integers a and B we can say a plus B is equal to B plus a you when we add zero to any whole number we get the same whole number zero is an additive identity for whole numbers similarly if we add zero to any integer we get the same integer in general for any integer a a plus zero is equal to a which equals zero plus a we know that multiplication of all numbers is repeated addition for example 2 plus 2 plus 2 is equal to 3 into 2 which equals 6 similarly multiplication of integers is also repeated addition for example -5 plus -5 plus -5 is equal to 3 into minus 5 therefore 3 into minus 5 is equal to minus 15 let's now find the product of a positive and a negative integer firstly multiply 6 by 4 and then put minus sign before the product obtained we get minus 24 similarly 5 into minus 4 is equal to minus 5 into 4 which equals minus 20 the product of two negative integers is a positive integer we multiply the two negative integers as whole numbers in general for any two positive integers a and B minus a into minus B is equal to a into B look at these examples in the first example two negative integers are multiplied in the second example three negative integers are multiplied in the third example four negative integers are multiplied and in the fourth example five negative integers are multiplied from these products we observe that the product of two negative integers is a positive integer the product of three negative integers is a negative integer the product of four negative integers is a positive integer we can see that in first and third examples the number of negative integers that are multiplied are even 2 & 4 respectively and the product obtained in 1 & 3 are positive integers and the number of negative integers that are multiplied in 2 & 4 are odd and the products obtained in 2 & 4 are negative integers hence we can say that if the number of negative integers in a product is even then the product is a positive integer and if the number of negative integers in a product is odd then the product is a negative integer the product of two integers is again an integer for example minus 20 into minus five is equal to 100 minus 15 into 17 is equal to minus 255 so we can say that integers are closed under multiplication in general a into B is an integer for all integers a and B now observe this example if we can say that 3 into minus 4 is equal to minus 4 into 3 and therefore multiplication is commutative for integers in general for any two integers a and B a into B is equal to B into a the product of a negative integer and zero is zero for example minus 3 into 0 equals 0 0 into minus 4 equals 0 minus 5 into 0 is equal to 0 0 into minus 6 is equal to 0 in general for any integer a a into 0 is equal to 0 into a which equals 0 when we multiply 1 with any integer we get the same integer for example minus 3 into 1 is equal to minus 3 1 into 5 is equal to 5 that means 1 is the multiplicative identity for integers in general for any integer a we have a into 1 is equal to 1 into a which equals a but when we multiply any integer with minus 1 we get additive inverse of the integer in these examples minus 3 has become 3 minus 6 has become 6 13 has become minus 13 and minus 25 has become 25 the product of three integers does not depend upon the grouping of integers and this is called the associative property for multiplication of integers for any three integers a B and C a into B into C is equal to a into B into C for example 7 into minus 6 into minus 4 is equal to 7 into minus 6 into minus 4 now let us move towards the distributive property of multiplication for any integers a B and C a into B plus C is equal to a into B plus a into C for example minus 2 into 3 plus 5 is equal to minus 2 into 3 plus minus 2 into 8 similarly for any three integers a B and C a into B minus C is equal to a into B minus a into C for example 4 into 3 minus 8 is equal to 4 into 3 minus 4 into 8 if we know that division is the inverse operation of multiplication since 3 into 5 is equal to 15 so 15 divided by 5 is equal to 3 and 15 divided by 3 is equal to 5 when we divide a negative integer by a positive integer we divide them as whole numbers and then put a minus sign before the caution thus we get a negative integer in general for any two positive integers a and B a divided by minus B is equal to minus a divided by B where B is not equal to zero when we divide a negative integer by a negative integer we first divide them as whole numbers and then put a positive sign in general for any two integers a and B minus a divided by minus B is equal to a divided by B where B is not equal to zero division is not commutative for integers for example minus 8 divided by minus 4 is not equal to minus 4 divided by minus 8 any integer divided by 0 is meaningless and 0 divided by an integer other than 0 is equal to 0 any integer divided by 1 gives the same integer for example minus 8 divided by 1 is equal to minus 8 minus 11 divided by 1 is equal to minus 11 but if any integer is divided by minus 1 it does not give the same integer for example minus 8 divided by minus 1 is equal to 8 11 divided by minus 1 is equal to minus 11 summary let us summarize what we have learned integers are a bigger collection of numbers which is formed by whole numbers and their negatives if a and B are any integers then a plus B and a minus B are again integers if a and B are any integers then a plus B is equal to B plus a for all integers a and B if a and B are any integers then a plus B plus C is equal to a plus B plus C for all integers a B and C integers zero is the identity under addition that is a plus zero is equal to zero plus a which equals a for any integer a product of a positive and the negative integer is a negative integer whereas the product of two negative integers is a positive integer product of even number of negative integers is positive whereas the product of odd number of negative integers is negative if a and B are any integers then a into B is an integer if a and B are any integers then a into B is equal to B into a the integer one is the identity and a multiplication that is 1 into a is equal to a into 1 for any integer a if a B and C are any integers then a into B into C is equal to a into B into C if a b and c are any integers a into B plus C is equal to a into B plus a into seen for any three integers a B and C when a positive integer is divided by a negative integer the quotient obtained is a negative integer and vice-versa Division of a negative integer by another negative integer gives a positive integer as quotient for any integer a we have a divided by zero is not defined and a divided by one is equal to a

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Saale tujhe sikhana nahi aata to kyu sikha raha hai

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Main 3rd example samjha nahin (-4)×(-3)×(-2)×(-1)

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Friends you can also visit to this mentioned link for this topic:

https://www.youtube.com/watch?v=qoTYWJCHuQg&index=2&list=PUwkHptkqI1OoKGys6yNHeKw

You don't FUKING explained it all

sanjay kumar

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it was helpful

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Silly

please hindi mein samjhaiye ✋✋✋✋

Another time you will say means what I will do I don't know

Meru waste Rami sir best

commutative property is wrong

Good example

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No he has two balls

3:58

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