Property of closure , Commutative , Associative , Distributive , Identity
Answers
Answer: Closure Property under Addition of Integers
If we add any two integers, the result obtained on adding the two integers, is always an integer. So we can say, that integers are closed under addition.
Let us say ‘a’ and ‘b’ are two integers, either positive or negative. When we add the two integers, their result would always be an integer, i.e (a + b) would always be an integer.
Example –
State whether (– 11) + 2 is closed under addition
Solution
– 11 + 2
– 9
Since both -11 and 2 are integers, and their sum, i.e (-9) is also an integer, we can say that integers are closed under addition.
Closure Property under Subtraction of Integers
If we subtract any two integers the result is always an integer, so we can say that integers are closed under subtraction.
Let us say ‘a’ and ‘b’ are two integers either positive or negative, their result should always be an integer, i.e (a + b) would always be an integer.
Example –
State whether (24 – 12) is closed under subtraction
Solution –
24 – 12
12
Since both 24 and -12 are integers, and their difference, i.e (12) is also an integer, we can say that integers are closed under subtraction.
Closure Property under Multiplication of Integers:
If we multiply any two integers the result is always an integer, so we can say that integers are closed under multiplication.
Let us say ‘a’ and ‘b’ are two integers either positive or negative, and if multiply it, their result should always be an integer, i.e [(-a) x b] and [a x (–b)] would always be an integer.
This means the two integers do not follow commutative property under subtraction.
Commutative Property under Division of Integers:
Commutative property will not hold true for division of whole number say (12 ÷ 6) is not equal to (6 ÷ 12). Let us consider for integers say, (-14) and (7), the division of two numbers are not always same.
[(-14) ÷ 7 = -2] and [7 ÷ (-14) = -0.5}, so the result of division of two integers are not equal so we can say that commutative property will not hold for division of integers.
In generalise form for any two integers ‘a’ and ‘b’
Associative Property of Integers
Associative Property under Addition of Integers:
As commutative property hold for addition similarly associative property also holds for addition.
In generalize form for any three integers say ‘a’, ’b’ and ‘c’
a + (b + c) = (a + b) + c
Associative Property under Subtraction of Integers:
On contradictory, as commutative property does not hold for subtraction similarly associative property also does not hold for subtraction of integers.
Associative Property under Multiplication of Integers:
As commutative property hold true for multiplication similarly associative property also holds true for multiplication.
The associative property of multiplication does not depend on the grouping of the integers.
In generalize form for any three integers say ‘a’, ’b’ and ‘c’
a x (b x c) = (a x b) x c
Example –
Distributive Property of Integers:
Distributive properties of multiplication of integers are divided into two categories, over addition and over subtraction.
1. Distributivity of multiplication over addition hold true for all integers.
In generalize form for any three integers say ‘a’, ’b’ and ‘c’
a x (b + c) = (a x b) + (a x c)
2. Distributivity of multiplication over subtraction hold true for all integers.
In generalize form for any three integers say ‘a’, ’b’ and ‘c’
a x (b – c) = (a x b) – (a x c)
Additive Identity:
When we add zero to any whole number we get the same number, so zero is additive identity for whole numbers. Similarly if we add zero to any integer we get the back the same integer whether the integer is positive or negative. In general for any integer ‘a’
pls mark me as the brainliest
Answer:
Hi,,.. Here your answer.
Hope this helps. Thank. you. Bye.
