*Explain the following and give two examples of each*
#Closure property of integers
#Commutative property of integers
#Associative property of integers
#Distributive property of integers#additive identity
#multiplicative identity
Answers
Answer: Property 1: Closure Property
Among the various properties of integers, closure property under addition and subtraction states that the sum or difference of any two integers will always be an integer i.e. if x and y are any two integers, x + y and x − y will also be an integer.
Example 1: 3 – 4 = 3 + (−4) = −1;
(–5) + 8 = 3,
The results are integers.
Closure property under multiplication states that the product of any two integers will be an integer i.e. if x and y are any two integers, xy will also be an integer.
Example 2: 6 × 9 = 54 ; (–5) × (3) = −15, which are integers.
Division of integers doesn’t follow the closure property, i.e. the quotient of any two integers x and y, may or may not be an integer.
Property 2: Commutative Property
The commutative property of addition and multiplication states that the order of terms doesn’t matter, the result will be the same. Whether it is addition or multiplication, swapping of terms will not change the sum or product. Suppose, x and y are any two integers, then
⇒ x + y = y + x
⇒ x × y = y × x
Example 4: 4 + (−6) = −2 = (−6) + 4;
10 × (−3) = −30 = (−3) × 10
But, subtraction (x − y ≠ y − x) and division (x ÷ y ≠ y ÷ x) are not commutative for integers and whole numbers.
Example 5: 4 − (−6) = 10 ; (−6) – 4 = −10
⇒ 4 − (−6) ≠ (−6) – 4
Ex: 10 ÷ 2 = 5 ; 2 ÷ 10 = 1/5
⇒ 10 ÷ 2 ≠ 2 ÷ 10
Property 3: Associative Property
The associative property of addition and multiplication states that the way of grouping of numbers doesn’t matter; the result will be the same. One can group numbers in any way but the answer will remain the same. Parenthesis can be done, irrespective of the order of terms. Let x, y and z be any three integers, then
⇒ x + (y + z) = (x + y) +z
⇒ x × (y × z) = (x × y) × z
Example 6: 1 + (2 + (-3)) = 0 = (1 + 2) + (−3);
1 × (2 × (−3)) =−6 = (1 × 2) × (−3)
Subtraction of integers is not associative in nature i.e. x − (y − z) ≠ (x − y) − z.
Example 7: 1 − (2 − (−3)) = −4; (1 – 2) – (−3) = −2
1 – (2 – (−3)) ≠ (1 − 2) − (−3)
Property 4: Distributive Property
The distributive property explains the distributing ability of operation over another mathematical operation within a bracket. It can be either distributive property of multiplication over addition or distributive property of multiplication over subtraction. Here, integers are added or subtracted first and then multiplied or multiply first with each number within the bracket and then added or subtracted. This can be represented for any integers x, y and z as:
⇒ x × (y + z) = x × y + x × z
⇒ x × (y − z) = x × y − x × z
Example 8: −5 (2 + 1) = −15 = (−5 × 2) + (−5 × 1)
Property 5: Identity Property
Among the various properties of integers, additive identity property states that when any integer is added to zero it will give the same number. Zero is called additive identity. For any integer x,
x + 0 = x = 0 + x
The multiplicative identity property for integers says that whenever a number is multiplied by 1 it will give the integer itself as the product. Therefore, 1 is called the multiplicative identity for a number. For any integer x,
x × 1 = x = 1 × x
If any integer multiplied by 0, the product will be zero:
x × 0 = 0 =0 × x
If any integer multiplied by -1, the product will be opposite of the number:
x × (−1) = −x = (−1) × x
Step-by-step explanation:
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Required Answer :
Closure Property :
All integers are closed under closure property for addition and subtraction. This means that whenever two integers are added or subtracted, the result is an integer itself.
Eg. (-1) + 2 = 1 which is an integer.
56 - 57 = -1 which is an integer.
Commutative property :
According to the commutative property of integers, two integers can be added or subtracted in whichever way possible. The result will always be the same.
Eg. 67 + (-8) = 59 = (-8) + 67
65 x 10 = 650 = 10 x 65
Associative property :
According to the associative property, three integers can be added or multiplied in whichever way possible. The sum/product will be the same.
Eg. (2 + 6) + 3 = 2 + (6 + 3) = 11
(5 x 2) x 3 = 5 x (2 x 3) = 30
Distributive property :
For three integers x, y and z,
x (y + z) = xy + yz and x (y - z) = xy - yz
Additive Identity:
Any number added to 0 is the number itself. Therefore 1 is the additive identity for all integers.
Eg. 1 + 0 = 1
5 + 0 - 5
Multiplicative identity :
Any number multiplied to 1 is the number itself. This means that 1 is the multiplicative identity for all integers.
6 x 1 = 6
90 x 1 = 90
Extra Bytes :
Integers is the collection of all positive and negative numbers along with zero. Thee collection is denoted by the letter Z.
Note:
Integers never contain fractions.