# prepare a chart properties exhibiting properties of integers closure property ,commutative property ,associative property ,distrib

prepare a chart properties exhibiting properties of integers closure property ,commutative property ,associative property ,distributive property with examples​

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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.

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 2: 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.

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 3: 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.

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 4: −5 (2 + 1) = −15 = (−5 × 2) + (−5 × 1)