Math
Make a book let on rules and properties of
integers with examples.
Answers
Properties of Integers
Integers have 5 main properties of operation which are:
- Closure Property
- Associative Property
- Commutative Property
- Distributive Property
- Identity Property
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,
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
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.
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