Properties of addition and subtraction of …

Integers have 5 main properties of operation which are:

• 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 : 3 – 4 = 3 + (−4) = −1;

  (–5) + 8 = 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 same. One can group numbers in any way but the answer will remain 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 : 1 + (2 + (-3)) = 0 = (1 + 2) + (−3);

• 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 + (−6) = −2 = (−6) + 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: −5 (2 + 1) = −15 = (−5 × 2) + (−5 × 1)

• 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 the number 1 it will give the integer itself as the product. Therefore, the integer 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