There are a few properties of integers which determine its operations. These principles or properties help us to solve many equations. To recall, integers are any positive or negative numbers, including zero. Properties of these integers will help to simplify and answer a series of operations on integers quickly.

All properties and identities for addition, subtraction, multiplication and division of numbers are also applicable to all the integers. Integers include the set of positive numbers, zero and negative numbers which are denoted with the letter Z.

Properties of Integers

Integers have 5 main properties of operation which are:

• Closure Property
• Associative Property
• Commutative Property
• Distributive Property
• Identity Property

The explanation of each of the integer properties is given below.

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.

Example 3: (−3) ÷ (−6) = ½, is not 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

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