Table of Contents

**What is the definition of integers?**

The set of all whole numbers, including negative numbers, is defined as integers. Integers are whole numbers and cannot be a per cent, decimal, or fraction.

Integers have a few qualities that determine how they are used. Any positive or negative number, including 0, is an integer. The properties of the integers assist in simplifying and answering a sequence of integer operations. Many equations can be solved using these ideas or attributes.

All integers have the same features and identities as the addition, subtraction, multiplication, and division of numbers. The set of positive, zero, and negative numbers indicated by the letter Z is called integers.

Z = {………-3, -2, -1, 0, 1, 2, 3, ……………}

**What are the Different Properties of Integers?**

The different properties relating to integers are five, namely associative, commutative, distributive, closure, and identity property. All the properties of integers have been given below in detail –

**1. Closure property:**

The closure property of 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 both integers, x y and x- y will both be integers.

Example: 8-3 =5

8 (-3) =5

The closure property of multiplication of integers states that the product of any two integers is also an integer.

i.e. if x and y are both integers, then their product xy is also an integer.

Example: If 6 and 4 are two integers, their product 6×4 =24 is also an integer.

Similarly, if -7 and 2 are integers, then product -7 x 2 =-14 is also an integer.

Division of integers does not hold for the closure property i.e. if x and y are two integers, then their quotient x /y may or may not be an integer.

Example: -4 /8 = -1/2 which is not an integer.

**2. Commutative property:**

The commutative property of addition and multiplication states that the order of terms does not affect the final result. The sum or product will not change if the terms are swapped, whether addition or multiplication. Assume that x and y are any two integers, then x+y =y+x and xy=yx.

Example: If 4 and 5 are two integers, then 4+5=5+4=9, .i.e, whether we add 4 to 5 or 5 to 4 result remains the same, i.e. 9.

Also, if 2 and -8 are two integers, then 2x(-8)=(-8)x2=-16, i.e. whether we multiply 2 with-8 or -8 with 2, the result remains the same.

The commutative property does not hold for subtraction and division.

**3. Associative property:**

The term “associative property” refers to the ability to group things. For addition and multiplication, associative property rules can be used. The result remains constant if the associative property for addition and multiplication is used regardless of the sequence in which they are grouped.

- Associative Property for Addition states that if

(x+y)+z = x+(y+z)

For example: (4+6)+5 = 4+(6+5) the answer for both the possibilities will be 15.

- Associative Property for Multiplication states that if

(rxs) x t = r x (sxt)

For example ( 5×2) x 6= 5 x ( 2×6), the answer for both the possibilities will be 60.

Thus we can apply the associative rule for addition and multiplication, but it does not hold for subtraction and division.

**4. Distributive property :**

To distribute, means to divide something given equally.

Distributive property means dividing the given operations on the numbers so that the equation becomes easier to solve. It states that “multiplication is distributed over addition.”

For example, take the equation l(m+n)

When applying distributive property, we have to multiply l with both m and n and add, i.e., l x m + l x n = l m+l n.

Example: 5 (2+8 )=5×10=50

By applying distributive property we can solve it as

5 (2+8 ) = 5 x 2 + 5 x 8 =10+40 =50

**5. Identity Property:**

When zero is added to a number, the result is the same. Zero is known as Additive identity.

If the integer k is any positive number, then

k 0 is the same as k.

Example: 5 0=5

When a number is multiplied by the number 1, the outcome is the integer itself, according to the multiplicative identity characteristic for integers. As a result, one is known as a number’s multiplicative identity.

If the integer k is any positive number, then

kx1=k=1xk

Example:2×1=2=1×2

The result of multiplying any integer by zero is zero.

Example : 5x 0 = 0.

**Solved Examples :**

Two solved examples have been given below for better understanding –

Example 1:Prove that 4 and -6 satisfy the commutative property of integers.

Ans. Let x =4 and y =-6.

According to commutative property of addition

x+y = y+x

L.H.S = x+y = 4+( -6 ) = -2

R.H.S = y+x = ( -6 )+ 4 = -2

Hence x+y = y+x.

i.e. Left Hand Side = Right Hand Side

Hence proved that integers follow the commutative property of addition

Example 2: Prove that (-8), 4, 6 follow associative property of addition of integers.

Ans. Let x= -8, y= 4 and z=6

According to associative property of addition of integers,

x+(y+z) = (x+ y)+ z

Left Hand Side

x+(y+z) = -8 (4+6) = -8+10 = 2

Right Hand Side

(x+y)+ z = (-8+4)+ 6 = -4+6 = 2

Left Hand Side = Right Hand Side

Hence proved that integers follow the associative property of addition.

**Conclusion**

The students should understand these properties carefully as integers and their properties along with different operations of Integers form the basics of Mathematics. A thorough study and a good amount of practice will help them to be well versed with Integers.

**Frequently Asked Questions**

1. What are the different operations for Integers?

Ans. There are four operations for Integers namely Addition, Subtraction, Division, and Multiplication.

2. Is zero an Integer?

Ans. Yes, zero is also an Integer.

3. What are the different properties relating to integers?

Ans. The different properties relating to integers are five, namely associative, commutative, distributive, closure, and identity property.

4. Does the division of integers hold for the closure property?

Ans. No, the division of integers does not hold for the closure property, i.e. if x and y are two integers, then their quotient x /y may or may not be an integer.

5. Does the associative property applies to all operations of integers?

Ans. The associative property is true for addition and multiplication, but it does not hold for subtraction and division.

6. For which operations of integers the commutative property does not hold?

Ans. The commutative property does not hold for subtraction and division.