Properties of Integers – Definition, Examples, Facts, FAQs

What Are the Properties of Integers?

Properties of integers in mathematics are certain rules that can be applied to simplify calculations when we perform arithmetic operations on them.

The set of integers are represented by the symbol ℤ.

ℤ $= \left\{…,-5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5,…\right\}$

The basic operation of addition, subtraction, multiplication, and division give rise to five main types of properties in integers:

1. Closure Property

2. Associative Property

3. Commutative Property

4. Distributive Property

5. Identity Property

Properties of Integers

Here’s a chart of properties of integers that gives an overview of each property.

Closure Property	$a + b \in$ ℤ $a \;-\; b \in$ ℤ $a \times b \in$ ℤ for any two integers a and b
Commutative Property	$a + b = b +a$ $a \times b = b \times a$ for any two integers a and b
Associative Property	$(a + b) + c = a + (b + c)$ $(a \times b) \times c = a \times (b \times c)$ for integers a, b, and c
Distributive Property	$a \times (b + c) = (a \times b) + (a \times c)$ $a \times (b\;-\;c) =(a \times b)\;-\;(a \times c)$ for integers a, b, and c
Identity Property	$a + 0 = a a \times 1 = 1 \times a = a$\ for any integer a

Let’s study all properties of integers in detail.

Closure Property of Integers

The closure property of integers states that the addition, subtraction, or multiplication of integers always results in an integer. 

For any two integers a and b, we have

• $a + b =$ integer
• $a \;-\; b  =$ integer
• $a \times b =$ integer

Examples:

• $-\;5 + 4 = -\;3$
• $8\;-\;5 = 3$
• $2 \times 3 = 6$

However, the closure property of integers does not work for the division of integers because the division of two integers may not be an integer.

For example: $15 \div 2 = 7.5$

Associative Property of Integers

According to the associative property of integers, if we add three or more integers, the result is not affected by the way we group the integers. Similarly, the product of three or more integers does not depend on how we group them.

Associative property of integers is true only for the addition and multiplication but not for division and subtraction.

For any three integers a, b, and c, we have

• Associative property of addition: $a + (b + c) = (a + b) + c$
• Associative property of multiplication: $a \times (b \times c) = (a \times b) \times c$

However, 

Examples: 

Operation	Result	Examples
Addition of integers	Associative property is satisfied.	$1 + (2 + 3) = 6$ $(1 + 2) + 3 = 6$
Multiplication of integers	Associative property is satisfied.	$4 \times (2 \times 3) = 24$ $(4 \times 2) \times 3 = 24$
Subtraction of integers	Associative property is not satisfied.	$1 \;-\; (2 \;-\; 3) = 2$ $(1\;-\;2)\;-\;3 = \;-4$
Division of integers	Associative property is not satisfied.	$8\div (4 \div 2) = 4$ $(8 \div 4) \div 2 = 1$

Commutative Property of Integers

The commutative property states that changing the order of integers in addition and multiplication does not change the result of the operation.

Interestingly, this property is true for only the addition and multiplication and not for division and subtraction.

For two integers a and b, we have

• Commutative property of addition: $a + b = b + a$
• Commutative property of multiplication: $a \times b = b \times a$

Examples:

Operation	Result	Example
Addition of integers	Commutative property is satisfied.	$1 + 2 = 3 = 2 + 1$
Multiplication of integers	Commutative property is satisfied.	$1 \times 2 = 3 = 2 \times 1$
Subtraction of integers	Commutative property is not satisfied.	$1\;-\;2 = 1\;-\;2$
Division of integers	Commutative property is not satisfied.	$1\;-\;2 = 1\;-\;2$

Distributive Property of Integers

The distributive property states that the multiplication of integers can be distributed over addition and subtraction to make the calculation easier.

So, for any three integers a, b, and c, we have

• Distributive property of multiplication over addition: $a \times (b + c) =(a \times b) + (a \times c)$
• Distributive property of multiplication over subtraction: $a \times (b\;-\;c) = (a \times b)\;-\;(a \times c)$

Example: 

• $5 \times (4 + 3) = (5 \times 4) + (5 \times 3) = 35$
• $\;-\; 2 \times (6 \;-\; 1) = (\;-\; 2 \times 6) + (\;-\; 2 \;-\;1) = \;-\; 10$

Identity Property of Integers

The identity property of an integer states that any integer added to zero will result in the same integer. Also, any integer multiplied by one will give the same integer.

0 is the additive identity of integers.

1 is the multiplicative identity of integers.

So, for any integer a, we have

• $a + 0 = 0 + a = a$
• $a \times 1 = 1 \times a = a$

Example: 

• $5 + 0 = 0 + 5 = 5$
• $5 \times 1 = 1 \times 5 = 5$

Facts about Properties of Integers

Conclusion

In this article, we have learned the various properties of integers. These properties of integers can be used to speed up and simplify a number of operations on integers. Now, let’s solve some examples and practice problems to understand the concept better. 

Solved Examples on Properties of Integers

1. Identify the correct properties of integers used in the following equations:

a) $a + (b + c) = (a + b) + c$

b) $a + 0 = 0 + a = a$

Solution: 

a) $a + (b + c) = (a + b) + c$  shows the associative property of integers.

It shows that the addition of three or more integers does not depend on how we group them.

b)  $a + 0 = 0 + a = a$  shows the identity property of integers

Addition of any integer and 0 is the integer itself.

2. Evaluate the expression: $(10 \times 5) + (10 \times 5)$ using the properties of integers.

Solution:

The given expression is $(10 \times 5) + (10 \times 4)$.

It can be solved by using the distributive property of integers, which states that,  

For any three integers a, b, and c:  

$(a \times b) + (a \times c) = a \times (b + c)$

$(10 \times 5) + (10 \times 4) = 10 \times (5 + 4)$

    $= 10 \times 9$

   $= 90$

Therefore, $(10 \times 5) + (10 \times 4) = 90$

3. If $a = \;-\;35$, $b = 10$ units and $c = \;-\;5$, verify that:

(i)  $a + (b + c) = (a + b) + c$

(ii) $a \times (b + c) = (a \times b) + (a \times c)$

Solution:

Given that $a = \;-\;35\;, b = 10\;, c = \;-\;5$

(i) $a + (b + c) = (a + b) + c$

L.H.S. $= a + (b + c) = (\;-\;35) + \left[10 + (\;-\;5)\right] = (\;-\;35) + 5 = \;-\;30$

R.H.S. $= (a + b) + c = \left[(\;-\;35) + 10\right] + (\;-\;5) = (\;-\;25) + (\;-\;5) = \;-\;(25 + 5) = \;-\;30$

Thus, L.H.S. = R.H.S.

Hence, verified.

(ii) $a \times (b + c) = (a \times b) + (a \times c)$

L.H.S. $= a(b + c)= (\;-\;35) \left[10 + (\;-\;5)\right] = (\;-\;35) 5 = \;-\;175$

R.H.S. $= (a \times b) + (a \times c) = (\;-\;35) \times 10 + (\;-\;35) \times (\;-\;5) = \;-\;350 + 175 = \;-\;175$

Thus, L.H.S. $=$ R.H.S.

4. State whether the following statements are true or false.

a) Closure property holds true for addition, subtraction and multiplication of integers.

b) The product of two negative integers is a negative integer.

c) According to commutative property of multiplication: $a \times (b \times c) = (a \times b) \times c$

Solution:

a) Closure property holds true for addition, subtraction and multiplication of integers.

True.

Closure property holds true for addition, subtraction and multiplication of integers. It fails for division of integers.

b) The product of two negative integers is a negative integer.

False.

The product of two negative integers is a positive integer.

c) According to commutative property of multiplication: $a \times (b \times c) = (a \times b) \times c$

False.

The commutative property of multiplication states that the order of integers in multiplication operation does not change the result of the operation. 

Commutative property of multiplication: $a \times b = b \times a$

Practice Problems on Properties of Integers

Frequently Asked Questions on Properties of Integers