What Are the Main Properties of Integers With Examples?
The concept of properties of integers plays a key role in mathematics and is widely applicable to both real-life situations and exam scenarios. Understanding these properties helps students solve problems faster and with confidence.
What Is Properties of Integers?
The properties of integers are a set of basic rules or laws that describe how integers behave under operations like addition, subtraction, multiplication, and division. These properties include closure, associative, commutative, distributive, and identity properties. You’ll find this concept applied in areas such as integer operations, number patterns, and algebraic calculations.
Key Properties of Integers
Here are the main properties of integers:
- Closure Property
- Commutative Property
- Associative Property
- Distributive Property
- Identity Property
Let’s understand each property in detail with examples.
Closure Property of Integers
The closure property states that when you add, subtract, or multiply any two integers, the result is always an integer. For division, this property does not always hold.
- Addition: \( -3 + 7 = 4 \)
- Subtraction: \( 8 - 12 = -4 \)
- Multiplication: \( -5 \times 6 = -30 \)
- Division (Not Always Closed): \( 6 \div 4 = 1.5 \), which is not an integer.
Commutative Property of Integers
The commutative property means the order of the numbers does not change the result in addition or multiplication.
- Addition: \( a + b = b + a \) (e.g., \( 4 + 9 = 9 + 4 \))
- Multiplication: \( a \times b = b \times a \) (e.g., \( -7 \times 2 = 2 \times -7 \))
Subtraction and division are not commutative.
Associative Property of Integers
The associative property tells us that when adding or multiplying three or more integers, the way we group them doesn’t affect the answer.
- Addition: \( (a + b) + c = a + (b + c) \) (e.g., \( (2 + 3) + 4 = 2 + (3 + 4) \))
- Multiplication: \( (a \times b) \times c = a \times (b \times c) \) (e.g., \( (2 \times 3) \times 5 = 2 \times (3 \times 5) \))
Again, subtraction and division are not associative.
Distributive Property of Integers
The distributive property connects multiplication with addition or subtraction:
\( a \times (b + c) = a \times b + a \times c \)
- Example: \( 3 \times (2 + 5) = 3 \times 2 + 3 \times 5 = 6 + 15 = 21 \)
Identity Property of Integers
There are two types of identity elements:
- Additive identity: 0 is the identity for addition (\( a + 0 = a \)).
- Multiplicative identity: 1 is the identity for multiplication (\( a \times 1 = a \)).
Properties Table with Examples
| Property | Definition | Example |
|---|---|---|
| Closure | Sum, difference, or product of any 2 integers is an integer (not always for division) | \( -4 + 6 = 2 \); \( 3 \times -2 = -6 \) |
| Commutative | Order doesn't change answer (addition, multiplication) | \( 5 + 8 = 8 + 5 \); \( -3 \times 7 = 7 \times -3 \) |
| Associative | Grouping doesn't change answer (addition, multiplication) | \( (1 + 2) + 3 = 1 + (2 + 3) \) |
| Distributive | Multiply over addition/subtraction | \( 2 \times (3 + 4) = 2 \times 3 + 2 \times 4 \) |
| Identity | Special number keeps integer same: 0 for addition, 1 for multiplication | \( a + 0 = a \); \( a \times 1 = a \) |
Step-by-Step Illustration: Example Problems
Commutative Property (Addition):
1. Consider \( a = -37 \) and \( b = 25 \)2. According to the commutative property: \( a + b = b + a \)
3. Calculate \( a + b = -37 + 25 = -12 \)
4. Calculate \( b + a = 25 + (-37) = -12 \)
5. Both answers are equal, confirming the commutative property under addition.
Associative Property (Addition):
1. Take \( a = -6 \), \( b = -2 \), \( c = 5 \)2. Left grouping: \( a + (b + c) = -6 + (-2 + 5) = -6 + 3 = -3 \)
3. Right grouping: \( (a + b) + c = (-6 + -2) + 5 = -8 + 5 = -3 \)
4. Both groupings give -3: the associative property is satisfied.
Speed Trick or Vedic Shortcut
To quickly check the closure or commutative property of integers for exam MCQs, just do a quick calculation in your mind. For distributive property, break the number like \( 4 \times 27 = 4 \times (20 + 7) = 80 + 28 = 108 \).
Trick: Multiply large numbers by splitting into round numbers to save time in exams!
Try These Yourself
- Prove the closure property for subtraction of integers with your own numbers.
- Show whether \( 7 \div 3 \) is closed under integers.
- Give one example of the associative property using three negative integers.
- Write a real-life situation where distributive property is useful.
Frequent Errors and Misunderstandings
- Assuming the closure property works for division of integers—it does not!
- Mixing up the commutative and associative properties.
- Forgetting zero only works as additive identity, not for multiplication.
Relation to Other Concepts
The idea of properties of integers connects to properties of whole numbers and properties of rational numbers. Mastering these helps in learning algebra, equations, and number systems.
Classroom Tip
A quick way to remember: "C-A-C-D-I" for Closure, Associative, Commutative, Distributive, Identity. Vedantu’s teachers use fun charts and memory aids so these properties stick for exams!
We explored properties of integers—from definitions, examples, mistakes to tried-and-tested tricks. Keep practicing these and explore more problem-solving on Vedantu to become quick and accurate with integer operations.
Explore Further
FAQs on Properties of Integers Explained With Solved Examples
1. What are the 5 properties of integers?
The 5 properties of integers are:
- Closure property: The sum or product of any two integers is also an integer. For any integers $a$ and $b$, $a + b$ and $a \times b$ are integers.
- Commutative property: The order of addition or multiplication does not affect the result. $a + b = b + a$ and $a \times b = b \times a$.
- Associative property: The grouping of numbers does not change the sum or product. $(a + b) + c = a + (b + c)$ and $(a \times b) \times c = a \times (b \times c)$.
- Distributive property: Multiplication distributes over addition. $a \times (b + c) = a \times b + a \times c$.
- Identity property: The sum of any integer and 0 is the integer itself; the product of any integer and 1 is the integer itself. That is, $a + 0 = a$ and $a \times 1 = a$.
2. What are the 7 properties of operations on integers?
The 7 properties of operations on integers include:
- Closure under addition
- Closure under multiplication
- Commutativity of addition
- Commutativity of multiplication
- Associativity of addition
- Associativity of multiplication
- Distributive property of multiplication over addition
3. What are the 4 basic properties of numbers?
The 4 basic properties of numbers are:
- Closure property
- Commutative property
- Associative property
- Distributive property
4. Can you use the commutative property with integers?
Yes, the commutative property applies to both addition and multiplication of integers. This means:
- For any integers $a$ and $b$, $a + b = b + a$
- For any integers $a$ and $b$, $a \times b = b \times a$
5. How do the closure and associative properties help solve integer problems?
Closure property ensures that the sum or product of any two integers will always be an integer, so calculations stay within the set of integers. The associative property allows us to group numbers in different ways without changing the result during addition or multiplication: $ (a + b) + c = a + (b + c)$ and $(a \times b) \times c = a \times (b \times c)$. These properties make complex calculations simpler—a topic covered with real-life examples in Vedantu’s live classes.
6. What is the difference between distributive property and commutative property in integers?
The distributive property relates multiplication and addition: $a \times (b + c) = a \times b + a \times c$. In contrast, the commutative property refers to the ability to change the order of addition or multiplication ($a + b = b + a$, $a \times b = b \times a$). While distributive property combines two operations, the commutative property applies to the same operation. Vedantu provides interactive problem-solving sessions to help students master these distinctions.
7. Why are properties of integers important in mathematics?
Understanding the properties of integers helps students solve equations more efficiently, recognize number patterns, and avoid common mistakes. These properties are the basis for algebra, number theory, and higher mathematics. At Vedantu, these concepts are explained in detail to build strong foundational skills in students.
8. How does Vedantu teach properties of integers to students?
Vedantu uses interactive classes, detailed theory lessons, and practical problem-solving sessions to teach the properties of integers. Tutors use examples, quizzes, and real-world problems so that students can master each property step-by-step according to their curriculum requirements.
9. What are some examples of the distributive property of integers?
An example of the distributive property is: $2 \times (3 + 5) = (2 \times 3) + (2 \times 5)$, which simplifies to $2 \times 8 = 6 + 10$ resulting in $16 = 16$. Such examples are a part of Vedantu’s student worksheets to enhance learning and confidence in applying mathematics concepts.
10. Are negative numbers included when discussing properties of integers?
Yes, negative numbers are integral to the set of integers. All properties—closure, commutative, associative, identity, and distributive—apply to both positive and negative integers. Vedantu’s course modules use diverse examples to help students understand how these properties work with negative, zero, and positive integers.
