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 main properties of integers?
The primary properties of integers are closure, commutative, associative, and distributive properties. These properties govern how integers behave under addition, subtraction, and multiplication. Understanding these properties simplifies calculations and problem-solving.
2. What is the closure property of integers?
The closure property states that the sum, difference, and product of any two integers is always another integer. In simpler terms, performing these operations on integers keeps the result within the set of integers. However, division of integers doesn't always result in an integer, so closure doesn't apply to division.
3. What is the commutative property of integers?
The commutative property says that the order of integers in addition and multiplication doesn't change the result. For example, 5 + 3 = 3 + 5 and 5 × 3 = 3 × 5. This property, however, does not hold true for subtraction and division.
4. Explain the associative property of integers.
The associative property states that the grouping of integers in addition and multiplication doesn't affect the outcome. For example, (2 + 3) + 4 = 2 + (3 + 4) and (2 × 3) × 4 = 2 × (3 × 4). Again, this property doesn't apply to subtraction or division.
5. What is the distributive property of integers?
The distributive property links multiplication with addition and subtraction. It states that a × (b + c) = (a × b) + (a × c) and a × (b - c) = (a × b) - (a × c). This allows us to simplify expressions by distributing multiplication over addition or subtraction.
6. What is the identity property of integers?
The identity property defines the additive identity (0) and the multiplicative identity (1). Adding 0 to any integer doesn't change its value (a + 0 = a), and multiplying any integer by 1 results in the same integer (a × 1 = a).
7. Do integers have an inverse property?
Yes, integers have an additive inverse. For every integer 'a', there exists an integer '-a' such that a + (-a) = 0. This means that adding an integer to its opposite always results in zero. There is no general multiplicative inverse for integers.
8. How are the properties of integers used in problem-solving?
The properties of integers are fundamental to simplifying complex expressions and solving equations. They allow us to rearrange terms, combine like terms, and apply various calculation techniques efficiently. They are the foundation for algebra and higher-level mathematics.
9. Are the properties of integers applicable to all number systems?
While some properties (like commutative and associative) apply to other number systems (such as rational numbers and real numbers), others (like closure) might have specific limitations. For example, closure under division only holds for rational numbers excluding zero, but not for integers.
10. Give examples of how the distributive property simplifies calculations.
The distributive property is very useful for mental math. For instance, to calculate 12 × 102, we can rewrite it as 12 × (100 + 2) = (12 × 100) + (12 × 2) = 1200 + 24 = 1224. This is often faster than direct multiplication.
11. Why is the closure property important for integers?
The closure property ensures that operations performed on integers always result in another integer (for addition, subtraction, and multiplication). This consistency simplifies calculations and allows us to work within a well-defined number system without unexpectedly encountering non-integer results during these operations.
12. How do integer properties help in real-life situations?
Integer properties are essential in numerous real-world applications, such as financial calculations (managing profits and losses), inventory management (tracking items), and programming (performing arithmetic operations). They form the basis for many algorithms and computational processes.
