Integer multiplication and division

Integer multiplication

Integer multiplication and division are key mathematical operations used to handle whole numbers, including positive and negative integers. The multiplication of integers involves repeated addition. For example, multiplying 5 × (-3) means adding -3 five times, resulting in -15. Similarly, when two integers with the same sign are multiplied, the result is positive, while the product of integers with opposite signs is negative. Division of integers involves splitting a number into equal parts. For instance, -12 ÷ 4 means dividing -12 into 4 equal groups, which results in -3. Division follows the same sign rules as multiplication: dividing integers with the same sign yields a positive quotient, while dividing integers with opposite signs gives a negative quotient. In this article, we will explore the rules, examples, and applications of integer multiplication and division in detail.

What is an Integer?

An integer is a whole number that can be positive, negative, or zero. Integers do not include fractions or decimals.

Examples of Integers:

• Positive integers: 1, 5, 19, 456
• Negative integers: -1, -19, -2587
• Zero: 0

Examples of Non-Integers:

• Fractions: 3/5, -7/12
• Decimals: 2.015, -14.133

A number line is a helpful tool to visualize integers, displaying positive numbers to the right of zero and negative numbers to the left.

Read the lesson ‘Adding and Subtracting Integers’

List of Integer Multiplication Rules

When multiplying integers, the sign of the product depends on the signs of the integers involved. The integer multiplication rules are as follows:

1. Rule #1 : Multiplying Two Positive Integers:
   • The product is positive.
   • Multiplying Two Positive Integer Example: (+6) × (+8) = +48

2. Rule #2 : Multiplying Two Negative Integers:
   • The product is positive.
   • Multiplying Two Negative Integer Example: (−6) × (−8) = +48

3. Rule #3 : Multiplying a Positive and a Negative Integer:
   • The product is negative.
   • Multiplying a Positive and a Negative Integer Example: (+4) × (−2) = −8

Summary of Multiplication Rules:

• Same Signs: Positive result
  • (+)×(+)=+
  • (−)×(−)=+
• Different Signs: Negative result
  • (+)×(−)=−
  • (−)×(+)=−

Division of Integers

The rules for dividing integers are similar to those for multiplication:

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List of Integer Division Rules

When dividing integers, the sign of the product depends on the signs of the integers involved. The integer division rules are as follows:

1. Rule #1 ; Dividing Two Positive Integers:
   • The quotient is positive.
   • Dividing Two Positive Integer Example: (+9) ÷ (+3) = +3

2. Rule #2 ; Dividing Two Negative Integers:
   • The quotient is positive.
   • Dividing Two Negative Integer Example: (−6) ÷ (−2) = +3

3. Rule #3 ; Dividing a Positive and a Negative Integer:
   • The quotient is negative.
   • Dividing a Positive and a Negative Integer Example: (+6) ÷ (−3) = −2

Summary of Division Rules:

• Same Signs: Positive result
  • (+)÷(+)=+(+)
  • (−)÷(−)=+(-)
• Different Signs: Negative result
  • (+)÷(−)=−(+)
  • (−)÷(+)=−(-)

Properties of Multiplication and Division of Integers

Understanding the properties of these operations can further enhance our comprehension:

1. What is Closure Property?
   • Multiplication: The product of any two integers is always an integer.
   • Division: The quotient of two integers may not always be an integer (e.g., 1÷2=0.5).

2. What is Commutative Property?
   • Multiplication: Changing the order of factors does not change the product.
     • Example: 5×(−6)=(−6)×5=−30
   • Division: Changing the order of operands changes the quotient.
     • Example: 15÷3=5 but 3÷15=0.2

3. What is Associative Property?
   • Multiplication: Changing the grouping of factors does not change the product.
     • Example: (2×3)×(−4)=2×(3×−4)=−24
   • Division: Changing the grouping of operands changes the quotient.
     • Example: (20÷5)÷2=2 but 20÷(5÷2)=8

4. What is Distributive Property?
   • Multiplication over Addition: a×(b+c)=(a×b)+(a×c)
     • Example: 3×(4+2)=(3×4)+(3×2)=18

5. What is Identity Property?
   • Multiplication: Multiplying any integer by 1 yields the same integer.
     • Example: 7×1=7
   • Division: Dividing any integer by 1 yields the same integer.
     • Example: 7÷1=7

Practical Examples

Let’s apply these rules to some examples:

1. Multiplication:
   • (−4)×5=−20
   • (−3)×(−7)=21
   • 6×(−2)=−12

2. Division:
   • (−15)÷3=−5
   • (−20)÷(−4)=5
   • 18÷(−6)=−3

Summary

1. Multiplication and Division Rules:
   • Same Signs: Result is positive.
   • Different Signs: Result is negative.

2. Key Properties
   • Closure: Multiplication is closed; division is not always closed.
   • Commutative: Applies to multiplication; does not apply to division.
   • Associative: Applies to multiplication; does not apply to division.
   • Distributive: Multiplication distributes over addition.
   • Identity: 1 is the identity element for both multiplication and division.

Understanding these rules and properties is essential for performing arithmetic operations involving integers accurately.