The distributive operation is a fundamental concept in mathematics that involves the distribution of one operation over another. It is commonly used to simplify expressions and perform calculations efficiently.
The concept of distributive operation can be traced back to ancient civilizations, where it was used in various mathematical systems. However, the formalization of this concept can be attributed to the development of algebra in the 16th century.
The distributive operation is typically introduced in elementary school, around 3rd or 4th grade, and is further reinforced in middle school and high school mathematics.
The distributive operation involves the distribution of multiplication or division over addition or subtraction. It can be explained step by step as follows:
Multiplication over addition: The distributive property states that for any three numbers a, b, and c, the product of a and the sum of b and c is equal to the sum of the products of a and b, and a and c. Mathematically, it can be expressed as a * (b + c) = (a * b) + (a * c).
Multiplication over subtraction: Similarly, the distributive property can be applied to multiplication over subtraction. For any three numbers a, b, and c, the product of a and the difference between b and c is equal to the difference between the products of a and b, and a and c. Mathematically, it can be expressed as a * (b - c) = (a * b) - (a * c).
Division over addition and subtraction: The distributive property can also be extended to division. For any three numbers a, b, and c, the quotient of a divided by the sum or difference of b and c is equal to the sum or difference of the quotients of a divided by b and a divided by c. Mathematically, it can be expressed as a / (b + c) = (a / b) + (a / c) and a / (b - c) = (a / b) - (a / c).
The distributive operation can be categorized into two types based on the operations involved:
Multiplication Distribution: This type involves the distribution of multiplication over addition or subtraction.
Division Distribution: This type involves the distribution of division over addition or subtraction.
The distributive operation exhibits the following properties:
Closure Property: The result of applying the distributive operation to any two numbers is always a number within the same mathematical system.
Associative Property: The order of applying the distributive operation does not affect the final result.
Commutative Property: The order of the numbers involved in the distributive operation can be interchanged without affecting the final result.
To find or calculate the distributive operation, follow these steps:
Identify the operation to be distributed (multiplication or division).
Identify the operation over which the distribution is to be performed (addition or subtraction).
Apply the distributive property formula based on the identified operations.
The distributive operation can be expressed using the following formulas:
Multiplication Distribution: a * (b + c) = (a * b) + (a * c) and a * (b - c) = (a * b) - (a * c).
Division Distribution: a / (b + c) = (a / b) + (a / c) and a / (b - c) = (a / b) - (a / c).
To apply the distributive operation formula, substitute the given values into the formula and perform the necessary calculations.
There is no specific symbol or abbreviation exclusively used for the distributive operation. However, the distributive property is often represented using parentheses or brackets to indicate the operation being distributed.
The distributive operation can be applied using various methods, including mental calculations, written calculations, and the use of calculators or computer software.
Example 1: Simplify the expression 3 * (4 + 2).
Solution: Using the distributive property, we have 3 * (4 + 2) = (3 * 4) + (3 * 2) = 12 + 6 = 18.
Example 2: Evaluate the expression 10 / (5 + 1).
Solution: Applying the distributive property, we get 10 / (5 + 1) = (10 / 5) + (10 / 1) = 2 + 10 = 12.
Example 3: Calculate the value of 5 * (8 - 3).
Solution: Using the distributive property, we have 5 * (8 - 3) = (5 * 8) - (5 * 3) = 40 - 15 = 25.
Simplify the expression 2 * (7 + 3).
Evaluate the expression 15 / (5 - 2).
Calculate the value of 4 * (6 - 2).
Question: What is the distributive operation?
Answer: The distributive operation involves the distribution of one operation over another, such as multiplication or division over addition or subtraction.