Problem 29 - Beginning and Intermediate Algebra

Question

Write an example that shows that division is not commutative.

Step-by-Step Solution

Answer

Division is not commutative as 10 ÷ 5 ≠ 5 ÷ 10.

Step 1: Choose Two Numbers

To demonstrate that division is not commutative, we need to select two numbers. Let's choose 10 and 5.

Step 2: Divide the First Number by the Second

Now, divide the first number by the second: \[ 10 \div 5 = 2 \]

Step 3: Reverse the Order and Divide the Second Number by the First

Next, reverse the order and divide the second number by the first: \[ 5 \div 10 = 0.5 \]

Step 4: Compare the Results

Compare the results from Steps 2 and 3. We have:- \( 10 \div 5 = 2 \)- \( 5 \div 10 = 0.5 \)Since 2 is not equal to 0.5, division is not commutative.

Key Concepts

Division

In mathematics, division is one of the four basic arithmetic operations, the others being addition, subtraction, and multiplication. Division involves splitting a number into equal parts or determining how many times one number is contained within another. For example, if you have 10 cookies and you want to divide them among 5 friends, each friend would receive 2 cookies. This operation is represented as \[ 10 \div 5 = 2 \].

However, division is a unique operation because, unlike addition or multiplication, dividing numbers does not always yield the same result when the order of the numbers is changed. This lack of interchangeability is why division is described as a non-commutative operation.

• Non-commutative means that changing the order of numbers changes the result.
• For division, \( a \div b \) is not the same as \( b \div a \).

By experimenting with different numbers, you can clearly see this principle in action.

Commutative Property

The commutative property is a fundamental principle in algebra that states you can change the order of numbers in an operation without changing the result. Simple examples include addition and multiplication, where this principle holds true:

• Addition: \( a + b = b + a \)
• Multiplication: \( a \times b = b \times a \)

However, this property does not extend to all mathematical operations. Division and subtraction do not follow the commutative property because changing the order of the numbers affects the outcome:
- Division: For example, \( 8 \div 4 eq 4 \div 8 \), because \( 8 \div 4 = 2 \) and \( 4 \div 8 = 0.5 \). This illustrates the non-commutative nature of division.
- Subtraction: Similarly, for subtraction, \( 5 - 3 eq 3 - 5 \).Understanding the commutative property helps in simplifying calculations and solving algebraic expressions, but it's important to remember which operations it applies to.

Basic Algebra Concepts

Basic algebra introduces fundamental concepts that lay the groundwork for understanding more complex mathematical theories. It provides the tools to represent real-life problems using numbers, symbols, and variables. Here are a few key concepts:

• Variables: Symbols like \( x \) or \( y \) are used to represent unknown values.
• Expressions: Combinations of variables, numbers, and operations (like \( 2x + 3 \)) make up algebraic expressions.
• Equations: Statements that express equality, such as \( 2x + 3 = 7 \), can be solved to find the value of the variables involved.
• Operations: Includes addition, subtraction, multiplication, and division.

Each concept in algebra builds upon the other, creating a comprehensive system that helps solve a wide variety of mathematical problems. For students, grasping these basic concepts is crucial, as they form the foundation for more advanced math topics. Recognizing properties such as commutativity, and understanding how operations like division differ, are essential steps in becoming proficient in algebra.