Problem 10 - Prealgebra

Question

Is division of whole numbers commutative? If not, give a counterexample.

Step-by-Step Solution

Answer

Division of whole numbers is not commutative. Counterexample: 6 ÷ 3 ≠ 3 ÷ 6.

Step 1: Define Commutative Property

The commutative property states that in an operation, changing the order of the numbers does not change the result. For example, for addition, \(a + b = b + a\), and for multiplication, \(a \times b = b \times a\).

Step 2: Analyze Division Operation

Division is the operation of distributing a number into equal parts. It is defined as \(a \div b\) or \(\frac{a}{b}\), which means 'a' is divided by 'b'. To check if division is commutative, we must see if \(a \div b = b \div a\) for all whole numbers.

Step 3: Test Using Specific Whole Numbers

Let's choose specific numbers: \(a = 6\) and \(b = 3\). Calculate \(6 \div 3\) which equals 2. Now switch the order: \(3 \div 6\), which equals \(0.5\).

Step 4: Conclusion from Calculation

Since \(6 \div 3 = 2\), but \(3 \div 6 = 0.5\), the results differ when the order of numbers is changed. This shows the operation is not commutative.

Step 5: Provide a Counterexample

The counterexample is using the numbers 6 and 3: \(6 \div 3 = 2\) is not equal to \(3 \div 6 = 0.5\). This proves that division of whole numbers is not commutative.

Key Concepts

Commutative Property

The commutative property is a fundamental concept in mathematics. It suggests that for certain operations, the order in which you perform the operation doesn't affect the final result. This property holds true for addition and multiplication.
For example:

• Addition: If you take any two numbers, say 4 and 5, and add them, you get the same result regardless of their order. That is, \(4 + 5 = 9\) and \(5 + 4 = 9\).
• Multiplication: Similarly, for multiplication, switching the order of multiplication gives the same product. For instance, \(3 \times 7 = 21\) is the same as \(7 \times 3 = 21\).

However, not all mathematical operations are commutative, and division is an example where this property does not apply. Understanding the commutative property helps grasp why certain operations behave differently.

Whole Numbers

Whole numbers are numbers without fractions or decimals. They include all the positive integers along with zero. Hence, they start from 0 and continue upwards: 0, 1, 2, 3, and so forth.
Whole numbers are used in various mathematical operations and are considered easier to handle because they do not involve dealing with parts of a number. The key characteristics of whole numbers are:

• They are always non-negative.
• They do not include fractions or decimals.
• They are integral in understanding basic operations like addition, subtraction, multiplication, and division.

In the context of the discussed exercise, understanding whole numbers is essential as it clarifies that when dividing, only whole portions, if any, can be shown when using decimals.

Math Operations

In mathematics, operations are processes you can perform on numbers. The primary operations we deal with at the basic level include addition, subtraction, multiplication, and division. Each of these operations has its own rules and properties.

• Addition and Multiplication: These are both commutative and associative, making them very versatile.
• Subtraction and Division: These operations are not commutative, and division is also not associative.

The focus in the original exercise is on division. Division represents taking a number and "splitting" it into a specified number of equal parts. It's crucial to know that for whole numbers, results can sometimes not be whole numbers themselves, as seen with \(3 \div 6 = 0.5\). This distinctiveness is what the counterexample aims to show.

Counterexample

A counterexample is used to demonstrate that a particular property or statement does not always hold true. In mathematics, identifying a counterexample helps in understanding the limitations of certain mathematical properties.
For instance, to assess whether division is commutative, consider two numbers: let's take 6 and 3. When you calculate their division in two orders:

• First: \(6 \div 3 = 2\)
• Second: \(3 \div 6 = 0.5\)

The results are different, thus showing that switching the order in division changes the outcome. This different outcome for the same pair of numbers but in reversed order serves as a counterexample, proving that division of whole numbers is not commutative. Using counterexamples is an excellent method in mathematics to test the validity of a property.