Problem 14 - Algebra 1: Concepts and Skills

Question

Find a counterexample to show that the statement is not true. If \(a, b,\) and \(c\) are nonzero real numbers, then \((a \div b) \div c=a \div(b \div c)\) (Note: The counterexample shows that the associative property does not hold for division.)

Step-by-Step Solution

Answer

A counterexample can be \(a = 2\), \(b = 3\), \(c = 4\). As the results of \( (2 \div 3) \div 4 \) equals \( \frac{1}{6} \) and \( 2 \div (3 \div 4) \) equals \( \frac{8}{3} \) are clearly not equal. This shows that the associative property does not hold for division.

Step 1: Choosing Real Numbers

Let's select any three nonzero real numbers, for instance \(a = 2\), \(b = 3\), \(c = 4\).

Step 2: Calculation For \( (a \div b) \div c \)

Substitute the selected numbers into the first equation \( (a \div b) \div c \). Then, calculate \( (2 \div 3) \div 4 \), which equals to \( \frac{2}{3} \div 4 = \frac{2}{12} = \frac{1}{6}.\)

Step 3: Calculation For \( a \div (b \div c) \)

Substitute the selected numbers into the second equation \( a \div (b \div c) \). Then, calculate \( 2 \div (\frac{3}{4}) = 2 \div 0.75 = \frac{8}{3}.\)

Step 4: Compare Both Results

Comparison of both results shows they are not equivalent. \( \frac{1}{6} \) is not equal to \( \frac{8}{3} \), hence the statement \( (a \div b) \div c = a \div (b \div c) \) is not always true. This proves that the associative property does not hold for division.

Key Concepts

Understanding Nonzero Real Numbers

When dealing with mathematical problems, especially those involving division, it is important to understand the concept of nonzero real numbers. Nonzero real numbers are simply all real numbers except for zero.
They can be either positive or negative, and include fractions and decimals as well. Why is it important that we discuss nonzero numbers? Because when you divide by zero, the operation becomes undefined. You can't perform division with zero as the divisor. This is why, in the provided exercise, we only use nonzero real numbers for calculation.
Real numbers can be:

• Whole numbers like 1, 2, 3
• Fractions such as 1/2, 2/3
• Decimals like 0.5, 3.75

By ensuring that our numbers are nonzero, we can safely perform division without running into any undefined conditions.

The Basics of Division

Division is one of the four fundamental arithmetic operations, and it essentially involves splitting a number into equal parts. Think of it as distributing items evenly. When you see notation like \(a \div b\), it means how many times \(b\) fits into \(a\).
Though division is generally straightforward, it possesses some intriguing properties:

• It is not commutative. Changing the order of division changes the result: \(a \div b eq b \div a\).
• Division is not associative, meaning that reorganizing terms can change the outcome, as seen in the exercise with the counterexample.

For example, in the problem with the numbers \(a = 2\), \(b = 3\), and \(c = 4\), observing careful division results in different outcomes for \((a \div b) \div c\) compared to \(a \div (b \div c)\). This illustrates that with division, 'grouping' order makes a difference!

What is a Counterexample?

A counterexample is a specific case used to show that a general statement is false.
In mathematics, it can be quite powerful because it takes only one counterexample to disprove a universal statement.In our exercise, we were given the statement: "If \(a, b,\) and \(c\) are nonzero real numbers, then \((a \div b) \div c = a \div (b \div c)\)."Using the numbers \(a=2\), \(b=3\), and \(c=4\), we found:

• \((2 \div 3) \div 4 = \frac{1}{6}\)
• \(2 \div (3 \div 4) = \frac{8}{3}\)

Since \(\frac{1}{6} eq \frac{8}{3}\), this serves as a counterexample showing that the associative property does not hold for division. Whenever you're asked to provide a counterexample, think of it as a puzzle! Find a case where the statement doesn’t fit, and that’s your key to breaking down the problem.