Problem 20 - Algebra 1

Question

Find a counterexample to show that the statement is not true. If \(a, b,\) and \(c\) are nonzero real numbers, then \((a \div b) \div c=a \div(b \div c)\)

Step-by-Step Solution

Answer

A counterexample to the statement is when \(a = 2\), \(b = 3\), and \(c = 4\). The calculated values in the LHS and RHS of the equation are not equal, hence demonstrating the statement is not true.

Step 1: Identify a, b, and c

Let's choose three nonzero real numbers a, b, and c. Here we can take \(a = 2\), \(b = 3\), \(c = 4\) for the ease of calculations.

Step 2: Calculate (a div b) div c

According to the order of operations, first solve \(a \div b = 2 \div 3 = 0.666666667\). Then solve \(0.666666667 \div 4 = 0.166666667\)

Step 3: Calculate a div (b div c)

Similar to step 2, first solve \(b \div c = 3 \div 4 = 0.75\). Then solve \(2 \div 0.75 = 2.666666667\)

Step 4: Comparing Results

Since the results in Step 2 and Step 3 are not the same, i.e, \(0.166666667 ≠ 2.666666667\). It proves that the given statement is not always correct. Hence, the chosen values are a legitimate counterexample.

Key Concepts

Understanding the Order of Operations

The order of operations is a fundamental concept in algebra that dictates the sequence in which arithmetic processes should be performed to accurately solve an expression. This rule is often abbreviated as PEMDAS, which stands for Parentheses, Exponents, Multiplication and Division (from left to right), and Addition and Subtraction (from left to right).

When faced with an algebraic expression such as \( (a \div b) \div c \) versus \( a \div (b \div c) \), the order in which you perform the divisions can significantly affect the outcome. This is why applying the order of operations is crucial in evaluating whether a given statement about arithmetic operations is true. In the context of our problem, by following the correct order, we are able to show that the initial assertion \( (a \div b) \div c = a \div (b \div c) \) is not generally valid, by providing a concrete counterexample.

Exploring Nonzero Real Numbers

In mathematics, real numbers encompass all the numbers that can be found on the number line, including both positive and negative integers, fractions, and irrational numbers. Nonzero real numbers are simply all the real numbers except for zero. The distinction of 'nonzero' is significant because division by zero is undefined – it's a mathematical impossibility. Thus, when solving algebraic equations or evaluating expressions, ensuring numbers are nonzero is essential to avoid undefined results.

For our counterexample, the condition of choosing nonzero real numbers for \(a, b,\) and \(c\) ensures that the division is valid and the mathematical laws can be properly tested. By selecting \(a = 2, b = 3,\) and \(c = 4\), we demonstrate that even with valid nonzero real numbers, the original statement doesn't hold, helping students understand the importance of counterexamples in challenging mathematical claims.

Arithmetic Operations and Their Properties

Arithmetic operations mainly include addition, subtraction, multiplication, and division. These operations have various properties that govern how they can be combined and manipulated. For instance, multiplication and addition are commutative, meaning that changing the order of the numbers does not change the result, as in \(a \times b = b \times a\) or \(a + b = b + a\). However, division and subtraction are not commutative.

Another property is associativity, which allows us to change the grouping of numbers without altering the result in the cases of addition and multiplication. However, in our example involving division, the lack of associativity is why the expression \( (a \div b) \div c \) gives us a different result from \( a \div (b \div c) \). This distinction is particularly crucial for students to understand that different arithmetic operations have unique properties that can significantly affect the outcomes of algebraic expressions.