Problem 28 - Algebra 1

Question

COUNTEREXAMPLES Decide whether the statement is true or false. If it is false, give a counterexample. (Review 2.1 for 12.8) The absolute value of a number is always positive.

Step-by-Step Solution

Answer

The statement 'The absolute value of a number is always positive' is false. A counterexample is \(|0| = 0\), which is not a positive number.

Step 1: Understanding Absolute Value

The absolute value of a number is the numerical value without regard to whether it is positive or negative. For a given number \( n \), the absolute value can be represented as \( |n| \).

Step 2: Analyzing the Statement

The statement asserts that the absolute value of a number is always positive. To verify this statement, it is necessary to consider what values could be produced by applying the absolute value operation to a number.

Step 3: Finding a Counterexample

When applying the absolute value operation to zero, the result is zero, not a positive number. Hence, zero serves as a counterexample invalidating the statement. Therefore, the statement is false.

Key Concepts

Counterexamples

Understanding the significance of counterexamples is crucial when dealing with mathematical statements. A counterexample is a special kind of example that disproves a statement by showing that it does not hold in every case.

When faced with the assertion 'The absolute value of a number is always positive', we employ the concept of counterexamples to test its validity. The method is simple: find just one instance where the absolute value is not positive. As the solution illustrates, the absolute value of zero ( |0| ) is neither positive nor negative; it is simply zero. This single case—zero—is a counterexample that clearly shows the original assertion is not always true.

Counterexamples serve as powerful tools in mathematics. They not only help in clarifying misconceptions but also ensure that one does not accept general statements without questioning their universal application. Thus, a good grasp of counterexamples emboldens critical thinking in students.

Numerical Value

The term 'numerical value' refers to the actual quantity represented by a number, disregarding its sign. When dealing with absolute values, the numerical value becomes central to understanding the concept.

Expressed by the symbol |n| , where 'n' is any real number, the absolute value represents the numerical value of 'n' without considering if 'n' were negative or positive. This measure is always zero or a positive number because it corresponds to the 'distance' a number is from zero on the number line, and distance cannot be negative.

The idea of numerical value is not just limited to the confines of absolute values. It emerges in various contexts across mathematics, economics, and science, representing quantities that may be compared or combined irrespective of their direction or polarity.

Positive Numbers

Positive numbers are all the real numbers greater than zero. They are to the right of zero on the number line and represent quantities with a magnitude and a direction—going away from zero. When discussing absolute values, one might say that absolute values usually result in positive numbers.

The confusion often arises with the special case of zero, which is neither positive nor negative. Hence, the absolute value of zero is zero itself, showing that it is an exception to the general outcome of the absolute value operation, which yields positive numbers. Understanding this distinction clarifies why the initial statement provided in the exercise—a misconception—is false.

In mathematics, acknowledging the unique status of zero vis-à-vis positive numbers is imperative, as it ensures accuracy in computation, logical consistency, and a deeper comprehension of number properties.