Problem 43 - Thomas Calculus

Question

Do not fall into the trap \(|-a|=a .\) For what real numbers \(a\) is this equation true? For what real numbers is it false?

Step-by-Step Solution

Answer

The equation is true for \( a \geq 0 \) and false for \( a < 0 \).

Step 1: Understanding Absolute Value

The absolute value of a number is its distance from zero on the number line, regardless of direction. Thus, for any real number \( a \), the absolute value \( |a| \) is always non-negative. It follows that \( |-a| \) is the absolute value of \( -a \), thus non-negative as well.

Step 2: Equation Analysis for Truth

The equation \( |-a| = a \) states that the absolute value of \( -a \) is equal to \( a \). Recognizing that both \( |a| \) and \( |-a| \) are equal, the current equation asserts that \( a \) itself must be non-negative, or \( a \geq 0 \). This ensures \( a \) equals its own absolute value form.

Step 3: Solving for True Conditions

For \( |-a| = a \) to be true, it must hold that \( a \geq 0 \). The simplest explanation for this is that when \( a \geq 0 \), both sides coincide, i.e., \( |-a| = a \equiv |a| = a \).

Step 4: Exploring Conditions of Falsehood

When \( a < 0 \), \( a \) itself is negative, which results in \( |-a| = -a \) being a positive equivalent. In contrast, the right-hand side \( a \) is negative, leading to inequality: \( |-a| eq a \). Thus, the equation does not hold true.

Key Concepts

Understanding Real Numbers

Real numbers are fundamental in mathematics as they include all numbers that can be found on the number line. This means they encompass various types of numbers, such as:

• Natural numbers (e.g., 1, 2, 3)
• Whole numbers, which incorporate zero along with natural numbers
• Integers, which include negative numbers and zero
• Rational numbers, which can be expressed as fractions
• Irrational numbers, such as pi or the square root of 2, which cannot be written as simple fractions

Understanding real numbers is crucial when analyzing equations since they provide a framework for visualizing the relationships and properties involving various types of numbers. Absolute value calculations often involve manipulating real numbers, and the challenge often lies in recognizing how they behave under different mathematical operations.

Exploring Inequalities with Absolute Values

Inequalities are mathematical expressions that compare two values, showing the relative size or order of the two values. For instance, using symbols like ">" and "<", inequalities allow us to express relationships such as one number being greater than or less than another. When absolute values are in play, it becomes a bit more involved, as absolute value describes the distance a number is from zero, without regard to whether it's positive or negative.

In the context of our original problem, we deal with the inequality when evaluating for which real numbers the equation |-a| = a is true or false. We see that:

• For the equation to be true, it requires that a \( \geq \) 0, meaning that the value must be equal to or greater than zero.
• On the other hand, if a is less than zero, the resulting inequality |-a| ≠ a arises, as the absolute value of a negative number is positive.

Careful handling of inequalities helps us understand when mathematical expressions, especially those involving absolute values, can be equal or unequal, thus setting the groundwork for solving such equations.

Equation Analysis of Absolute Value Expressions

Analyzing an equation involves breaking down its components to understand what makes it true or false. In the case of absolute value expressions like |-a| = a, one must dissect the equation thoroughly.

• The absolute value function \(| a |\) yields the same result for both positive and negative inputs by returning the non-negative magnitude.
• For negative inputs, like -a, the absolute value operation essentially flips the sign of the input, which is where the confusion often arises in inequalities and equation analysis.

To solve \(|-a| = a\) correctly, recognize that this equation is valid only when \( a \geq 0 \), as in this scenario, \(|-a|\) produces a non-negative number equivalent to \( a\). If \( a < 0 \), the equation becomes invalid because \(|-a|\) results in a positive value whereas \( a\) is inherently negative in such cases. This detailed examination is crucial when evaluating the truthfulness and the implications of absolute value equations across all real numbers.