Problem 10 - A Primer of Real Analysis

Question

Show that for any \(a \in \mathbb{R},-|a| \leq a \leq|a|\).

Step-by-Step Solution

Answer

The inequality \(-|a| \leq a \leq |a|\) holds true for any real number \(a\).

Step 1: Understanding the Absolute Value

The absolute value of a real number \(a\), denoted \(|a|\), is defined as the non-negative value of \(a\). Essentially, \(|a|\) is equal to \(a\) if \(a\) is positive or zero, and \(-a\) if \(a\) is negative.

Step 2: Considering Case 1: When \(a\) is Non-Negative

For some \(a \geq 0\), we know \(|a| = a\). Thus, \(-|a| = -a\). Consequently, the inequality simplifies to \(-a \leq a \leq a\), which holds true because \(-a\) is non-positive.

Step 3: Considering Case 2: When \(a\) is Negative

For some \(a < 0\), we know \(|a| = -a\). Therefore, \(-|a| = -(-a) = a\). The inequality becomes \(a \leq a \leq -a\), which is clearly satisfied because the first part \(a \leq a\) holds, and as \(a\) is negative, \(a < -a\), fulfilling \(a \leq -a\).

Step 4: Combining Both Cases

Both conditions prove that in either scenario (whether \(a\) is non-negative or negative), the inequality \(-|a| \leq a \leq |a|\) holds true. We conclude that the statement is valid for any \(a \, \in \, \mathbb{R}\).

Key Concepts

Absolute Value

The concept of absolute value is central to understanding inequalities in mathematics, especially when dealing with real numbers. The absolute value of a real number \( a \), denoted as \( |a| \), represents the distance of \( a \) from zero on the real number line. This distance is always non-negative.

• When \( a \) is positive or zero, the absolute value of \( a \) is simply \( a \) itself, i.e., \( |a| = a \).
• If \( a \) is negative, the absolute value becomes the negation of \( a \), which turns it into a positive quantity: \( |a| = -a \).

Understanding absolute value is crucial because it defines the bounds on the number line from zero to the point \( a \). This concept is often used in mathematical inequalities and is foundational in determining ranges and expressing distances among numbers.

Inequalities in Mathematics

Inequalities provide a way to express the size or order relations between numbers. They state that one value is greater or lesser, or at times equal to another. In mathematics, common symbols used are \(<, \leq, >, \geq\). An inequality is what connects these values within a certain constraint, often graphically represented on a number line.

• A basic inequality form we encounter here with absolute values is that \(-|a| \leq a \leq |a|\).
• This statement essentially says that for any real number \( a \), its absolute value serves as a boundary, meaning \( a \) cannot be greater than or less than its own absolute value.

These inequalities require considering different cases based on whether \( a \) is positive, negative, or zero. Inequality statements are pivotal because they help describe intervals, solve equations, and understand the feasibility of solutions in real problems.

Real Numbers

Real numbers \( \mathbb{R} \) encompass a vast set that includes all rational and irrational numbers. This set forms a continuous line, often referred to as the real number line. Each point on this line correspondingly represents a real number, and this continuity is what allows for a comprehensive examination of properties like absolute value and inequalities. Real numbers can be categorized as:

• Rational numbers: Numbers that can be expressed as a fraction, \( \frac{p}{q} \), where \( p \) and \( q \) are integers and \( q eq 0 \). Examples include 1/2, 3, -4.
• Irrational numbers: Numbers that cannot be written as a simple fraction, whose decimal expansions are non-repeating and non-terminating. Examples include \( \sqrt{2} \), \( \pi \), and \( e \).

Understanding real numbers is fundamental, as it forms the basis for calculus, analysis, and various applications within mathematics. They are used to measure continuous quantities and are essential in ensuring that inequalities cover every possible scenario when providing a range for a variable like \( a \).