Problem 8 - A Primer of Real Analysis

Question

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

Step-by-Step Solution

Answer

For any rational number \(a\), the inequality \(-|a| \leq a \leq |a|\) holds true.

Step 1: Understanding Absolute Value

The absolute value of a number, denoted as \[|a|,\]is the distance from zero on the number line. For any rational number \(a\), the absolute value \(|a|\) is always non-negative. It can be defined piecewise as follows:\[|a| = \begin{cases} a, & \text{if } a \geq 0; \ -a, & \text{if } a < 0. \end{cases} \]

Step 2: Establish Inequality Conditions

Let's consider both possible cases for \(a\): when \(a\) is non-negative (\(a \geq 0\)) and when \(a\) is negative (\(a < 0\)).

Step 3: Case 1: Non-negative \(a\)

When \(a \geq 0\), we know:\[ |a| = a. \]In this case, we need to show:\[-|a| \leq a \leq |a|\]which translates to:\[-a \leq a \leq a\].The first part \(-a \leq a\) simplifies to \(0 \leq 2a\), which is true as \(a \geq 0\). The second part \(a \leq a\) is trivially true.

Step 4: Case 2: Negative \(a\)

When \(a < 0\), we know: \[|a| = -a.\]In this situation, we need to verify:\[-|a| \leq a \leq |a|,\]which translates to:\[a \leq a \leq -a.\]The first part \(a \leq a\) is obvious. Now, the second part \(a \leq -a\) simplifies to \(0 \leq 2a\).Since \(a < 0\), multiplying by 2 yields \(2a < 0\), confirming the condition as all negative numbers satisfy the inequality.

Step 5: Combining Results for All \(a\) in \(\mathbb{Q}\)

As both cases - when \(a\) is non-negative and when \(a\) is negative - satisfy the inequality:\[-|a| \leq a \leq |a|,\]we conclude that the statement holds true for any rational number \(a\).

Key Concepts

Absolute Value

Absolute value is an essential concept in mathematics. It represents the "size" or "magnitude" of a number regardless of its sign. In simpler terms, absolute value measures how far a number is from zero on a number line without considering which direction it goes. For example:

• The absolute value of a positive number is just the number itself, since it is already greater than zero.
• For a negative number, the absolute value is the number's opposite (positive) counterpart. This means the negative sign is removed.

Mathematically, the absolute value of a number \(a\), written as \(|a|\), is defined as:- \( a \) if \( a \geq 0 \)- \( -a \) if \( a < 0 \)This simple piecewise function makes it clear that no matter what rational number you choose, its absolute value will always be non-negative—it is either positive or zero. By understanding and correctly using absolute values, we can solve many mathematical problems involving distance and magnitude.
Absolute value provides a solid foundation for tackling a variety of mathematical concepts, such as inequalities and number lines, which are just a few taps away as you continue learning.

Inequality

Inequality is a fundamental part of mathematics that involves finding the relationship between numbers, expressions, or functions that are not equal. It describes the way in which one quantity is larger or smaller than another and is key to understanding and solving mathematical problems that don't have exact equality.

In this context, let's break down the statement \[-|a| \leq a \leq |a|\] and understand its importance:

• When \(a\) is positive (\(a \geq 0\)), consider that the absolute value \(|a|\) equals \(a\) itself. Here, the inequality becomes \(-a \leq a \leq a\), which simplifies to \(0 \leq 2a\). Clearly, this is always true for any non-negative number.
• When \(a\) is negative (\(a < 0\)), \(|a|\) becomes \(-a\). In this case, the inequality becomes \(a \leq a \leq -a\), translating to \(0 \leq 2a\). Since multiplying a negative number by two still results in a negative, this also holds true.

These inequalities show that for any rational number \(a\), the number itself is always sandwiched between its own negative absolute value and positive absolute value. Understanding inequalities helps in visualizing and conceptualizing complex mathematical relationships.
With practice, mastering inequalities will make solving mathematical problems much more intuitive.

Number Line

The number line is a visual tool that helps us understand numbers better, especially when dealing with concepts like absolute values and inequalities. It is a horizontal line that stretches in both directions, showing positive numbers to the right of zero and negative numbers to the left.

When working with absolute values, the number line helps us to clearly see how far a number is from zero, irrespective of its direction. Here's how:

• For any rational number \(a\), its absolute value \(|a|\) represents its distance from zero, which is always a non-negative number on the line.
• The inequality \(-|a| \leq a \leq |a|\) graphically demonstrates that \(a\) is always positioned somewhere between the negative and positive absolute values of \(a\) on the line.

Using a number line helps to visualize the position of numbers relative to each other and can be a powerful tool in solving mathematical problems. It simplifies the understanding of how different numbers, absolute values, and inequalities relate to one another.
With practice, number lines provide an intuitive hint to solving equations and developing spatial understanding of numerical concepts.