Problem 65 - Elementary Algebra

Question

Write the algebraic definition of the absolute value of the number \(a\).

Step-by-Step Solution

Answer

Answer: The algebraic definition of the absolute value of a number \(a\) is written as \(|a|\), and it can be defined as: $|a| = \begin{cases} a & \text{if } a \geq 0\\ -a & \text{if } a < 0 \end{cases}$ This means that the absolute value of \(a\) is equal to \(a\) itself if \(a\) is greater than or equal to zero, and it is equal to \(-a\) if \(a\) is less than zero.

Step 1: Recall the definition of absolute value

The absolute value of a number is its distance from 0 (zero) on the number line. It is always positive or equal to zero.

Step 2: Write down the algebraic definition of absolute value

The algebraic definition of the absolute value of a number \(a\) is written as \(|a|\). To define it algebraically, we must consider when \(a\) is positive, negative, or zero. The absolute value of \(a\) can be defined as: $|a| = \begin{cases} a & \text{if } a \geq 0\\ -a & \text{if } a < 0 \end{cases}$ In other words, the absolute value of \(a\) is equal to \(a\) itself if \(a\) is greater than or equal to zero, and it is equal to \(-a\) if \(a\) is less than zero.

Key Concepts

Algebraic Definition

The algebraic definition of absolute value provides clarity on how to handle numbers of any kind using mathematical symbols. Absolute value, denoted as \(|a|\), represents the distance of a number \(a\) from zero. This distance is always considered as non-negative. Hence, it simplifies mathematical problems by focusing solely on the magnitude of numbers, without considering their direction.

• For any positive number or zero, the absolute value is the number itself.
• For any negative number, the absolute value is the number made positive (multiplied by -1).

By using the piecewise function: \[|a| = \begin{cases} a & \text{if } a \geq 0\-a & \text{if } a < 0\end{cases}\]we cover all possible real numbers. This mathematical rule makes absolute values reliable in calculations related to distances or magnitudes.

Number Line

A number line is a simple, visual representation of numbers positioned at equal intervals along a straight line. This tool is essential in understanding absolute value, as it clearly illustrates how far a particular number is from zero, regardless of its sign.
On a number line:

• Zero is the central point, usually marked clearly to discern positive and negative ends.
• Positive numbers extend to the right, increasing away from zero.
• Negative numbers extend to the left, decreasing away from zero.

When considering absolute value, imagine measuring distance from the zero point. Whether a number lies to the left or right on the number line, the distance is always positive and represents the absolute value. For instance, both \(+3\) and \(-3\) are three units away from zero, and thus, have an absolute value of \(|3|\). This visualization solidifies the concept that absolute value is about distance, not direction.

Positive and Negative Numbers

Understanding positive and negative numbers is foundational when working with absolute values. These numbers not only signify direction relative to zero but impact how absolute value is calculated.

• Positive numbers, found to the right of zero, naturally result in positive absolute values, simply because they are already at the actual distance measurement we need.
• Negative numbers, found to the left of zero, also result in positive absolute values. This may seem counterintuitive but remember, absolute value is about distance, not the direction. To convert a negative number to its absolute value, simply remove the negative sign.
• Zero is special because it is neither positive nor negative. Its absolute value is zero because it is perfectly at the origin of our number line.

By recognizing the role of positive and negative numbers, it's easier to apply the algebraic definition of absolute value and solve equations effectively. Emphasizing the distance from zero, rather than the sign, helps simplify calculations and aids in understanding fundamental algebraic principles.