Problem 10 - Fundamentals of Mathematics

Question

Use the algebraic definition of absolute value to find the following values. $$ |-5| $$

Step-by-Step Solution

Answer

The absolute value of \(-5\) is 5.

Step 1: Understanding Absolute Value

The absolute value of a number is its distance from zero on the number line, without considering which direction from zero the number lies in. It is always a non-negative number.

Step 2: Applying the Definition to -5

The algebraic definition says that for any real number \( x \), the absolute value \( |x| \) is defined as:\[ |x| = \begin{cases} x & \text{if } x \geq 0, \ -x & \text{if } x < 0 \end{cases} \]Since \(-5\) is less than 0, we apply the second case: \(|-5| = -(-5)\).

Step 3: Calculating Result

According to the applied definition, we now compute: \(-(-5) = 5\). Thus, the absolute value of \(-5\) is 5.

Key Concepts

Real Numbers

Real numbers are an essential concept in mathematics and encompass all the numbers that we use in day-to-day life. These include:

• Natural numbers like 1, 2, 3, and so on
• Whole numbers, which are natural numbers including zero
• Integers such as -1, -2, 0, 1, 2
• Rational numbers, which are fractions like \( \frac{1}{2} \) or \( -\frac{2}{3} \)
• Irregular numbers known as irrational numbers, such as \( \sqrt{2} \) or \( \pi \)

Real numbers can be thought of as points on an infinitely long number line. This line contains all sorts of numbers, represented continuously without any breaks. They play a crucial part in understanding mathematical concepts, including absolute value. Absolute value treats both positive and negative real numbers similarly in terms of distance from zero.

Number Line

The number line is a mathematical tool that gives us a visual representation of real numbers. Imagine a straight horizontal line extending in both directions.
The center of this line is zero. This zero point acts as a reference for understanding absolute value.

We use this line to depict:

• Positive numbers, which are on the right side of zero
• Negative numbers, which are to the left of zero

To find the absolute value of a number, we measure the distance from zero. The direction does not matter. For instance, both +5 and -5 are five units away from zero, hence both have the same absolute value. This shows how the number line helps us visualize the concept of absolute value.

Non-Negative Numbers

Non-negative numbers are pretty straightforward—they're all the numbers that are zero or positive. In other words, they don’t have a negative sign. These numbers include:

• Zero itself
• Positive integers, like 1, 2, 3, etc.
• Positive fractions and decimals, such as \( 1.5 \) or \( \frac{3}{4} \)

The absolute value of a real number is always a non-negative number. This is because the absolute value represents distance. And distance in mathematics does not have a direction. Therefore, it cannot be negative. Whether we apply absolute value to -5 or +5, both result in the non-negative number 5. Understanding non-negative numbers helps clarify why the absolute value strips away the sign of a number.