Problem 35 - Prealgebra

Question

Find each of the following absolute values. $$|2|$$

Step-by-Step Solution

Answer

The absolute value of 2 is 2.

Step 1: Understanding Absolute Value

The absolute value of a number is its distance from zero on the number line, regardless of direction. This means the absolute value is always non-negative.

Step 2: Evaluate the Expression

To find the absolute value of 2, consider the distance of 2 from zero on the number line. Since 2 is already positive, its absolute value is the same as the number itself.

Step 3: Write Down the Result

Based on the evaluation, we can state that \(|2| = 2\).

Key Concepts

Number Line

The number line is a straight line where every point corresponds to a real number, placed in a sequential manner. It's like a ruler that helps you visualize numbers. The center point of this line is always zero. Positive numbers are found to the right of zero, and negative numbers are to the left. This makes it easier to see the relationship between numbers, especially when considering absolute value.

Visualizing numbers on a number line is beneficial because it gives you a clear sense of order and magnitude. It helps in understanding many mathematical concepts, including the idea of absolute value. By seeing where a number is located, you can quickly determine its distance from zero, an essential step in calculating absolute values.

Distance from Zero

The concept of 'distance from zero' is crucial in understanding absolute value. Imagine standing at zero on the number line, and you take steps toward any number; the amount of steps you take is the distance from zero. Absolute value is concerned with this distance, not with which direction you travel.

• If a number is positive, its distance from zero is simply its value.
• If a number is negative, its distance from zero is its positive equivalent.

For example, both the numbers 2 and -2 are 2 units away from zero, so their absolute values are equal, \(|2| = |-2| = 2\). This is why absolute value always remains non-negative—it focuses only on the distance, never direction.

Non-Negative Numbers

Non-negative numbers include all positive numbers and zero. These numbers are important in absolute values because they define the result of any absolute value operation. The absolute value of any real number is always a non-negative number.

• For a positive number like 5, its absolute value is 5 because it's already non-negative.
• For a negative number like -5, its absolute value changes to 5, making it non-negative.
• For zero, the absolute value remains zero, which is inherently non-negative.

Understanding that absolute value leads to non-negative results helps build the foundation for solving complex equations where knowing the sign of a result is critical. The emphasis on non-negative outcomes ensures a consistent and predictable way to measure distance from zero on the number line.