Problem 25 - Fundamentals of Mathematics

Question

Determine each of the values. $$ |-26| $$

Step-by-Step Solution

Answer

The value is 26.

Step 1: Understand the Absolute Value

Absolute value refers to the distance of a number from zero on the number line, without considering which direction from zero the number lies. It is always non-negative.

Step 2: Apply the Absolute Value to -26

Since the absolute value is the distance from zero, take the positive counterpart of -26. Thus, \(|-26| = 26\).

Key Concepts

distance on the number line

When we talk about distance on the number line, we are referring to how far a number is from zero, regardless of its direction. This is essential in understanding absolute value. Imagine standing on a number line at zero, the center point. From here, you can move in both the positive direction (right) and the negative direction (left).

The distance is how many steps you take from zero. For instance, if you move to -26, you've taken 26 steps to the left. On the number line, only the number of steps—your distance from zero—matters. Therefore, no matter if you move left (negative) or right (positive), the distance is simply 26. This is why the absolute value is always shown as a positive number because it reflects just the distance, not direction.

positive counterpart

A positive counterpart refers to the positive equivalent of a number. The concept of absolute value emphasizes this, as it represents just the size of the number without negative or positive attributes.

When dealing with absolute value, you disregard the negative sign and use the positive counterpart of the number. In the case of -26, its positive counterpart is 26. This is because -26 is 26 steps away from zero on the number line. The concept of a positive counterpart is crucial because absolute value is all about measuring size, not direction.

Using absolute value transforms numbers into their positive versions, like turning -26 into 26. This gives a consistent measure of magnitude, vital in many mathematical calculations.

non-negative value

A non-negative value is a number that is zero or greater. It’s an important concept because it underlines the idea that absolute values are never negative. On a number line, whether you are on the left side (negative) or the right side (positive), the absolute value is the same.

The absolute value of any real number is always non-negative. For example, |-26| becomes 26 because it’s all about the distance from zero. Distances—like 26 steps from zero—cannot be negative, no matter the direction traveled on the number line.

This idea ensures that the absolute value function returns reliable results that don't depend on the sign of the number. Whether you begin with -26 or +26, the absolute value is always 26, emphasizing that distance is inherently non-negative.