Problem 19 - Fundamentals of Mathematics

Question

Determine each of the values. $$ |-1| $$

Step-by-Step Solution

Answer

The absolute value of \(-1\) is 1.

Step 1: Understand Absolute Value

The absolute value of a number is the distance of that number from zero on the number line, without considering direction. It is always a non-negative number. For any real number \( x \), the absolute value is denoted by \( |x| \).

Step 2: Determine Absolute Value of the Given Number

Consider the given expression \(|-1|\). Since absolute value measures distance from zero, and distances are always positive (or zero), the absolute value of \(-1\) is the positive version of \(-1\).

Step 3: Apply Absolute Value Rule

For a negative number \( x \), \(|x| = -x\). Therefore, for our case: \(|-1| = -(-1) = 1\). The absolute value of \(-1\) is \(1\).

Key Concepts

Distance from Zero

When dealing with absolute value, the concept of "distance from zero" is crucial. Imagine you are standing on a number line. The absolute value of a number tells you how far away you are from the zero point, ignoring which direction you are facing.

Absolute values are always non-negative since distance can't be negative. Thus, when we calculate the absolute value of \(-1\), we find that \(-1\) is one unit away from zero. This is why \(|-1| = 1\).

Understanding this helps to simplify absolute value calculations. Whether you are dealing with \(-1\) or \(+1\), both have the same distance from zero. Their absolute values are equal.

Number Line

The number line is a visual representation that helps us understand numbers and their absolute values. It is a horizontal line with zero placed at the center.

Positive numbers are located to the right of zero, while negative numbers are to the left. When you think about absolute values, imagine measuring the distance of a point from zero along this line.

• For example, the point \(-1\) lies one step to the left of zero. Its distance from zero is just the number of steps away, irrespective of the direction.
• Positive numbers like \(+1\) lie one step to the right and have the same distance from zero as \(-1\), yielding an absolute value of 1.

This visualization makes it clear that absolute values reflect distance, not orientation on the number line.

Non-negative Number

The absolute value of any real number, whether positive or negative, is always a non-negative number. The concept of non-negative numbers means numbers that are either positive or zero.

When we take the absolute value of a number, we are stripping it of any negative sign. This is because absolute values focus solely on how far a value is from zero without regard to direction.

• If you take the absolute value of -5, it becomes 5, because it is 5 units away from zero.
• The absolute value of \(3\) is also \(3\), as it is already a non-negative number.

This universal approach works for all real numbers. The concept ensures we avoid negative values when measuring distances, aligning with the nature of distance as a non-negative attribute.