Problem 53 - Algebra and Trigonometry

Question

Evaluate each expression. $$ \begin{array}{ll}{\text { (a) }|100|} & {\text { (b) }|-73|}\end{array} $$

Step-by-Step Solution

Answer

(a) 100, (b) 73.

Step 1: Understanding Absolute Values

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

Step 2: Evaluate |100|

To find the absolute value of 100, consider how far it is from zero on the number line. Because 100 is positive and is 100 units away from zero, \(|100| = 100\).

Step 3: Evaluate |-73|

The number -73 is 73 units away from zero on the number line, disregarding the negative sign. Therefore, the absolute value is \(|-73| = 73\).

Key Concepts

Number Line

The number line is a visual representation of numbers arranged in order, used to illustrate concepts such as positive and negative numbers, as well as absolute values. Imagine a straight horizontal line where each point corresponds to a real number. Numbers to the right of zero are positive, and numbers to the left are negative. Zero sits at the center, acting as a neutral point between positive and negative numbers.
The number line is incredibly useful for understanding distances between numbers. When we talk about distance on a number line, we are often referring to the concept of absolute value. Each number on the line is equidistant from its neighboring numbers by a value of one unit. This makes the number line a handy tool for visual learners who want to grasp the concept of distance in a straightforward manner.

Positive and Negative Numbers

Positive numbers are those greater than zero, and they appear on the right side of a number line. Negative numbers, on the other hand, are less than zero and appear on the left side. Both types of numbers indicate magnitude but in opposite directions. For instance, 5 is positive and -5 is negative, existing on opposite sides of zero with equal distance.

• Zero is neither positive nor negative; it's neutral.
• Positive numbers are represented without a sign or with a plus (+).
• Negative numbers always have a minus (-) sign.

Understanding positive and negative numbers helps in determining absolute values. For absolute value, the sign is disregarded, focusing solely on the distance from zero.

Distance from Zero

Distance from zero is a core concept related to absolute value. It measures how far a number is from zero on a number line, without considering direction. This is why absolute values are always non-negative.
The concept can be illustrated by considering both positive and negative numbers:

• The distance from 0 to 3 is 3, so \( |3| = 3 \).
• The distance from 0 to -5 is also 5, even though -5 is negative, so \( |-5| = 5 \).

The absolute value signs remove the negative aspect, effectively turning the number into its positive counterpart. By focusing on distance alone, absolute value provides a uniform measure that simplifies calculations and helps compare quantities regardless of direction on the number line.