Problem 61 - Algebra and Trigonometry

Question

\(61-66=\) 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: Understand Absolute Value

The absolute value of a number is its distance from 0 on the number line, regardless of direction. It is always a non-negative number. This concept can be represented as \(|a|\), which means the absolute value of \(a\).

Step 2: Calculate \(|100|\)

The number 100 is positive, so its absolute value is the same as the number itself. Thus, \(|100| = 100\).

Step 3: Calculate \(|-73|\)

The number -73 is negative, so its absolute value is its positive counterpart. Therefore, \(|-73| = 73\).

Key Concepts

Number Line

Imagine a long, straight line with numbers placed at equal intervals along it. This is called a number line. The idea is pretty simple: it helps us to visually represent numbers and their relationships.

• The number line extends infinitely in both directions - to the right for positive numbers and to the left for negative numbers.
• The middle point, or the starting point on this line, is 0. This is an important reference point because it separates positive numbers from negative numbers.

On the number line, the absolute value of a number is essentially the distance from that number to 0 without considering the direction. For example, if you're standing at -73, you'd take 73 steps to reach 0. And if you're at 100, you still need 100 steps to reach 0. This illustrates that absolute value always gives you a non-negative outcome since it reflects distance, which can't be negative.
Visualizing these concepts on a number line helps understand absolute values better.

Positive Number

Let's talk about positive numbers. These are the numbers you would find to the right of 0 on the number line. Positive numbers are greater than 0 and indicate values larger than nothing.

• Examples of positive numbers include 1, 2, 50, and 100.
• They are generally referred to as natural numbers when considering whole values, like counting numbers.

A crucial point is that the absolute value of a positive number is the number itself. For instance, the absolute value of 100 is 100. This happens because the distance to 0 doesn't change the direction for positive numbers.
Therefore, positive numbers in the absolute value context stay unaltered since they are already non-negative.

Non-Negative Number

Non-negative numbers are numbers that are either greater than or equal to 0. This includes all whole numbers as well as 0 itself. It's like a broader category that contains positives but also includes neutral ground - 0.

• Examples are 0, 0.5, 3, and 15.
• 0 is special because it is neither positive nor negative but is still non-negative.

When we talk about absolute values, we only ever end up with non-negative numbers. This is because absolute value measures distance, and like the number line shows, distance never has a negative value.
Thus, whether starting with a negative, like -73, or a positive number like 100, calculating absolute value will always keep you in the realm of non-negative numbers, simplifying many calculations and concepts in mathematics.