Problem 37 - Fundamentals of Mathematics

Question

Determine each value. $$ |9| $$

Step-by-Step Solution

Answer

The absolute value of 9 is 9.

Step 1: Understanding Absolute Value

The absolute value of a number is its distance from zero on a number line, without considering direction. Absolute value is always non-negative.

Step 2: Evaluate the Absolute Value

To find the absolute value of any number, simply remove any negative sign that might be in front of the number. For a positive number, the absolute value is the number itself.

Step 3: Apply the Concept

Given the number is 9, which is already positive, its absolute value is the same. Therefore, the absolute value of 9 is 9.

Key Concepts

Positive Numbers

In mathematics, positive numbers are numbers greater than zero. They can appear with or without a "+" sign, but by default, they are assumed to be positive if no sign is shown. Positive numbers are primarily used to represent quantities with physical meaning, like length, time, money, and more. They are located to the right of zero on a number line.

• For example, if you have 5 apples, this can be expressed with the positive number 5.
• Positive numbers add up in straightforward ways; for instance, 2 + 3 equals 5.

Understanding positive numbers is key when dealing with absolute values because the concept involves turning any number into its non-negative counterpart. For positive numbers, this means they remain unchanged when their absolute value is considered.

Number Line

A number line is a straight, horizontal line that visually represents numbers as points along its length. The center point of a number line is zero, and numbers are evenly spaced on either side. It is a simple yet powerful tool to understand numerical concepts.

• To the right of zero, numbers increase in positive magnitude (1, 2, 3, ...).
• To the left of zero, numbers decrease and are negative (-1, -2, -3, ...).

Using a number line helps in seeing why the absolute value of a number is its distance from zero. When we say that the absolute value of 9 is 9, it’s because 9 units separate the number 9 from zero on this line. The direction doesn't matter, only the distance.

Distance from Zero

Distance from zero is a concept tied to absolute value. It measures how far a number is from zero on a number line, ignoring its direction. This is why absolute values are always non-negative. Even if you're dealing with negative numbers, like -5, its distance from zero is a positive one (5 units away).

• To find the distance from zero: treat both positive and negative numbers in the same way. They both end up as the same non-negative absolute value.
• This also shows how absolute values are useful in real-world problems. Picture negative temperatures where conduction usually involves positive numbers for easier computation.

In mathematical operations, knowing the distance from zero helps in comparisons, calculations, and understanding the overall magnitude of numbers without worrying about which side of zero they are on.