Problem 9 - Fundamentals of Mathematics

Question

Find each value. |-16|

Step-by-Step Solution

Answer

The value is 16.

Step 1: Understand Absolute Value

The absolute value of a number is the distance between that number and zero on a number line, without considering direction. It is always a non-negative value.

Step 2: Apply Absolute Value Definition to -16

Identify the number given, which is -16, and calculate its absolute value. The distance from -16 to 0 is 16, regardless of its negative sign.

Step 3: Conclusion

Thus, | -16 | = 16, since absolute value measures distance, which is always positive or zero.

Key Concepts

Understanding the Number Line

A number line is a visual representation where we arrange numbers in order from least to greatest, usually from left to right. Each position on this line corresponds to a number. The center of a basic number line is usually the number zero.
It's a simple but powerful tool in mathematics. On the number line:

• Numbers to the right of zero are positive.
• Numbers to the left of zero are negative.

These positions help us to easily visualize the concept of absolute value. The absolute value of any number is essentially the distance of that number from zero on the number line. For example, both -16 and 16 have an absolute value of 16, because each is 16 units away from zero.

Non-Negative Values

Non-negative values are numbers that are either zero or positive. In the context of absolute value, this is a crucial concept. Absolute value is always non-negative because it measures the distance from zero, and distance cannot be negative.
When you compute the absolute value of a number, you are essentially removing any negative sign, focusing only on the magnitude of the number:

• For a positive number, its absolute value is the number itself. For example, |16| is 16.
• For a negative number, its absolute value is the positive counterpart. As in our exercise, |-16| is 16.
• The absolute value of zero is zero, because zero is 0 units away from itself on the number line.

Mathematical Operations with Absolute Value

In mathematics, operations involving absolute values are handled with care due to their nature of converting negative values into non-negative ones.
Here's how you often use absolute values in operations:

• When comparing numbers, absolute values help determine magnitudes rather than signs.
• Equations may involve absolute values, where solving them requires considering both the positive and negative scenarios.
• Inequalities with absolute values often need splitting into two separate inequalities for different scenarios.
• When adding or subtracting absolute values, ensure that the results make sense by aligning this operation within the context of the absolute value properties.

Through understanding these operations, you can solve more complex problems involving absolute values, enhancing your mathematical reasoning and skills.