Problem 41 - Fundamentals of Mathematics

Question

Determine each value. $$ -|-2| $$

Step-by-Step Solution

Answer

The value is -2.

Step 1: Understand the Absolute Value

The absolute value of a number is its distance from zero on the number line, without regard to direction. In mathematical terms, for any number \( x \), ~~it is represented by \(|x|\). The absolute value of a positive number is the same number, and the absolute value of a negative number is the same number but positive.

Step 2: Calculate the Inner Absolute Value

We first calculate \(|-2|\). Since \(-2\) is negative, its absolute value is the positive version of \(-2\), which is 2. Therefore, \(|-2| = 2\).

Step 3: Apply Negative Sign to the Result

Now that we have calculated \(|-2| = 2\), we apply the negative sign from the original expression. Thus, we have \(-|2| = -2\).

Key Concepts

Understanding the Number Line

A number line is a visual representation that helps us understand numbers, their order, and their relationships. Picture it like a straight horizontal line. In the center of this line is the number zero. To the right are positive numbers, like 1, 2, 3, and so on. Each step to the right increases the number value.
On the other hand, to the left of zero, you'll find negative numbers, like -1, -2, -3, stretching endlessly. As you go left, the numbers get smaller. A number line with its separation of positive and negative numbers is a simple yet powerful tool.
It visually represents mathematical concepts and helps with calculations like absolute value, ensuring we clearly see the distance of any number from zero.

Exploring Negative Numbers

Negative numbers are numbers less than zero. They appear on the left side of the number line. These numbers are commonly used in real-life situations like debts or temperatures below zero.
Negative numbers have a distinctive property that makes them unique: when added to their positive counterpart, they zero each other out. For instance, -2 + 2 = 0.

• Knowledge of negative numbers helps in comprehending arithmetic operations such as subtraction and addition spanning zero.
• They are also important when dealing with absolute value, where signs switch to demonstrate distance, but not direction.

Understanding negatives helps us grasp broader concepts in mathematics, like absolute value used in this exercise.

Clarifying Mathematical Terms

Mathematical terms can sometimes sound intimidating, but they are simpler than they seem. In this exercise, absolute value is a key term. It refers to the distance a number is from zero on a number line, without looking at direction. For instance, the absolute value of -2 is simply 2.
When we say distance, think of it as how far you are from a point, zero in this case, without worrying if you're to the left or right.

• Absolute value is always positive, since distance cannot go wrong way.
• The symbol |x| represents absolute value, where x is any number.

This concept is central in helping us understand expressions like -|-2|, where we apply operations after finding the absolute value of a number.