Problem 30 - Fundamentals of Mathematics

Question

Determine each of the values. $$ -(-|-42|) $$

Step-by-Step Solution

Answer

-42

Step 1: Understand Absolute Value

The absolute value of a number is the distance of that number from zero on the number line, without considering the direction. It is always a non-negative number. So, to find \(-|-42|\), we first determine \(|-42|\).

Step 2: Calculate the Absolute Value

The absolute value of \(-42\) is 42 because we ignore the negative sign and take the magnitude. Hence, \(|-42| = 42\).

Step 3: Handle the Negation

Now, we handle the outer negative sign. We need to negate the result of the absolute value, which means we find \(-42\) when we apply the negation to the result from Step 2.

Step 4: Conclusion: Final Result

Therefore, the expression \(-(-|-42|)\) simplifies and evaluates to \(-42\).

Key Concepts

Negation

When we talk about negation in mathematics, we often refer to the process of reversing the sign of a number. This means turning a positive number into its negative counterpart or vice versa. For example:

• The negation of 5 is -5.
• The negation of -7 is 7.

In our exercise, after calculating the absolute value, we encountered an outer negative sign. This negation reverses the sign of the result from the absolute value calculation. So, if the absolute value gives us a positive number, applying negation switches it to negative. Understanding negation is crucial because it allows us to transform numbers depending on the context needed, like switching a financial gain to a loss or a temperature rise to a drop. This concept plays a key role in understanding expressions involving multiple layers of operations, such as our example of \( -(-|-42|) = -42.\)

Number Line

A number line is a simple and effective way to visually represent numbers. It's a horizontal line where:

• Zero is positioned at the center.
• Positive numbers are placed to the right of zero.
• Negative numbers stretch to the left of zero.

Using a number line helps in understanding absolute value because it shows the distance of any number from zero. For example, both -3 and 3 are three units away from zero, helping us see why their absolute values are equal. It is a handy tool for visualizing basic arithmetic operations, including addition and subtraction, which involve moving along this line. When we see expressions involving absolute values or negation, like in this exercise, imagining them on a number line can make these concepts more intuitive. It aids in recognizing that negation is like flipping positions over the zero point, switching sides, and thereby altering the direction or sign of the number.

Negative Numbers

Negative numbers are numbers less than zero, and they are usually represented with a minus sign. They are an essential part of mathematics, enabling us to express loss, deficiency, and reversal of direction among other things. They appear leftward on the number line and have the following properties:

• When added to a positive number, they can reduce the total, potentially resulting in zero, or another negative number.
• Subtracting a negative number is effectively adding the positive equivalent.
• Multiplying or dividing two negative numbers results in a positive number. However, mixing negative and positive numbers in multiplication or division yields a negative outcome.

In the context of our exercise, -42 represents a value below zero. The expression \(-(-|-42|)\) shows how the concepts of negative numbers and negation can interact. First, the absolute value operation turns -42 into 42, which is positive. The final negation then transforms this result back into -42, demonstrating that handling negative numbers often involves layers of operations that are conceptually simple but powerful.