Problem 73 - Prealgebra

Question

If \(n\) is a negative number, is \(-n\) positive or negative?

Step-by-Step Solution

Answer

\(-n\) is positive if \(n\) is negative.

Step 1: Understanding the Problem

We are given that \( n \) is a negative number and need to determine the sign of \( -n \). A negative number is any number less than zero. We will analyze what happens when we multiply the negative number by -1.

Step 2: Apply the Concept of Multiplication by -1

Multiplying a negative number by -1 changes its sign. This is because the operation \( -n = (-1) imes n \) reflects the number across zero on the number line, moving it from a negative to a positive.

Step 3: Determine the Sign of -n

Since multiplying \( n \) (which is negative) by -1 changes the sign, \(-n\) will be a positive number. This is because the sign of a number changes when it is multiplied by -1.

Key Concepts

Negative Numbers

Negative numbers are numbers less than zero. They lie to the left of zero on the number line. You can think of negative numbers as representing things like debt or a loss of points. Often denoted with a minus sign (e.g., -3), they are the opposite of positive numbers, which are greater than zero and lie to the right.
Working with negative numbers can sometimes be tricky, as they have properties that might be unfamiliar. For example:

• Adding two negative numbers results in a more significant negative number.
• Subtracting a negative number is like adding its positive counterpart.

Becoming comfortable with the behavior of negative numbers on the number line helps in visualizing their transformations.

Multiplication by -1

Multiplying by -1 is a special operation that reverses the sign of a number. It's like flipping the direction from which you view the number on the number line. For example, if you have -4 and multiply it by -1, you get 4. The negative sign moves from one side of zero to the other.
This function of changing signs is because:

• Multiplying a positive number by -1 results in a negative number.
• Multiplying a negative number by -1 results in a positive number.

So, when multiplying any number by -1, assess the number's original sign to determine the outcome.

Number Line

A number line is a visual representation that helps us understand numbers' positions and their relationships to each other. It is a straight line where numbers are placed at equal intervals, where 0 is typically in the center. On the number line:

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

Understanding the number line helps us visualize operations like addition, subtraction, and multiplication. For instance, when you multiply a number by -1, on the number line, it mirrors across zero, showing the change from one side (positive or negative) to the opposite side.