Problem 74 - Prealgebra

Question

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

Step-by-Step Solution

Answer

-n is negative because it is the opposite of positive n.

Step 1: Understand the Problem

We are given a positive number represented by \(n\), and we need to determine if its opposite, \(-n\), results in a positive or negative number.

Step 2: Evaluate the Properties of Opposites

The opposite of any number is that number with its sign flipped. If a number is positive, its opposite is negative, and vice versa.

Step 3: Apply the Concept to \(n\)

Since \(n\) is a positive number, according to the properties of numbers, \(-n\) is the opposite and hence is negative.

Key Concepts

Opposite Numbers

When we talk about opposite numbers, we mean numbers that have the same value but different signs. Let's break it down a bit more.
For any given number, its opposite is simply obtained by changing its sign. Here's what this looks like:

• If you start with a positive number, its opposite is negative.
• Conversely, if you start with a negative number, its opposite will be positive.

To determine the opposite of a given number, you can think of a horizontal number line. Standing at a point for a number, its opposite is equidistant from zero but in the reverse direction.
Understanding opposites helps you to quickly determine how one value relates to its counterpart.

Properties of Numbers

Numbers possess several properties that form the basis for arithmetic operations. Here’s an overview:

• Additive Inverse: Every number has an additive inverse, which is its opposite. The sum of a number and its additive inverse is zero, expressed as: \[ n + (-n) = 0 \]
• Multiplicative Identity: For any number, multiplying by 1 keeps the number unchanged:\[ n imes 1 = n \]
• Commutative Property of Addition: You can add numbers in any order, and you'll get the same result:\[ a + b = b + a \]

Recognizing these properties can simplify problem-solving and improve your ability to manipulate numbers within expressions and equations.

Positive Numbers

Positive numbers are numbers that are greater than zero. They are located on the right side of zero on the number line. Understanding positive numbers is essential because:

• They represent quantities that can accumulate or grow, such as money or population.
• Basic operations like addition and multiplication with positive numbers help develop fundamental arithmetic skills.

When teaching yourself or solving problems with positive numbers, remember:

• Adding two positive numbers always results in another positive number.
• Multiplying two positive numbers also yields a positive result.
• The opposite of a positive number is a negative number, which helps balance equations.

Embracing these essentials about positive numbers builds a strong foundation for exploring more complex mathematical concepts.