Problem 7 - Elementary Algebra

Question

Find the opposite of each real number. $$ 8 $$

Step-by-Step Solution

Answer

Answer: The opposite of the real number 8 is -8.

Step 1: Identify the given number's sign

The given number is 8, which is a positive number because it is greater than 0.

Step 2: Change the sign of the given number to find its opposite

Since the given number is positive, we will change its sign by making it negative. Therefore, the opposite of 8 is -8.

Key Concepts

Opposite Numbers

In the realm of real numbers, understanding opposite numbers is crucial. Opposite numbers are simply numbers that are equidistant from zero on the number line, but on opposite sides. To find the opposite of a given real number, you simply change its sign. For example, consider the number 8. Since 8 is positive, its opposite would be -8, which is the negative counterpart. Similarly, the opposite of -2.5 is 2.5. The opposite of zero is zero itself, as it does not have a positive or negative direction on the number line.

• The opposite of a positive number is the same number with a negative sign.
• The opposite of a negative number is the same number with a positive sign.
• The opposite of zero remains zero.

Positive and Negative Numbers

Real numbers can be categorized into positive and negative numbers, along with zero. These categories are based on their position relative to zero on the number line:

• Positive Numbers: These are numbers greater than zero. They are usually found on the right side of the number line. Examples include 3, 8, and 15.
• Negative Numbers: These numbers are less than zero and appear on the left side of the number line. Examples include -1, -5, and -12.

The addition or subtraction of these numbers is essential in mathematics. When you add a positive number, you move to the right on the number line, while adding a negative number moves you to the left. This movement relative to zero helps in understanding operations involving positive and negative numbers.

Number Properties

Real numbers have several essential properties that play a significant role in arithmetic and algebra. Some of the critical properties of real numbers include:

• Additive Inverse: For every real number \( a \), there exists an opposite or additive inverse \( -a \) such that \( a + (-a) = 0 \).
• Closure Property: The sum or product of any two real numbers is a real number. For example, adding two positive numbers or multiplying a positive and a negative number results in a real number.
• Distributive Property: This property states that \( a(b + c) = ab + ac \). It allows for the simplification and rearrangement of expressions in algebra.

Understanding these properties is foundational to solving equations, simplifying expressions, and performing various mathematical operations with real numbers.