Problem 47 - Beginning and Intermediate Algebra

Question

Find (a) the additive inverse and (b) the absolute value. 8

Step-by-Step Solution

Answer

(a) The additive inverse of 8 is -8. (b) The absolute value of 8 is 8.

Step 1 - Understand the Additive Inverse

The additive inverse of a number is what you add to that number to get a sum of zero. In other words, the additive inverse of a positive number is its negative counterpart, and vice versa.

Step 2 - Find the Additive Inverse

For the number 8, the additive inverse is -8, because 8 + (-8) = 0. So, the additive inverse of 8 is -8.

Step 3 - Understand the Absolute Value

The absolute value of a number is the distance between that number and zero on the number line, without considering the direction. It is always a non-negative number.

Step 4 - Find the Absolute Value

For the number 8, the absolute value is 8 because it is 8 units away from zero on the number line: \(|8| = 8\).

Key Concepts

Additive Inverse

In mathematics, understanding the additive inverse is crucial. The additive inverse of a number is what you add to that number to get a sum of zero. In simpler terms, it's the opposite of the number. Think of it as the 'negative counterpart' for positive numbers, and the 'positive counterpart' for negative numbers. For instance, if you have the number 8, its additive inverse is -8 because when you add them together, you get zero: 8 + (-8) = 0. This concept helps in balancing equations and solving algebraic problems.

Absolute Value

Absolute value is another fundamental concept in mathematics. It represents how far a number is from zero on the number line, without considering if it’s in the positive or negative direction. Essentially, it’s always a non-negative number. For example, the absolute value of both 8 and -8 is 8 since both are 8 units away from zero, but in different directions. The symbol used to denote absolute value is two vertical bars around the number, like this: \(|8| = 8\). Absolute values are used in real-world scenarios such as calculating distances irrespective of the direction.

Positive and Negative Numbers

Positive and negative numbers are the fundamental building blocks in the number system. Positive numbers are greater than zero and can be found to the right of zero on the number line. They represent quantities like profits, assets, or elevations above sea level. On the other hand, negative numbers are less than zero and are located to the left of zero on the number line. They represent quantities like debts, losses, or elevations below sea level. Understanding how to work with both positive and negative numbers is crucial for solving complex math problems and for real-life financial calculations.

Number Line

A number line is a visual representation of numbers on a straight horizontal line. Each point on the line corresponds to a real number. It helps to understand various concepts like addition, subtraction, absolute values, and integer operations. The center of the number line is zero. Positive numbers are placed to the right of zero, and negative numbers are placed to the left. The number line makes it easier to visualize the additive inverse and absolute value. For instance, you can easily see that the additive inverse of 8 is -8, as they are symmetrical around zero. Similarly, you can measure the distance from zero to any number to find its absolute value.