Problem 66 - Beginning and Intermediate Algebra

Question

Find each additive inverse or opposite. See Example \(9 .\) $$ 4 $$

Step-by-Step Solution

Answer

The additive inverse of 4 is -4.

Step 1: Understanding Additive Inverse

The additive inverse of a number is what you add to the original number to get zero. For any number \( a \), the additive inverse is \( -a \).

Step 2: Identify the Original Number

In this exercise, the original number is given as \( 4 \).

Step 3: Find the Additive Inverse

To find the additive inverse, simply take the negative of the original number. Therefore, the additive inverse of \( 4 \) is \( -4 \).

Key Concepts

Opposite Numbers

In mathematics, opposite numbers are pairs that sit equal distance from zero on a number line, but on opposing sides. These numbers are quite important as they help in understanding the concept of balance when dealing with equations and expressions. Here's a closer look:

• If you have a number, say, 4, its opposite would be -4. They both are 4 units away from zero.
• Opposite numbers always add up to zero, a key component in integer operations, often expressed as the principle of zero.

Understanding opposite numbers is vital in mastering integer operations and algebra basics. This foundational concept leads directly into the exploration of additive inverses, which you'll often encounter in various algebraic problems.

Algebra Basics

Algebra is a branch of mathematics that deals with variables and numbers along with operations. It forms the bedrock of more complex mathematical concepts. Let's break it down simply:

• Variables: Symbols, usually letters, that represent numbers whose values can change.
• Constants: Numbers that have a fixed value.
• Expressions: Combinations of variables and numbers formed using operations like addition, subtraction, multiplication, and division.

A common principle is the additive inverse, which implies that any number added to its opposite results in zero. For instance, 4 plus its opposite, -4, results in 0. This concept is fundamental in solving equations where you might need to isolate a variable on one side of the equation by "canceling out" terms.

Integer Operations

Integer operations involve basic actions like addition, subtraction, multiplication, and division but specifically deal with whole numbers, both positive and negative. Key factors to remember include:

• Addition: Combining numbers, like adding opposites (4 + -4), results in zero.
• Subtraction: Taking one integer away from another can be thought of as adding its opposite.
• Multiplication: Product of two integers, be it positive or negative.
• Division: Splitting an integer into parts, considering rules for signs (e.g., a negative divided by a positive is negative).

The use of additive inverses is particularly common in addition and subtraction operations with integers. Mastering these basics paves the way for more advanced mathematical and algebraic concepts.