Problem 9 - Fundamentals of Mathematics

Question

Find the opposite of each number. -6

Step-by-Step Solution

Answer

The opposite of -6 is 6.

Step 1: Understand the Concept of Opposite Numbers

Opposite numbers, also called additive inverses, are pairs of numbers that sum up to zero. For a given number, its opposite is the number equidistant from zero on the number line but in the opposite direction.

Step 2: Identify the Number

In this exercise, we are given the number -6.

Step 3: Find the Opposite

To find the opposite, you switch the sign of the number. The opposite of -6 is 6, since -6 + 6 = 0.

Key Concepts

Additive Inverses

Additive inverses, often referred to as opposite numbers, play a crucial role in mathematics. The main idea is that when you add a number to its additive inverse, the result is always zero. This is because they cancel each other out.

For example, consider the number \(-6\). Its additive inverse is \(6\) because \(-6 + 6 = 0\). This property is very useful in simplifying algebraic expressions and solving equations.

• The additive inverse of a positive number is negative and vice versa.
• Decimals, fractions, and whole numbers all have additive inverses.
• The additive inverse of zero is zero itself, since \(0 + 0 = 0\).

The concept is simple but fundamental and serves as a building block for more advanced topics in mathematics.

Number Line

The number line is a visual representation of numbers in a straight line. Here, numbers are placed at equal intervals, and they extend infinitely in both the positive and negative directions. The number line provides a way to easily grasp the concept of opposite numbers.

Important features of the number line include:

• Zero is at the central point, serving as a reference.
• Positive numbers lie to the right of zero.
• Negative numbers extend to the left of zero.

When locating the opposite of a number, find its position on the number line, then locate the same distance from zero in the opposite direction. For example, to find the opposite of \(-6\), look six units to the left of zero, then count six units to the right. This helps in quickly identifying and understanding additive inverses visually.

Integers

Integers are a set of numbers that include all positive whole numbers, negative whole numbers, and zero. These numbers do not have fractions or decimals.

• Examples include \(-3, 0, 7\).
• Integers can be categorized into positive integers (e.g., 1, 2, 3), negative integers (e.g., \(-1, -2, -3\)), and zero.
• Every integer has an opposite, or additive inverse, within the integer set.

In the exercise of finding the opposite, we are essentially identifying the opposite integer. For \(-6\), the integer \(6\) acts as its additively inverse, balancing it out to zero. Working with integers is a foundational skill that appears in many areas of mathematics, especially in solving equations and inequalities.