Problem 59 - Algebra 1: Concepts and Skills

Question

Find the opposite of the number. $$ 5.6 $$

Step-by-Step Solution

Answer

The opposite of \(5.6\) is \(-5.6\).

Step 1: Understand the Concept of Opposite

In mathematical terms, 'opposite' means the number that is the same distance from 0 on the number line, but in the opposite direction. For any real number, the opposite is just the negation of that number.

Step 2: Identify the Given Number

The given number in this exercise is \(5.6\).

Step 3: Find the Opposite of the Given Number

The opposite of \(5.6\) can be found by negating \(5.6\).

Key Concepts

Negation

Negation in mathematics is the process of changing the sign of a number. Every real number has a counterpart, which is its opposite, and this opposite is the result of negating the original number. When you negate a number, you move it to the opposite side of the number line relative to zero.

For instance, the negation of a positive number, such as 5.6, is negative. Hence, \[ -5.6\] is the negation of 5.6. This transformation can be thought of as flipping the number across the origin of the number line. Here are a few key points about negation:

• The opposite of a positive number is negative.
• The opposite of a negative number is positive.
• Zero is its own negation, as \( 0 = -0 \).

Understanding negation is crucial for finding opposite numbers and is fundamental in operations involving real numbers.

Number Line

A number line is a visual tool used in mathematics to represent numbers as points along a horizontal line. It helps us understand the relationship between different numbers, including their opposites.

Imagine a straight line with zero placed at the center. Numbers to the right of zero are positive, whereas numbers to the left are negative. The concept of opposite numbers on the number line is simple: they are equidistant from zero, one on the positive side and one on the negative side. For example:

• 5.6 is represented at a point 5.6 units to the right of zero.
• -5.6 is at a point 5.6 units to the left of zero.

The number line is a powerful aid because it provides a visual representation, making it easier to understand concepts such as negation, distance, and direction.

Real Numbers

Real numbers include all the numbers that exist on the number line. They encompass rational numbers, like integers and fractions, and irrational numbers, such as \( \sqrt{2} \).

Real numbers are continuous, meaning there are no gaps between them on the number line. This continuity allows for every point on the number line to represent a real number and vice versa.

• Every real number has an opposite, which is a direct application of negation.
• They can be positive, negative, or zero.
• Understanding real numbers involves recognizing their infinite range.

The concept of real numbers is fundamental to all areas of mathematics, whether you're dealing with simple arithmetic or more complex functions.