Problem 112 - Prealgebra

Question

Give the opposite of each number. $$-5$$

Step-by-Step Solution

Answer

The opposite of -5 is 5.

Step 1: Understand the concept of opposites

The opposite of a number is what you add to a number to get zero. For any given number, its opposite will be the same distance from zero on the number line but in the opposite direction. In general, the opposite of a negative number is a positive number of the same magnitude.

Step 2: Determine the opposite of -5

Given the number is \(-5\). According to the concept of opposites, the opposite of a negative number is a positive number. Thus, the opposite of \(-5\) is \(5\).

Key Concepts

Negative Numbers

Negative numbers are numbers that are less than zero. They are usually represented on the number line to the left of zero, marked with a minus (-) sign. For instance, numbers like

• -1
• -2
• -3
• -5

are all examples of negative numbers.
Negative numbers have interesting properties, especially when used in arithmetic operations:

• Adding a negative number is essentially the same as subtracting a positive number.
• For example, adding a number like -3 is similar to subtracting 3.
• Multiplying two negative numbers results in a positive number.
• So, if you multiply -2 by -3, the result is 6.
• However, if you multiply a negative number by a positive one, the result is negative.

These properties can help in understanding concepts like opposites, because the opposite of a negative number is indeed a positive number, which visually shifts to the right on the number line.

Positive Numbers

Positive numbers are all the numbers that are greater than zero. You can find them on the number line to the right of zero, and they do not have any sign before them. Some examples include 1, 2, 5, and 20.
Positive numbers play a central role in many areas of mathematics and everyday life.

• They represent quantities that are more than nothing, like 5 apples or 3 meters.
• Positive numbers are often used in addition and multiplication.
• For instance, adding 5 + 3 results in 8, and multiplying 4 by 2 yields 8.

When thinking about opposites and number lines, a positive number is the opposite of a negative number. For example, if you have -5, the opposite, or positive counterpart, is 5.
Understanding these basics helps when trying to determine the opposite of numbers in mathematics.

Number Line

A number line is a visual representation of numbers on a straight line, used to illustrate the position of numbers relative to each other. Zero is often placed in the center.
To the right of zero, positive numbers increase as you move farther to the right. Meanwhile, to the left, negative numbers decrease moving further left.

Characteristics of a Number Line:

• The number line extends indefinitely in both directions.
• It is generally incremented uniformly.
• It helps in understanding addition, subtraction, multiplication, and division.

Using a number line to find opposites is simple. Take the number you want the opposite of, and jump the same number of spaces in the opposite direction.
For example, if you have -5, moving 5 spaces to the right from zero will land you at 5. Thus, 5 is the opposite of -5.
Similarly, if positive 3 is chosen, delivering 3 spaces in reverse direction will steer you to -3.
Number lines are a straightforward, helpful tool for visual learners, giving a clear image of the concepts of numerical opposites.