Problem 51 - Prealgebra

Question

Give the opposite of each of the following numbers. $$3$$

Step-by-Step Solution

Answer

The opposite of 3 is -3.

Step 1: Understanding Opposites

The opposite of a number is what you would add to the original number to get zero. For any number, its opposite has the same magnitude but an opposite sign.

Step 2: Finding the Opposite of 3

To find the opposite of the number 3, you need to change its sign. Since 3 is positive, its opposite will be negative. Therefore, the opposite of 3 is -3.

Key Concepts

Integers

Integers are a crucial part of mathematics and help us understand whole numbers more completely. Essentially, integers are a set of numbers that include all whole numbers and their opposites. This means that integers include:

• Positive numbers like 1, 2, 3, and so on,
• Negative numbers like -1, -2, -3, etc.,
• And zero.

Unlike fractions or decimals, integers are whole numbers without any fractional or decimal component.

In mathematics, integers are denoted by the letter \(\mathbb{Z}\), stemming from the German word 'Zahlen,' which means numbers. Understanding integers is crucial for grasping concepts about the balance, opposites, and how numbers relate to each other in algebra and many real-world applications.

For example, when we consider daily temperatures or financial credits and debits, we often use positive and negative numbers alongside zero, making these practical instances where the understanding of integers is applied.

Positive and Negative Numbers

Positive and negative numbers form a foundational concept in mathematics, representing balances, elevations, and directions.

Here’s how they work:

• Positive numbers are numbers greater than zero. They denote quantities, gains, or heights. Typical examples include 5, 10, 232, and so on.
• Negative numbers are numbers less than zero. They illustrate losses, debts, or depths, such as -5, -1, -10, etc.

Whenever two numbers are opposites, like 3 and -3, they have the same distance from zero but are on different sides of the number line.

The idea of opposites is incredibly important. For example, the opposite of a loss (which is negative) represents a gain (which is positive), and vice versa. This pairing is fundamental to understanding how addition and subtraction operate, particularly because adding a number and its opposite equals zero.

Number Line

A number line is a visual representation that helps in understanding numbers as it lays out integers in a straight line, each at equal distances from one another. The number line extends infinitely in both directions.

• The numbers to the right of zero are positive numbers.
• The numbers to the left of zero are negative numbers.

This visual tool is crucial for grasping the concept of arithmetic operations. It intuitively shows how numbers add, subtract, and how opposites are visualized.

For instance, starting at zero, move three spaces to the right for the number 3 and three spaces to the left for its opposite, -3.

By using a number line, students can easily solve problems involving the distance between numbers and their opposites by counting steps, simplifying the understanding of such fundamental operations.