Problem 3 - Algebra 1: Concepts and Skills

Question

Find the opposite of the number. $$ 1 $$

Step-by-Step Solution

Answer

The opposite of the number 1 is -1.

Step 1: Identify the Number

The given number is 1.

Step 2: Find the Opposite

The opposite of a number is obtained by changing its sign. Since the given number is a positive 1, changing its sign gives the opposite, which is -1.

Key Concepts

Positive and Negative Numbers

In the world of mathematics, numbers are classified primarily into two categories: positive and negative. Understanding these classes is crucial because they form the foundation of more complex concepts. Positive numbers are greater than zero and can be found to the right of zero on a number line.

• For example, 1, 2, and 10 are all positive numbers.
• They are often used to represent quantities, benefits, or assets.

Negative numbers, on the other hand, are less than zero and lie to the left of zero on the number line. They are characterized by a minus sign (-) before the numeral.

• Examples include -1, -3, and -7.
• They typically represent debts, decreases, or losses.

The concept of opposite numbers is directly linked here. The opposite of a positive number is negative, and vice versa.

Integer Properties

Integers are a set of numbers composed of positive numbers, negative numbers, and zero. They do not include fractions or decimals. This makes them easier to visualize and work with in calculations.

• An important property of integers is that for any integer \( n \), there is an opposite integer, \( -n \).
• The integer \( 0 \) is unique because it is neither positive nor negative. It is its own opposite.

Considering integer properties helps us identify opposites in practical terms:

• For an integer \( 1 \), its opposite is \( -1 \).
• Similarly, the opposite of \( -5 \) is \( 5 \).

These properties not only assist with finding opposites but also with operations like addition, where the sum of a number and its opposite always equals zero.

Mathematical Operations

Mathematical operations often involve applying rules for adding, subtracting, multiplying, or dividing numbers. When dealing with opposite numbers, these operations can help clarify their use.

• Addition and Subtraction: Using opposite numbers, we observe that equal magnitudes of opposite numbers sum to zero. This is a powerful tool for simplifying calculations.
• Multiplication: Multiplying two numbers with differing signs results in a negative product. For instance, multiplying \( 1 \) by \( -1 \) yields \( -1 \).
• Division: When a positive number divides into a negative number, or vice versa, the quotient is negative. For example, \( 6 \div (-3) = -2 \).

These operations highlight how opposite numbers balance situations in equations, enabling us to alter and solve them effectively.