Problem 109 - Prealgebra

Question

Give the opposite of each number. $$9$$

Step-by-Step Solution

Answer

The opposite of 9 is -9.

Step 1: Understanding Opposite Numbers

An opposite (or additive inverse) of a number is what you add to a number to get zero. For example, the opposite of a positive number is a negative number with the same absolute value and vice versa.

Step 2: Identify the Given Number

The given number is 9, which is a positive integer.

Step 3: Find the Opposite

To find the opposite of the positive number 9, change it to negative. The opposite of 9 is -9.

Key Concepts

Additive Inverse

The term "additive inverse" might sound complex, but it has a straightforward meaning. The additive inverse of any number is the number that, when added to the original number, results in zero.

For example, if you have the number 3, its additive inverse is -3. When you add 3 and -3, the result is 0. This is a core concept in arithmetic and algebra as it helps in solving many mathematical problems.

• The additive inverse of a positive number is negative.
• Conversely, the additive inverse of a negative number is positive.

Recognizing the additive inverse helps in various mathematical operations, such as simplifying equations by canceling out terms. Understanding this concept ensures that you can always "balance" an equation or an operation.

Positive and Negative Integers

Integers include all whole numbers, both positive and negative, and zero. Understanding the distinction between positive and negative integers is crucial for grasping how numbers work on a number line.

Positive integers are numbers greater than zero, such as 1, 2, and 3. These are numbers you would naturally associate with counting.

Negative integers, on the other hand, start from -1 and go downwards, such as -2 and -3. These numbers indicate a sense of opposition or deficiency compared to zero.

• The number zero itself is neither positive nor negative.
• It serves as the midpoint on the number line, marking a transition between positive and negative values.

Recognizing positive and negative integers helps when dealing with real-life scenarios, such as financial transactions like credits and debits.

Absolute Value

Absolute value relates to the "magnitude" or size of a number without considering its sign. It's like asking, "How far is this number from zero on the number line?" The absolute value signifies this distance regardless of direction.

For any number, positive or negative, the absolute value is its non-negative version.

• For positive numbers, the absolute value is the number itself.
• For negative numbers, the absolute value is the number without its negative sign.

Mathematically, it is denoted by two vertical bars, like this: \(|-5| = 5\) or \(|3| = 3\).

Understanding absolute value is essential when you need to measure real-world distances or when simplifying expressions in algebra. It highlights how the concept of distance is independent of direction.