Problem 85 - Beginning and Intermediate Algebra

Question

Fill in the table with the opposite (additive inverse), and the reciprocal (multiplicative inverse). Assume that the value of each expression is not 0 $$ 8 $$

Step-by-Step Solution

Answer

Opposite: -8; Reciprocal: \( \frac{1}{8} \).

Step 1: Understand the Opposite (Additive Inverse)

The opposite, also known as the additive inverse, of a number is the value that, when added to the original number, results in zero. For any number \( a \), the opposite is \( -a \).

Step 2: Find the Opposite of 8

For the number \( 8 \), the opposite is calculated by taking its negative. Thus, the opposite of \( 8 \) is \( -8 \).

Step 3: Understand the Reciprocal (Multiplicative Inverse)

The reciprocal, or multiplicative inverse, of a number is the value that, when multiplied by the original number, results in one. For any non-zero number \( a \), the reciprocal is \( \frac{1}{a} \).

Step 4: Find the Reciprocal of 8

For the number \( 8 \), the reciprocal is found by taking \( \frac{1}{8} \).

Key Concepts

Additive Inverse

In algebra, the additive inverse is an important concept that helps us understand how to neutralize a number using addition. When we talk about the additive inverse, we're referring to the value that, when added to the original number, gives us zero. This is why it's also often called the "opposite".
To find the additive inverse of any number, you simply take the negative of that number. For example, if you have a number \( a \), its additive inverse is \( -a \).

• For positive numbers: The additive inverse will be negative.
• For negative numbers: The additive inverse will be positive.

In the exercise you were given the number 8, and you needed to find its additive inverse. By simply taking the negative of 8, you find that its additive inverse is \( -8 \). This means if you add 8 and \(-8\) together, they cancel each other out and you get 0, which is a fundamental property of opposites in arithmetic.

Multiplicative Inverse

The multiplicative inverse, also known as the reciprocal, is a value that when multiplied by the original number, results in one. This concept is crucial when solving algebraic equations, particularly those involving fractions and rational expressions.
For any non-zero number \( a \), its multiplicative inverse is \( \frac{1}{a} \). Essentially, you're flipping the number upside-down, which is a straightforward way to think about it.

• The reciprocal of a fraction \( \frac{b}{c} \) is \( \frac{c}{b} \).
• For whole numbers, like 8, the reciprocal is \( \frac{1}{8} \).

In our exercise, the number provided was 8, a whole number. Thus, its multiplicative inverse, or reciprocal, is \( \frac{1}{8} \). When you multiply 8 by \( \frac{1}{8} \), the result is 1, affirming their role as multiplicative inverses.

Reciprocal

Reciprocals are an integral part of understanding fractions and rational calculations. When we talk about reciprocals in mathematics, we're specifically pointing out how two numbers, when multiplied together, result in 1.
The idea is simple: you're looking for another number that performs a full circle to 1 through multiplication. This is often applied in algebra when working to simplify expressions or solve equations.

• For any fraction \( \frac{a}{b} \), the reciprocal is \( \frac{b}{a} \).
• If dealing with a whole number, the reciprocal involves flipping it to be a fraction over 1.

Remember, the number 0 does not have a reciprocal, since division by zero is undefined. But for any other number, like 8 used in the example, the reciprocal strategy involves simply forming \( \frac{1}{8} \). This approach ensures that when they are multiplied, they bring back the unity of 1, confirming their reciprocal nature.