Problem 33 - Algebra 2

Question

Identify the additive inverse and multiplicative inverse for each number. $$ -4 \frac{3}{5} $$

Step-by-Step Solution

Answer

Additive inverse: \(4 \frac{3}{5}\), Multiplicative inverse: \(-\frac{5}{23}\).

Step 1: Understanding the Additive Inverse

The additive inverse of a number is the number that, when added to the original number, yields zero. To find the additive inverse of \(-4 \frac{3}{5}\), we must change its sign.

Step 2: Finding the Additive Inverse

The additive inverse of \(-4 \frac{3}{5}\) is \(+4 \frac{3}{5}\), because \(-4 \frac{3}{5} + 4 \frac{3}{5} = 0\).

Step 3: Understanding the Multiplicative Inverse

The multiplicative inverse of a number is the number that, when multiplied by the original number, equals one. For a fraction, this means swapping the numerator and the denominator. Our number in improper form is \(-\frac{23}{5}\).

Step 4: Finding the Multiplicative Inverse

For \(-\frac{23}{5}\), the multiplicative inverse is \(-\frac{5}{23}\). This is because \(-\frac{23}{5} imes -\frac{5}{23} = 1\).

Key Concepts

Additive Inverse

The additive inverse of a number is a concept rooted in basic arithmetic. While the term might sound a bit complex, it is actually quite simple. Think of the additive inverse as a number's opposite that cancels it out. It is the number you can add to your original number to reach zero. If you have a number like \(-4 \, \frac{3}{5}\), its additive inverse would be \(+4 \, \frac{3}{5}\). Why is this useful? Because when you add \(-4 \, \frac{3}{5}\) and \(+4 \, \frac{3}{5}\) together, their sum is zero: \(-4 \, \frac{3}{5} + 4 \, \frac{3}{5} = 0\).
This simple operation helps in balancing equations and simplifying expressions that frequently pop up in algebra and higher mathematics.

Multiplicative Inverse

The multiplicative inverse, also known as the reciprocal, is slightly different from the additive inverse. It answers the question: "What do I multiply this number by to get one?" Let's look at an example to understand it better. For the fraction \(-\frac{23}{5}\), we need to find its multiplicative inverse.
Here's how to do it:

• Switch the places of the numerator and the denominator.
• If the sign is negative in the original number, it remains negative.

The multiplicative inverse of \(-\frac{23}{5}\) becomes \(-\frac{5}{23}\). That's because multiplying these two numbers gives you one: \(-\frac{23}{5} \times -\frac{5}{23} = 1\).
This concept is crucial for solving equations and working with fractions, particularly when dividing fractions.

Improper Fractions

Improper fractions behave a bit differently than proper fractions because their numerators (the top number) are larger than their denominators (the bottom number).
For example, the improper fraction \(-\frac{23}{5}\) results from converting the mixed number \(-4 \, \frac{3}{5}\).

• The numerator 23 indicates that it has more parts than the whole denominator 5.
• This usually means that you have a mixed number with a full number part and a fraction part mixed together.

Converting between improper fractions and mixed numbers helps in operations like addition and subtraction because it simplifies your understanding of the value. Understanding improper fractions can also aid in finding multiplicative inverses, as fractions love to be in improper formats.