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Chapter 7: Problem 355

In the following exercises, find the multiplicative inverse. $$ -5 $$

Short Answer

Expert verified
The multiplicative inverse of -5 is \(-\frac{1}{5}\).

Step by step solution

01

Understand the Multiplicative Inverse

The multiplicative inverse of a number is another number which, when multiplied with the original number, gives a product of 1. It is also known as the reciprocal.
02

Identify the Given Number

The given number is -5.
03

Find the Reciprocal

The reciprocal of a number is found by flipping the numerator and the denominator. For any number like \-5, the reciprocal becomes \(-\frac{1}{5}\).
04

Verify

Verify if \(-5 \times -\frac{1}{5} = 1\): \(-5 \times -\frac{1}{5} = 1\), which confirms that \(-\frac{1}{5}\) is indeed the multiplicative inverse of -5.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Reciprocal
The key idea behind the multiplicative inverse is the concept of a reciprocal. When we talk about the reciprocal of a number, we mean the number which, when multiplied by the original number, yields 1. Simple examples include the number 2, whose reciprocal is \(\frac{1}{2}\). The reciprocal of a fraction like \(\frac{3}{4}\) is its upside-down version, \(\frac{4}{3}\). In other words, if you have any number \ (x)\, its reciprocal is \(\frac{1}{x}\). Remember, not all numbers have a reciprocal 鈥� zero doesn't have one because dividing by zero is undefined.
Negative Numbers
Finding the reciprocal of negative numbers follows the same rules as finding the reciprocal of positive ones, but with an extra step to consider the negative sign. If our original number is -5, its reciprocal would be \(-\frac{1}{5}\). Notice how the negative sign is preserved. This doesn't change the mathematical operation; it simply means the reciprocal is also negative. So, for any negative number \(-n\), its reciprocal is \(-\frac{1}{n}\). This ensures the product of the number and its reciprocal is still 1.
Verification of Inverse
After finding the multiplicative inverse, it's crucial to verify that our solution is correct. Verification involves checking if the product of your original number and its proposed inverse equals 1. For example, let's verify \(-5 \times -\frac{1}{5}\):

When you multiply them, you get \(-5 \times -\frac{1}{5} = 1\). Since the result is 1, this confirms that \(-\frac{1}{5}\) is indeed the multiplicative inverse of -5.

Verification serves as a double-check to ensure the correctness of your work. Remember, if the result is anything other than 1, then the reciprocal is not accurate.

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