Problem 238 - Intermediate Algebra

Question

Explain how you find the reciprocal of a negative number.

Step-by-Step Solution

Answer

The reciprocal of -5 is \( \frac{1}{-5} \) or -0.2.

Step 1: Identify the Number

First, identify the negative number for which the reciprocal is needed. For example, let's choose -5 as the number.

Step 2: Definition of Reciprocal

The reciprocal of a number is defined as 1 divided by that number. So, the reciprocal of any number 'x' is given by \( \frac{1}{x} \).

Step 3: Apply the Reciprocal Definition

Using the definition, find the reciprocal by dividing 1 by the negative number. For -5, the reciprocal is \( \frac{1}{-5} \).

Step 4: Simplify the Expression

Simplify the fraction if necessary. In this case, \( \frac{1}{-5} \) is already in its simplest form. Hence, the reciprocal of -5 is -0.2.

Key Concepts

reciprocal

The concept of a reciprocal is fundamental in mathematics. A reciprocal of a number is essentially what you multiply that number by to get 1. For a given number 'x', its reciprocal is expressed as \( \frac{1}{x} \). This means that when you multiply \( x \) by \( \frac{1}{x} \), the result will always be 1.

In simpler terms, if you have a number, you simply flip it. For instance:

• The reciprocal of 2 is \( \frac{1}{2} \).
• The reciprocal of 3 is \( \frac{1}{3} \).
• The reciprocal of 0.5 is 2 (since \( \frac{1}{0.5} = 2 \)).

Understanding reciprocals is essential, especially when dealing with fractions and division.

negative numbers

Negative numbers are simply numbers with a '-' sign in front of them, indicating that they are less than zero. They commonly appear in various calculations and are the opposite of positive numbers.

When dealing with reciprocals of negative numbers, the process remains the same as with positive numbers. The negative sign will stay with the reciprocal. For example, the reciprocal of -4 is \( \frac{1}{-4} \), which is also written as \( -\frac{1}{4} \) or -0.25. The process doesn't change; the negative sign simply carries over.

fractions

Fractions are numerical quantities that are not whole numbers. They are represented as a ratio of two numbers, where the top number is the numerator, and the bottom number is the denominator. For example, \( \frac{3}{4} \) is a fraction where 3 is the numerator and 4 is the denominator.

When working with reciprocals, you often end up with fractions. Especially when finding the reciprocal of a whole number, you get a fraction. For instance, the reciprocal of 5 is \( \frac{1}{5} \). This is because 5 can be seen as \( \frac{5}{1} \) and flipping it gives \( \frac{1}{5} \).

simplification

Simplifying fractions involves reducing them to their simplest form, where the numerator and denominator have no common factors other than 1. This basic concept ensures that your fraction is as clear and concise as possible. When dealing with reciprocals, simplification might not always be necessary.

For example, the reciprocal of -5 is \( \frac{1}{-5} \). In this case, there's no need for further simplification because it's already in its simplest form. However, if you had a fraction like \( \frac{4}{-8} \), you can simplify it by dividing both the numerator and the denominator by their greatest common divisor, which is 4, to get \( \frac{1}{-2} \).