Problem 54 - Elementary Algebra

Question

Determine the reciprocal of the following numbers. $$ -43 $$

Step-by-Step Solution

Answer

The reciprocal of \(-43\) is \( \frac{1}{-43} \).

Step 1: Understanding Reciprocals

The reciprocal of a number is obtained by taking one divided by that number. Mathematically, if the number is denoted as \( a \), then its reciprocal is \( \frac{1}{a} \). This concept applies to all non-zero numbers whether they be whole numbers, fractions, or negatives.

Step 2: Applying the Concept

For the number \(-43\), we apply the rule of reciprocals to find that the reciprocal is \( \frac{1}{-43} \). This transformation takes the negative whole number and turns it into a negative fraction.

Key Concepts

Understanding Negative Numbers

Negative numbers are an essential part of the number system. They are numbers less than zero and are typically used to represent loss, debt, or decrease. In mathematics, a negative number is indicated by a minus sign (-) in front of it. For example, -5 denotes five units lower than zero. Negative numbers can be used in various mathematical operations like addition, subtraction, multiplication, and division. When dealing with negative numbers:

• Adding a negative number is like subtracting its absolute value.
• Subtracting a negative number results in adding its absolute value.
• The multiplication or division of two negative numbers results in a positive number.

Negative numbers maintain their nature during the reciprocal process, as seen when transforming a number like -43 into \( \frac{1}{-43} \). This transformation still reflects the characteristics of negative numbers.

What are Fractions?

Fractions represent parts of a whole and are expressed as \( \frac{a}{b} \), where \( a \) is the numerator indicating the number of parts we have, and \( b \) is the denominator indicating the total number of equal parts the whole is divided into. They allow us to express values between integers, like, \( \frac{1}{2} \), which is half of one.

• If the numerator is less than the denominator, the fraction is called a proper fraction (e.g., \( \frac{3}{4} \)).
• If the numerator is greater than or equal to the denominator, it is an improper fraction (e.g., \( \frac{5}{3} \)).
• A mixed number combines a whole number with a fraction (e.g., 2 \( \frac{1}{3} \)).

When calculating reciprocals, whole numbers and fractions both transform into fractions. For instance, the reciprocal of -43 (a whole number) becomes \( \frac{1}{-43} \), which is a fraction representing part of a whole.

Basic Mathematical Operations

Mathematical operations include addition, subtraction, multiplication, and division. These operations are the foundation of all calculations in mathematics and allow us to manipulate numbers to reach desired outcomes. Let's break it down:

• Addition (+): Combining two numbers to get their total.
• Subtraction (−): Calculating the difference between two numbers.
• Multiplication (×): Repeated addition of a number by the number of times specified by another.
• Division (÷): Splitting a number into specified equal parts.

When finding the reciprocal of a number, division is the key operation. We divide 1 by the given number, such as \( -43 \), resulting in its reciprocal \( \frac{1}{-43} \). This process demonstrates applying basic operations to find reciprocals effectively.