Problem 58 - Beginning and Intermediate Algebra

Question

Find each reciprocal or multiplicative inverse. $$ -\frac{6}{13} $$

Step-by-Step Solution

Answer

The reciprocal of \(-\frac{6}{13}\) is \(-\frac{13}{6}\).

Step 1: Understanding the Reciprocal Concept

The reciprocal or multiplicative inverse of a number is a value which, when multiplied by the original number, gives one. For a fraction \( a/b \), its reciprocal is \( b/a \).

Step 2: Determine the Reciprocal

Given the fraction \( -\frac{6}{13} \), interchange the numerator and the denominator. This gives the reciprocal \( -\frac{13}{6} \). The sign of the fraction remains the same since both the original number and its reciprocal are negative.

Step 3: Verify the Reciprocal

To ensure correctness, multiply the original fraction \( -\frac{6}{13} \) by its reciprocal \( -\frac{13}{6} \). The multiplication is \( -\frac{6}{13} \times -\frac{13}{6} = \frac{6 \times 13}{13 \times 6} = 1 \), confirming that \( -\frac{13}{6} \) is indeed the correct reciprocal.

Key Concepts

Multiplicative Inverse

The multiplicative inverse, often called the reciprocal, is a fundamental concept in mathematics. It refers to a number which, when multiplied by a given number, results in the product being one. For any number expressed as a fraction in the form \( \frac{a}{b} \), the multiplicative inverse is simply \( \frac{b}{a} \). It's important to keep in mind that this definition holds true even if the numbers involved are negative or include variables. This concept is crucial in solving equations and simplifying algebraic expressions, making it invaluable in both basic math and more advanced algebraic operations.

• To find the reciprocal of a fraction, swap its numerator and denominator.
• The multiplicative inverse of an integer is \( \frac{1}{a} \) where \( a eq 0 \).
• Both the original and its reciprocal share the same sign.

This means if you start with \( -\frac{6}{13} \), the reciprocal is \( -\frac{13}{6} \), as they both are multiplied to yield one.

Fractions

Fractions represent parts of a whole and are essential in everyday mathematics. A fraction is written as \( \frac{a}{b} \), where \( a \) is the numerator—the part—and \( b \) is the denominator—the whole. Understanding fractions means knowing that they're essentially division problems. Here are some key points to remember:

• The numerator tells us how many parts we have.
• The denominator shows how many parts make up a whole.
• When the numerator is greater than the denominator, the fraction is greater than one, known as an improper fraction.

Fractions appear in many forms; they can be positive, negative, proper (numerator less than the denominator), or improper (numerator greater than the denominator). In the case of negative fractions, the negative sign can be with either the numerator or the denominator but not both as they cancel out to a positive.

Algebra Concepts

Algebra involves the study of mathematical symbols and the rules for manipulating these symbols. It's a unifying thread of almost all of mathematics and provides a foundation for a logical approach to problem-solving. Within algebra, the use of fractions and finding reciprocals plays an integral role. For instance, solving equations often requires isolating a variable, and one method is multiplying both sides by the multiplicative inverse.

• Understanding algebra requires grasping the properties of numbers, such as commutative, associative, and distributive laws.
• Equations involving fractions can be simplified by multiplying all terms by the least common denominator (LCD).
• The reciprocal is used to divide fractions; you multiply by the reciprocal of the divisor.

Finding reciprocals is a straightforward process that greatly simplifies the manipulation of fractions within algebraic expressions, making them easier to solve and interpret. This forms a bridge between basic arithmetic and more advanced mathematical processes.