Problem 40 - Introductory and Intermediate Algebra for College Students 4th

Question

In Exercises \(35-42,\) find the multiplicative inverse of each number. $$-12$$

Step-by-Step Solution

Answer

The multiplicative inverse of -12 is -1/12.

Step 1: Understanding the concept of multiplicative inverse

This property is fundamental in mathematics. It states that for every integer, fraction, or real number \( a \) except zero, there exists \( b = 1/a \) such that \( a * b = 1 \). The number \( b \) is called the multiplicative inverse of \( a \). So to solve this exercise, the multiplicative inverse of -12 needs to be calculated.

Step 2: Calculate the multiplicative inverse of -12

The multiplicative inverse of -12 is found by taking the reciprocal of -12. The reciprocal of any number \( a \) is \( b = 1/a \). Given that \( a = -12 \), \( b = 1/(-12) = -1/12.

Key Concepts

Reciprocal

The concept of a reciprocal is an essential aspect of understanding multiplicative inverses. In mathematics, the reciprocal of a number is simply one divided by that number. This means if you have a number, say \( a \), then its reciprocal is \( \frac{1}{a} \). For example, the reciprocal of 2 is \( \frac{1}{2} \). Reciprocals are crucial because they allow us to "reverse" a multiplication. Specifically:

• If you multiply a number by its reciprocal, the result is always 1. For instance, \( 2 \times \frac{1}{2} = 1 \).
• Reciprocals are used when solving equations because they help in isolating variables.

For negative numbers, such as -12, the reciprocal is simply \( \frac{1}{-12} \) or \( -\frac{1}{12} \). This demonstrates the principle that a number and its reciprocal will always produce a product of 1, regardless of whether the number is positive or negative.

Inverse Operations

Inverse operations are operations that "undo" each other. They are like mathematical opposites. For multiplication, the inverse operation is division, and vice versa.Consider these important properties:

• Multiplication and its invert, division, are inverse operations. This means if you multiply a number by another number, you can divide by the same number to return to the original value.
• In the context of the exercise, finding the multiplicative inverse means you're looking for a number that can multiply with the original to yield 1, effectively "undoing" the multiplication.

For instance, when you find the multiplicative inverse of -12, you seek a number that, when multiplied by -12, equals 1. The reciprocal, \( -\frac{1}{12} \), fulfills this role since:\[-12 \times -\frac{1}{12} = 1\]This balanced equation illustrates how inverse operations are utilized to achieve equilibrium in mathematical expressions.

Real Numbers

Real numbers are comprehensive and form the foundation of most arithmetic operations you encounter daily. They include numerous types of numbers such as:

• Integers like -3, 0, and 5
• Fractions like \( \frac{1}{2} \)
• Decimals like 3.14
• Irrational numbers like \( \sqrt{2} \)

Because real numbers encompass both positive and negative values, as well as zero, they are incredibly versatile in performing various mathematical operations, including addition, subtraction, multiplication, and division.In the context of finding multiplicative inverses, it's important to note that every real number, except zero, has such an inverse. Zero is excluded because there cannot exist a real number that multiplied by zero equals 1. This is a key concept when considering solutions in algebra and calculus, ensuring each non-zero real number has a reciprocal that preserves the balance of mathematical operations.