Problem 39 - Introductory and Intermediate Algebra for College Students 4th

Question

In Exercises \(35-42,\) find the multiplicative inverse of each number. $$-10$$

Step-by-Step Solution

Answer

The multiplicative inverse of -10 is -0.1.

Step 1: Understand the Concept of Multiplicative Inverse

The multiplicative inverse of a number is the number that we can multiply with the original number to get the product as 1. Essentially, for a number \(a,\) its multiplicative inverse is \(\frac{1}{a}.\)

Step 2: Apply the Concept to Our Problem

Now let's apply this to our problem. The number given is -10. Therefore, the multiplicative inverse of -10 is simply \(\frac{1}{-10}.\)

Step 3: Simplify the Result

The calculated multiplicative inverse \( \frac{1}{-10} \) simplifies to -0.1.

Key Concepts

Reciprocal

To understand what a reciprocal is, think of it as a special number you multiply by another number to get the result of 1. It’s a simple yet essential math concept. The reciprocal of a number is often called the multiplicative inverse. Here’s how it works:

• For a positive or negative number, the reciprocal is 1 divided by that number.
• For example, the reciprocal of 5 is \( \frac{1}{5} \).
• In our example, the reciprocal of -10 is \( \frac{1}{-10} \).

When you multiply a number by its reciprocal, the result is always 1. In mathematical terms, if you have a number \( a \), its reciprocal is \( \frac{1}{a} \). This is true for any non-zero number, whether it’s positive or negative.Remember, finding the reciprocal is fundamental when dealing with division, fractions, and algebraic equations. By swapping the positions of numerator and denominator, you can easily uncover the reciprocal. Practicing with various numbers helps reinforce this concept, making math exercises easier to tackle.

Negative Numbers

Negative numbers can seem a bit tricky at first, but they follow the same rules as positive numbers when dealing with reciprocals. Here’s the catch:

• The reciprocal of a negative number is also negative. It’s simply the opposite sign of the positive counterpart.
• For instance, if you know the reciprocal of 10 is \( \frac{1}{10} \), then the reciprocal of -10 is \( \frac{1}{-10} \).

Working with negative numbers is straightforward. They are numbers less than zero, and they show up in math to express values in the opposite direction or below a reference point. It's crucial to notice that the rules for multiplication and division are consistent for both positive and negative numbers:

• Multiplying two negative numbers results in a positive number.
• Multiplying a negative and a positive number results in a negative number.

When finding the multiplicative inverse for a negative number, just place a negative sign in front of the reciprocal of the positive version of the number. Keeping these rules in mind helps you solve math problems with confidence.

Simplification

Simplification is about making mathematical expressions easier to understand by breaking them down into their simplest form. When simplifying a reciprocal, you often translate the fraction into a decimal where necessary.Let's take the example of -10:

• The reciprocal is \( \frac{1}{-10} \).
• To simplify, divide 1 by -10, which results in -0.1.

Simplification can help you better visualize and handle numbers, especially when dealing with complex calculations or equations. It's a process where you aim to make expressions clear and neat without changing their value.Here are some tips for simplification:

• Perform any division to convert fractions to decimal form, if required.
• Always keep the negativity of numbers (if any) intact during simplification.
• Try to use approximations only when exact values aren't necessary, to avoid errors.

Working on simplifying numbers effectively ensures you communicate mathematical ideas more clearly and minimizes the risks of errors during calculations. Practice this skill often, as it’s incredibly helpful in achieving accurate solutions.