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Chapter 1: Problem 125

Explain how to find the multiplicative inverse of a number.

Short Answer

Expert verified
The multiplicative inverse of a number is found by calculating 1 divided by that number. It is a number which when multiplied by the original number yields 1.

Step by step solution

01

Understanding the concept of multiplicative inverse

The multiplicative inverse of a number, also known as the reciprocal, is a number which when multiplied with the original number gives the product as 1. For any non-zero number 'a', the multiplicative inverse is 1/a.
02

Calculating the multiplicative inverse

To find the multiplicative inverse of a number, simply calculate 1 divided by that number. For example, to find the multiplicative inverse of 5, calculate 1/5 which equals 0.2.
03

Checking the result

To verify if the result , simply multiply the original number by its multiplicative inverse. The product should be 1. For the example above, verify the result by multiplying 5 (original number) and 0.2 (its multiplicative inverse). The result is 1, which proves that 0.2 is indeed the multiplicative inverse of 5.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Understanding the Reciprocal
The reciprocal, often termed the multiplicative inverse, is a key concept in mathematics that is incredibly simple once you grasp it. Imagine you have a non-zero number. The reciprocal of this number is another number which, when multiplied by the original number, results in the product of 1. It acts as a sort of mirror image in multiplication terms. For instance, for a number \( a \), its reciprocal is \( \frac{1}{a} \). This concept ensures that math equations and manipulations stay balanced and coherent, especially when solving algebraic equations that involve division or fractions.
Finding the Multiplicative Inverse
Finding the multiplicative inverse, or reciprocal, can be straightforward if you follow a simple rule. To find the inverse of any given non-zero number, just divide 1 by that number.

For instance:
  • For the number 5, calculate \( \frac{1}{5} \), which equals 0.2.
  • For the fraction \( \frac{3}{4} \), its reciprocal is \( \frac{4}{3} \).
The concept holds true for integers, fractions, and even decimals. As long as the number is not zero (since division by zero is undefined), you can always find its reciprocal. This calculation plays a crucial role in solving equations and simplifying expressions in algebra.
Verification of the Inverse
After finding a multiplicative inverse, it is essential to verify its correctness.

This verification step guarantees that no errors were made during calculation. The process is simple:
  • Take the original number and multiply it by the reciprocal you found.
  • If the result is 1, then you have correctly identified the multiplicative inverse.
Consider this example: For the number 5, we found the reciprocal to be 0.2. Multiply 5 by 0.2:
\[5 \times 0.2 = 1\]Since the result is 1, our calculation is confirmed. This verification is a handy check, ensuring that your reciprocal is accurate and will work correctly in algebraic manipulations.
Importance of Algebra Concepts
In algebra, understanding core concepts like the reciprocal is instrumental. These foundational ideas are often utilized in more complex operations and problem-solving scenarios. Across various problems, knowing how to find and verify the multiplicative inverse simplifies many processes, be it solving equations or managing fractions.

Here's why it matters:
  • Simplifies solving equations by reversing operations.
  • Assists in understanding and working with ratios and proportions.
  • Helps in simplifying complex algebraic expressions effectively.
Grasping these algebra concepts builds confidence and capability, aiding your journey through higher-level math with much greater ease and understanding.

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Most popular questions from this chapter

In Palo Alto, California, a government agency ordered computer-related companies to contribute to a pool of money to clean up underground water supplies. (The companies had stored toxic chemicals in leaking underground containers.) The mathematical model $$ C=\frac{200 x}{100-x} $$ describes the cost, \(C,\) in tens of thousands of dollars, for removing \(x\) percent of the contaminants. Use this formula to solve. a. Find the cost, in tens of thousands of dollars, for removing \(60 \%\) of the contaminants. b. Find the cost, in tens of thousands of dollars, for removing \(90 \%\) of the contaminants. c. Describe what is happening to the cost of the cleanup as the percentage of contaminants removed increases.

Will help you prepare for the material covered in the next section. In each exercise, a subtraction is expressed as addition of an opposite. Find this sum, indicated by a question mark. $$-8-13=-8+(-13)=?$$

Write each phrase as an algebraic expression. a loss of half of an investment of \(d\) dollars

Find this sum, indicated by a question mark. \(4(-3)=(-3)+(-3)+(-3)+(-3)=?\)

Explain how to add two numbers with a number line. Provide an example with your explanation.
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