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Chapter 1: Problem 8

Give the reciprocal of each number. $$ 1 / a $$

Short Answer

Expert verified
The reciprocal of \( \frac{1}{a} \) is \( a \).

Step by step solution

01

Understand the Reciprocal

The reciprocal of any number is the number you get when you divide 1 by that number. It's essentially the inverse related to multiplication.
02

Identify the Given Number

The problem presents the number as \( \frac{1}{a} \). We are going to find its reciprocal.
03

Invert the Fraction

To find the reciprocal of \( \frac{1}{a} \), you simply invert the fraction, turning the numerator into the denominator and the denominator into the numerator.
04

Write the Reciprocal

The reciprocal of \( \frac{1}{a} \) is \( a \). Once inverted, the fraction becomes \( a \). This is because multiplying \( \frac{1}{a} \) by its reciprocal \( a \) gives you 1.

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Key Concepts

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

Fractions
Fractions are essential building blocks of mathematics. A fraction represents a part of a whole.
It's written as two numbers separated by a slash, like this: \(\frac{1}{a}\).
  • The top number is called the numerator.
  • The bottom number is called the denominator.
With fractions, you can perform various operations such as addition, subtraction, multiplication, and division. Understanding fractions is key to solving problems that involve parts of a whole.
For instance, \(\frac{1}{a}\) is a fraction where '1' is the numerator and 'a' is the denominator. Here, \(a\) represents the total parts available, and '1' represents one part of those total parts.
Inverse operations
Inverse operations are operations that reverse the effect of the original operation. They are essential in solving equations and understanding the relationships between numbers. When you think of finding a reciprocal, you're dealing with an inverse operation related to multiplication.
When you multiply a number by its reciprocal, the result is 1. For example:
  • If you have \(\frac{1}{a}\), then its reciprocal is \(a\).
  • Multiplying \(\frac{1}{a} \times a\) gives you 1.
These operations are fundamental in algebra, allowing you to "undo" operations and solve for unknown variables. Recognizing and applying inverse operations aids in comprehending more complex mathematical concepts.
Multiplication
Multiplication is a basic arithmetic operation that involves combining equal groups together. It's an essential skill for working with fractions and their reciprocals. When you multiply fractions, you simply multiply the numerators with each other and the denominators with each other.
To understand the multiplication of a number and its reciprocal, think about it like flipping the fraction and multiplying it with the original. For example:
  • The reciprocal of \(\frac{1}{a}\) is \(a\).
  • When you multiply \(\frac{1}{a}\times a\), you multiply the numerators together (1 and \(a\)) and the denominators together (\(a\) and 1).
  • This equals 1, which confirms the identity property of multiplication.
Multiplication is straightforward but incredibly powerful, especially in simplifying equations and solving mathematical problems with ease.

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Most popular questions from this chapter

Show that each of the following statements is an identity by transforming the left side of each one into the right side. \(\frac{\csc \theta}{\cot \theta}=\sec \theta\)

\(\sin \theta(\csc \theta-\sin \theta)=\cos ^{2} \theta\)

Write each of the following in terms of \(\cos \theta\) only: \(\sec \theta\)

Add and subtract as indicated. Then simplify your answers if possible. Leave all answers in terms of \(\sin \theta\) and/or \(\cos \theta\). $$ \cos \theta+\frac{1}{\sin \theta} $$

Show that each of the following statements is an identity by transforming the left side of each one into the right side. \(\sin \theta \cot \theta=\cos \theta\)
See all solutions

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