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Chapter 10: Problem 22

Express each of the numbers in Exercises \(19-26\) as the ratio of two integers. \(0 . \bar{d}=0 . d d d d \ldots, \quad\) where \(d\) is a digit

Short Answer

Expert verified
The repeating decimal \(0.\overline{d}\) is \(\frac{d}{9}\).

Step by step solution

01

Understand the Problem

We need to convert the repeating decimal \(0.\overline{d}\), where the digit \(d\) repeats indefinitely, into the form of a fraction (ratio of two integers). This type of problem typically requires setting up an equation to eliminate the repeating part.
02

Set Up the Equation

Let \( x = 0.\overline{d} \). This means \( x = 0.ddd\ldots \). To eliminate the repeating decimal, multiply \( x \) by 10: Then, we have \(10x = d.ddd\ldots \).
03

Subtract to Eliminate Repeating Part

We now have:\[\begin{align*}x &= 0.ddd\ldots \10x &= d.ddd\ldots \ \end{align*}\] Subtract \(x\) from \(10x\): \[10x - x = d \Rightarrow 9x = d\]
04

Solve for x

Divide both sides of the equation by 9 to solve for \(x\): \[x = \frac{d}{9}\] This shows that the repeating decimal \(0.\overline{d}\) can be expressed as the fraction \(\frac{d}{9}\).

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Key Concepts

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

Ratio of Integers
A ratio of integers is essentially a fraction where both the numerator and denominator are whole numbers. For example, \( \frac{3}{2} \) is a ratio of integers. The idea behind expressing a number as a ratio of two integers is transforming it into a format that allows us to easily manipulate and understand it numerically. When we encounter a repeating decimal, we aim to convert it into this form because fractions are much simpler to work with in mathematical equations and reasoning.
  • The numerator and denominator must both be integers.
  • There should ideally be no common factor other than 1, in simplest form.
  • The denominator cannot be zero as division by zero is undefined.
Understanding the concept of a ratio of integers is crucial because it forms the foundation for handling repeating decimals effectively.
Converting Decimals to Fractions
Converting decimals, especially repeating ones, into fractions might seem a bit challenging at first, but it becomes much simpler once you familiarize yourself with the steps. The goal is to reach the ratio of integers format which is a more natural state for mathematical expressions.Firstly, let's consider a simple non-repeating decimal, like 0.75. This can be written as \( \frac{75}{100} \), which simplifies to \( \frac{3}{4} \).For a repeating decimal such as \( 0.\overline{d} \), here's how the conversion happens:
  • Set the repeating decimal equal to \( x \). For example, \( x = 0.\overline{d} \).
  • Multiply by a power of 10 to shift the decimal point. For instance, \( 10x = d.ddd\ldots \).
  • Subtract the original equation from this new one to eliminate the repeating part, which simplifies the expression into a basic equation.
  • Solve this equation to express \( x \) as a fraction.
With these steps, you can convert repeating decimals into fractions efficiently, paving the way for easier calculations.
Eliminating Repeating Decimals
Repeating decimals, although they appear complex at first, can be eliminated with a clever process that involves an understanding of multiplication and subtraction.The technique is straightforward but requires careful execution:
  • Identify the repeating pattern. In our case, \( 0.\overline{d} \), which repeats indefinitely.
  • Set this repeating decimal equal to a variable, like \( x \).
  • Multiply \( x \) by a power of 10 so that the digits after the decimal point align when subtracted.
  • Write the two equations side by side and use subtraction to cancel out the repeating part.
  • The resulting equation typically leaves a simple expression to solve for \( x \).
For example, having \( 10x - x = d \) simplifies to \( 9x = d \), and ultimately, \( x = \frac{d}{9} \).This process simplifies the repeating decimal into a clear, concise fraction, which is much more user-friendly in mathematical operations.

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Most popular questions from this chapter

Determine if the sequence is monotonic and if it is bounded. \(a_{n}=\frac{2^{n} 3^{n}}{n !}\)

If \(\sum a_{n}\) converges absolutely, prove that \(\Sigma a_{n}^{2}\) converges.

Use a geometric series to represent each of the given functions as a power series about \(x=0,\) and find their intervals of convergence. a. \(f(x)=\frac{5}{3-x}\) b. \(\quad g(x)=\frac{3}{x-2}\)

a. Let \(P\) be an approximation of \(\pi\) accurate to \(n\) decimals. Show that \(P+\sin P\) gives an approximation correct to \(3 n\) decimals. (Hint: Let \(P=\pi+x .)\) b. Try it with a calculator.

Use a CAS to perform the following steps for the sequences. a. Calculate and then plot the first 25 terms of the sequence. Does the sequence appear to be bounded from above or below? Does it appear to converge or diverge? If it does converge, what is the limit \(L\) ? b. If the sequence converges, find an integer \(N\) such that \(\left|a_{n}-L\right| \leq 0.01\) for \(n \geq N .\) How far in the sequence do you have to get for the terms to lie within 0.0001 of \(L ?\) \(a_{n}=\frac{\ln n}{n}\)
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