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Chapter 7: Problem 114

Determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false. Every decimal with a repeating pattern of digits is a rational number.

Short Answer

Expert verified
The statement is True. Every decimal with a repeating pattern of digits is a rational number, because it can be expressed as the quotient of two integers.

Step by step solution

01

Understand the definition of a rational number

A rational number is defined as a number that can be expressed as the quotient of two integers, where the denominator is not zero. In other words, if a number can be written in a fraction form, it is a rational number.
02

Understand the characteristic of repeating decimals

A repeating decimal is a kind of decimal notation in which after some point, the same pattern of digits repeats indefinitely. This can also be expressed in fraction form, where the numerator and denominator are integers.
03

Validate the statement

Based on the understanding from Step 1 and Step 2, it's clear that a decimal number with a repeating pattern can indeed be expressed as the quotient of two integers - this is evident from the method of converting repeating decimals to fractions. Therefore, a decimal with a repeating pattern is indeed a rational number.

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Most popular questions from this chapter

Find all values of \(x\) for which the series converges. For these values of \(x,\) write the sum of the series as a function of \(x\). $$ \sum_{n=1}^{\infty}\left(\frac{x^{2}}{x^{2}+4}\right)^{n} $$

(a) write the repeating decimal as a geometric series and (b) write its sum as the ratio of two integers $$ 0 . \overline{9} $$

Determine the convergence or divergence of the series. $$ \sum_{n=1}^{\infty} \ln \frac{1}{n} $$

Find the sum of the convergent series. $$ 4-2+1-\frac{1}{2}+\cdots $$

Use a graphing utility to graph the function. Identify the horizontal asymptote of the graph and determine its relationship to the sum of the series. $$ \frac{\text { Function }}{f(x)=2\left[\frac{1-(0.8)^{x}}{1-0.8}\right]} \frac{\text { Series }}{\sum_{n=0}^{\infty} 2\left(\frac{4}{5}\right)^{n}} $$
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