Question

Consider the number \(0.555555 \ldots,\) which can be viewed as the series \(5 \sum_{k=1}^{\infty} 10^{-k} .\) Evaluate the geometric series to obtain a rational value of \(0.555555 .\) b. Consider the number \(0.54545454 \ldots\), which can be represented by the series \(54 \sum_{k=1}^{\infty} 10^{-2 k} .\) Evaluate the geometric series to obtain a rational value of the number. c. Now generalize parts (a) and (b). Suppose you are given a number with a decimal expansion that repeats in cycles of length \(p,\) say, \(n_{1}, n_{2} \ldots ., n_{p},\) where \(n_{1}, \ldots, n_{p}\) are integers between 0 and \(9 .\) Explain how to use geometric series to obtain a rational form for \(0 . \overline{n_{1}} n_{2} \cdots n_{p}\) d. Try the method of part (c) on the number \(0 . \overline{123456789}=0.123456789123456789 \ldots\) e. Prove that \(0 . \overline{9}=1\)

Short answer

Question: Apply the general method from part (c) to determine the rational value of the following repeating decimal: 0.3678245823 Answer: To apply the general method to the given number, we first need to identify the length of the repeating part. In this case, the repeating part is "3678245823", which is of length 10. Now let's apply the general formula to the given number \(0.\overline{3678245823}\): $$x = \cfrac{3678245823}{10^{10} - 1} = \cfrac{3678245823}{9999999999}.$$ The rational value of \(0.\overline{3678245823}\) is \(\cfrac{3678245823}{9999999999}\).

Step-by-step solution

Calculate the Sum of the Series for 0.555555 (Part a)

Using the geometric series formula, the sum of the series \(5\sum_{k=1}^{\infty} 10^{-k}\) can be calculated: $$S = \cfrac{5}{1-\frac{1}{10}} = \cfrac{5}{\cfrac{9}{10}} = \cfrac{50}{9}.$$ The rational value of 0.555555 is \(\cfrac{50}{9}\).

Calculate the Sum of the Series for 0.54545454 (Part b)

Using the geometric series formula, the sum of the series \(54\sum_{k=1}^{\infty} 10^{-2k}\) can be calculated: $$S = \cfrac{54}{1-\frac{1}{100}} = \cfrac{54}{\cfrac{99}{100}} = \cfrac{5400}{99}.$$ The rational value of 0.54545454 is \(\cfrac{5400}{99}\).

Generalize the Method for any Repeating Decimal (Part c)

Let's denote the repeating decimal as \(0.\overline{n_1n_2\cdots n_p}\), where \(n_i\) are integers between 0 and 9, and the length of the repeating part is p. We can write the number as the following geometric series: $$x = \sum_{k=1}^{\infty} \cfrac{n_1n_2\cdots n_p}{10^{kp}} = (n_1n_2\cdots n_p) \sum_{k=1}^{\infty} \cfrac{1}{10^{kp}}.$$ Now let's find the sum of this geometrical series: $$S = \cfrac{n_1n_2\cdots n_p}{1-\frac{1}{10^p}}.$$ The rational value of the repeating decimal is: $$x = \cfrac{n_1n_2\cdots n_p}{10^p-1}.$$

Apply the General Method to 0.123456789 (Part d)

Now let's apply the general formula to the given number \(0.\overline{123456789}\): $$x = \cfrac{123456789}{10^9 - 1} = \cfrac{123456789}{999999999}.$$ The rational value of \(0.\overline{123456789}\) is \(\cfrac{123456789}{999999999}\).

Prove that 0.9 = 1 (Part e)

Let's apply the general formula to the given number \(0.\overline{9}\): $$x = \cfrac{9}{10^1 - 1} = \cfrac{9}{9} = 1.$$ From the calculation, it is proved that \(0.\overline{9} = 1\).

Key concepts

Geometric Series

Understanding the concept of a geometric series is crucial when dealing with repeating decimals. A geometric series is a sequence of numbers where each term after the first is found by multiplying the previous one by a fixed non-zero number, known as the common ratio. For example, in the series 2, 6, 18, 54, ..., each term is 3 times the previous term, thus the common ratio is 3.

When we express a repeating decimal like the one given, \(0.555555\ldots\), in terms of a geometric series, we leverage the common ratio of \(\frac{1}{10}\) for each subsequent term. In the case of repeating decimals, this series doesn't stop, which makes it an infinite geometric series. The beauty of mathematics allows us to sum this infinite series to find a precise rational number that corresponds to the repeating decimal.

To calculate the sum of an infinite geometric series, we use the formula \(S = \frac{a}{1-r}\), where \(a\) is the first term, and \(r\) is the common ratio. The key takeaway is that if the common ratio's absolute value is less than one, the series converges to a rational number, which allows us to make sense of those never-ending decimals.

Calculus

Calculus, often deemed as the mathematics of change, gives us powerful tools to analyze and solve problems involving continuous phenomena. It might seem abstract at first, but when studying repeating decimals, calculus notions come in handy, especially when dealing with infinite series.

Through the lens of calculus, an infinite series like the sum \(\sum_{k=1}^{\text{infinity}} \frac{1}{10^k}\) can be rigorously analyzed to determine convergence or divergence and its ultimate sum if it converges. The convergence test, derived from calculus, ensures that the geometric series corresponding to a repeating decimal will indeed sum up to a rational value. But calculus isn't just about proving the point; it also enables us to generalize formulas and apply them to various functions and sequences, giving us a comprehensive understanding of their behavior and how to succinctly represent infinitely repeating processes with finite expressions.

Rational Value of Decimals

When working with numbers, it's often necessary to understand the type of number we're dealing with. Rational numbers, those that can be expressed as a fraction of two integers, are particularly interesting when we consider repeating decimals. A repeating decimal is an infinite decimal that, after a certain point, consists of a finite sequence of digits repeated indefinitely.

For instance, the repeating decimal \(0.555555\ldots\) has a rational equivalent, which can be arrived at by summing its corresponding geometric series. This is where a fundamental understanding of how to convert a repeating decimal to a rational number becomes pivotal. The given exercises showcase this conversion by transforming the repeating figures into fractions with integers in the numerator and the denominator. This has profound implications not only in academic settings but also in real-world applications where precise calculations are necessary, and approximations just won't do.

Infinite Series

An infinite series is a sum of infinitely many terms. While this concept may seem counterintuitive, certain infinite series converge to a finite value, meaning that their sum approaches a certain number as more and more terms are added. This convergence is essential for making sense of repeating decimals since every repeating decimal represents an infinite series of digits.

The trick to working with infinite series is to determine if, and to what value, the series converges. In the context of repeating decimals, we focus specifically on infinite geometric series, where each term is derived from the last by multiplying by a constant ratio. By applying the formula for the sum of an infinite geometric series, we can convert these never-ending decimals into precise and neatly packaged rational numbers. Understanding infinite series isn't just a neat mathematical trick—it is a fundamental skill in interpreting and representing continuous quantities in discrete terms, a skill that finds applications in fields ranging from finance to physics.