Problem 49 - Essential Calculus

Question

(a) write the repeating decimal as a geometric series and (b) write its sum as the ratio of two integers $$ 0 . \overline{81} $$

Step-by-Step Solution

Answer

\( 0.8181 \ldots = \frac{81}{99} \)

Step 1: Express the decimal as a geometric series

Let \( x = 0.8181 \ldots \). Multiplying this by 100 gives \( 100x = 81.8181 \ldots \) . Subtracting the first equation from this results in \( 99x = 81 \), so \( x = \frac{81}{99} \) as a geometric series this can be expressed as \( \frac{8}{10} + \frac{1}{100} + \frac{8}{10000} + \frac{1}{1000000} + \ldots \)

Step 2: Find the sum of the geometric series

The sum to infinity of a geometric series can be calculated using the formula \( S = \frac{a}{1-r} \), where \( a \) is the first term and \( r \) is the common ratio. In this case, \( a = \frac{8}{10} \) and \( r = \frac{1}{100} \). Substituting these values into the formula gives \( S = \frac{\frac{8}{10}}{1-\frac{1}{100}} = \frac{81}{99} \), which is the ratio of two integers.

Key Concepts

Repeating Decimals

Repeating decimals are numbers with a sequence of digits that repeats indefinitely after the decimal point. They can be expressed in two ways: using a bar over the repeating digits, as in \(0.\overline{81}\), or by listing the sequence once followed by an ellipsis, as in 0.8181... Repeating decimals can be represented in a more mathematical form through a geometric series. This series involves adding a sequence of terms where each term gets increasingly smaller. For instance, in \(0.\overline{81}\), the pattern of 81 repeats every two decimal places, allowing us to express it as a sum like \(\frac{8}{10} + \frac{1}{100} +\frac{8}{10000} + \frac{1}{1000000} + \ldots\) Understanding repeating decimals as series can greatly simplify computations and translations into other mathematical forms, like fractions.

Sum of Infinite Series

The sum of an infinite series might sound like a paradox, but when a sequence's terms are getting smaller, you can calculate it using specific formulas. A geometric series, where each term is a constant multiple of the previous one, is especially useful. To find 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 of the series. It's crucial that \(|r| < 1\) for the formula to apply, meaning the terms are decreasing. When applying this to our repeating decimal example, we have \(a = \frac{8}{10}\) and \(r = \frac{1}{100}\). Plugging these values into the formula gives us \(S = \frac{\frac{8}{10}}{1-\frac{1}{100}} = \frac{81}{99}\). Calculating these sums helps turn repeating decimals into single, rational expressions.

Rational Numbers

Rational numbers are numbers that can be expressed as the quotient of two integers, \(\frac{p}{q}\), where \(q eq 0\). They are a core component of mathematics, encompassing both integers and fractions. Repeating decimals, like \(0.\overline{81}\), can always be converted into a rational number because their recurring nature can be expressed as a fraction. In our example, through solving the geometric series, we found that \(0.\overline{81} = \frac{81}{99}\). Rational numbers are immensely valuable in math because they provide a concrete way to describe numbers with infinite decimal places using finite means. Understanding how repeating decimals relate to rational numbers helps in converting complex decimal expressions into simple, usable forms.