Problem 22 - Thomas Calculus

Question

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

Step-by-Step Solution

Answer

The number \(0.\overline{d}\) is equal to \(\frac{d}{9}\).

Step 1: Understand the Repeating Decimal

The given number is \(0.\overline{d}\), which means the decimal \(d\) repeats indefinitely. For example, if \(d=3\), the number becomes \(0.3333\ldots\).

Step 2: Set Up the Equation

Let \(x = 0.\overline{d}\). Then \(x = 0.ddd\ldots\). We can write this as \(x = d/10 + d/100 + d/1000 + \ldots\).

Step 3: Recognize the Geometric Series

The repeating part \(ddd\ldots\) is a geometric series with the first term \(a = d/10\) and the common ratio \(r = 1/10\).

Step 4: Apply the Formula for Infinite Geometric Series

Recall the formula for an infinite geometric series: \(S = \frac{a}{1-r}\). Substituting \(a = d/10\) and \(r = 1/10\) gives \(x = \frac{d/10}{1-1/10} = \frac{d/10}{9/10}\).

Step 5: Simplify the Expression

Simplifying \(\frac{d/10}{9/10}\), we have \(x = \frac{d}{9}\). This means \(0.\overline{d}\) is expressed as the ratio \(\frac{d}{9}\) in terms of integers.

Key Concepts

Geometric Series

A geometric series is a series of numbers with a specific pattern. Each term after the first is found by multiplying the previous term by a fixed number called the common ratio. This common ratio can be greater than or less than one, or even negative. For example, the series \(1, 2, 4, 8, 16\ldots\) has a common ratio of 2, because each term is obtained by multiplying the previous term by 2.
Geometric series are everywhere around us. They are used in calculating interest in finance, determining population growth in biology, and are fundamental in understanding repeating decimals like \(0.\overline{d}\).
These series are characterized by a few key components:

• First Term \(a\): This is the starting number of the series.
• Common Ratio \(r\): This is the constant factor between consecutive terms.
• Sum of \(n\) Terms: The formula for the sum of the first \(n\) terms is \(S_n = a \frac{1-r^n}{1-r}\), if \(r eq 1\).

The geometric series is a foundation for solving repeating decimals, since repeating decimals can be expressed as series.

Infinite Geometric Series

An infinite geometric series is a special type of geometric series that goes on forever, or infinitely. It has an infinite number of terms. The series continues as long as there is a constant ratio between the terms.
An important aspect of an infinite geometric series is whether it converges or diverges. Converges means that as you add more and more terms, the series approaches a specific value. Diverges means it does not approach a single value.
For an infinite geometric series to converge, its common ratio \(r\) must satisfy \( |r| < 1 \). When this condition is met, you can use the formula for the sum \(S\) of an infinite series:

• \( S = \frac{a}{1-r} \)

Here, \(a\) is the first term and \(r\) is the common ratio.
For the example of repeating decimals, \(0.\overline{d}\), we treat them as infinite geometric series. The repeating part can be decomposed into terms like \(d/10, d/100, d/1000\ldots\), making it manageable to sum them using this formula. This gives us a simple way of converting a repeating decimal into a fraction.

Ratio of Integers

The ratio of two integers is simply a fraction. Fractions are one of the basic ways to express parts of a whole, and are especially useful when dealing with repeating decimals.
When you see a repeating decimal like \(0.\overline{d}\), it might look complex, but it can be converted into a simpler form. This simpler form is a fraction, which is the ratio of two integers.
Here's the general approach to convert a repeating decimal into a ratio:

• Recognize the repeating pattern in the decimal.
• Express the repeating decimal as a geometric series.
• Use the formula for the sum of an infinite geometric series to find the equivalent fraction.

In our specific case with \(0.\overline{d}\), the first term is \(d/10\), and the common ratio \(r\) is \(1/10\). By substituting into the formula \( S = \frac{a}{1-r} \), we get \( \frac{d/10}{1/10} \), which simplifies to \(\frac{d}{9}\). Thus, the repeating decimal \(0.\overline{d}\) is expressed as \(\frac{d}{9}\), a simple ratio of two integers.