Problem 29 - Essential Calculus Early Transcendentals

Question

Let \(x=0.99999 \ldots\) (a) Do you think that \(x<1\) or \(x=1 ?\) (b) Sum a geometric series to find the value of \(x\) . (c) How many decimal representations does the number 1 have? (d) Which numbers have more than one decimal representation?

Step-by-Step Solution

Answer

(a) \(x = 1\). (b) Sum is 1. (c) Infinitely many. (d) Numbers with finite decimals.

Step 1: Understanding the Infinite Decimal

When we write \(x = 0.99999\ldots\), we mean that \(x\) is a repeating decimal that continues infinitely. Intuitively, \(0.99999\ldots\) appears extremely close to 1. Our task is to determine whether \(x\) is actually less than 1 or equal to 1.

Step 2: Convert to Series Representation

The number \(0.99999\ldots\) can be expressed as an infinite geometric series: \(0.9 + 0.09 + 0.009 + \ldots\). This series has a first term \(a = 0.9\) and a common ratio \(r = 0.1\).

Step 3: Sum the Infinite Geometric Series

The formula for the sum \(S\) of an infinite geometric series \(a + ar + ar^2 + \cdots\) is \(S = \frac{a}{1-r}\) when \(|r| < 1\). Here, \(a = 0.9\) and \(r = 0.1\), so the sum is \(\frac{0.9}{1-0.1} = \frac{0.9}{0.9} = 1\). Thus, \(x = 1\).

Step 4: Consider Decimal Representations

Decimal numbers can sometimes have multiple representations. For instance, \(1.0000\ldots = 0.99999\ldots\), demonstrating the idea that a number can be represented as a terminating decimal or an equivalent repeating decimal.

Step 5: Identify Numbers with Multiple Representations

In general, numbers that have a finite decimal representation (like \(1.0000\ldots\)) may also have a repeating representation with 9s. These are exactly the numbers that are decimals terminating in zeros.

Key Concepts

Infinite Series

An infinite series is a sum of an endless sequence of terms. It doesn't have a finite number of terms, which means it continues without stopping. When dealing with decimals like 0.99999..., we're actually looking at an infinite geometric series.
\[0.99999\ldots = 0.9 + 0.09 + 0.009 + \cdots\]
This is a classic example of how infinite series can be used to represent infinite repeating decimals. The terms ( * First term ( * 0.9 in our example )* Common ratio ( * The factor by which each term is multiplied to get the next, 0.1 in this case ))These are key components for understanding geometric series. The sum of the series in this context can be calculated using the sum formula for infinite geometric series:\[S = \frac{a}{1-r}\]where \(a\) is the first term and \(r\) is the common ratio.

Decimal Representation

Decimal representation is how we write numbers in the base-10 number system. This includes numbers with fractional parts, which are often shown after a decimal point.
Every number can be represented in this system, even if it means having a repeating block of digits. For example, 0.3333... represents one-third, or 1/3. In this exercise, we're exploring the representation of 1 as a recurring decimal, 0.99999....

• Terminating decimals have a finite number of digits after the decimal point (e.g., 0.25).
• Non-terminating, repeating decimals continue indefinitely with a repeating pattern (e.g., 0.99999...).

In the decimal system, a number like 1 can have multiple representations. The number 1 can be written as 1.0000... or as 0.9999.... Both representations are valid and indeed the same numerically.

Recurring Decimals

Recurring decimals, also known as repeating decimals, are decimals in which a digit or group of digits repeat infinitely. This happens when division leads to a remainder that repeats in a cycle. For instance, 1/3 = 0.3333..., with 3 repeating forever.
Understanding recurring decimals helps clarify how a number like 0.99999... can be equal to 1. It highlights that appearances can be deceiving in math, and numeric value doesn't change with its representation.
Numbers with more than one decimal representation tend to be those with terminating decimals. Any number that can end in a string of zeros (like 1.0000...) can also be expressed as repeating decimals with 9s (like 0.9999...).
Such expressions underscore the flexibility of the decimal system, where:

• Recurring decimals offer alternative representations for the same value.
• Different representations can make calculations or concepts easier to understand.

This concept demonstrates the elegance of mathematical equality in different forms.