Problem 48 - Calculus Early Transcendentals: Pearson New International Edition

Question

. Is there a number between \(0.9999 \ldots\) (repeating \(9 \mathrm{~s}\) ) and 1? How do you resolve this with the statement that between any two different real numbers there is another real number?

Step-by-Step Solution

Answer

No, because 0.9999... is equal to 1, so they are not distinct numbers.

Step 1: Understanding Repeating Decimals

The number given is expressed with repeating nines: \( 0.9999 ext{...} \). In decimal notation, this indicates an infinitely repeating sequence of 9s after the decimal point.

Step 2: Convert Repeating Decimal to Fraction

To understand \( 0.9999 ext{...} \), we convert it into a fraction. We set \( x = 0.9999 ext{...} \). Multiply both sides by 10 to shift the decimal: \( 10x = 9.9999 ext{...} \). Now subtract \( x = 0.9999 ext{...} \) from this: \[ 10x - x = 9.9999 ext{...} - 0.9999 ext{...} \]This simplifies to \( 9x = 9 \), so \( x = 1 \). Thus, \( 0.9999 ext{...} = 1 \).

Step 3: Apply the Property of Real Numbers

The mathematical property states that between any two distinct real numbers, there is another real number. However, since \( 0.9999 ext{...} \) is exactly equal to 1, there are no two different real numbers here. This means the property does not apply because we have two equivalent numbers rather than distinct ones.

Step 4: Conclusion

Since \( 0.9999 ext{...} = 1 \), there is effectively no gap between them, and thus no real number can exist 'between' \( 0.9999 ext{...} \) and 1 as they are the same number.

Key Concepts

Repeating Decimals

Repeating decimals occur when a number has one or more digits after the decimal point that repeat indefinitely. A common example is 0.3333... where the '3' repeats endlessly. Understanding repeating decimals is key because they link directly to our decimal and fraction conversion process.
Repeating decimals provide a rich insight into how we express numbers differently. Instead of writing an endless sequence of a repeating number, mathematicians often use a bar above the repeating digits (e.g., \(0.\overline{9}\) for \(0.9999...\)). This notation indicates the repeating segment, making it easier to work with such numbers in calculations. Repeating decimals are not as straightforward as they seem, as they require recognizing how they align with fractions and other number types.

Decimal to Fraction Conversion

Decimal to fraction conversion involves transforming a decimal into its equivalent fraction form. The process is particularly fascinating when dealing with repeating decimals. Take any repeating decimal like \(0.9999...\). By setting it equal to a variable (say \(x\)), we can manipulate this value by moving the decimal point through multiplication.
Here's the step-by-step conversion of \(0.9999...\) to \(1\), showing the relationship between decimals and fractions:

• Assign \(x = 0.9999...\).
• Multiply both sides by 10: \(10x = 9.9999...\).
• Subtract \(x = 0.9999...\) from \(10x = 9.9999...\).
• This results in \(9x = 9\).
• Solve for \(x\): \(x = \frac{9}{9} = 1\).

Thus, the repeating decimal \(0.9999...\) can be precisely expressed as the fraction \(1\). This reflects the idea that many repeating decimals are simply another form of whole numbers.

Properties of Real Numbers

Real numbers are the entire set of numbers that include both rational and irrational numbers, covering everything from whole numbers to fractions and decimals. One of the crucial properties of real numbers is that between any two distinct real numbers, there exists another real number. This property emphasizes the concept of density in real numbers.
However, a curious case arises with numbers like \(0.9999...\) and \(1\). As demonstrated through decimal to fraction conversion, \(0.9999... = 1\), meaning they are not distinct but equivalent. Hence, the property of density does not apply since we are dealing with two representations of the same number, not two different numbers.
Understanding this highlights a fundamental aspect of real numbers: equivalent representations can often appear different, yet belong to the same point on the number line. This knowledge not only clarifies tricky scenarios but also enriches our comprehension of the real number system as a whole.