Problem 44 - Calculus Early Transcendentals: Pearson New International Edition

Question

Show that any rational number \(p / q\), for which the prime factorization of \(q\) consists entirely of \(2 \mathrm{~s}\) and \(5 \mathrm{~s}\), has a terminating decimal expansion.

Step-by-Step Solution

Answer

If \(q\)'s prime factors are only 2s and 5s, \(\frac{p}{q}\) terminates.

Step 1: Understanding the Problem

To determine if a rational number \(\frac{p}{q}\) has a terminating decimal expansion, we need to analyze the prime factorization of its denominator \(q\). According to the problem, \(q\) is composed only of the prime factors 2 and 5.

Step 2: Criteria for Terminating Decimal

A rational number in its simplest form \(\frac{a}{b}\) can be expressed as a terminating decimal if and only if the prime factorization of the denominator \(b\) contains only powers of 2 and/or 5. This means that \(b\) could be expressed as \(2^m \times 5^n\), where \(m\) and \(n\) are non-negative integers.

Step 3: Applying the Criteria

According to the problem statement, the prime factorization of \(q\) consists solely of 2s and 5s. This confirms the form \(q = 2^m \times 5^n\). Therefore, \(\frac{p}{q}\) meets the criteria for having a terminating decimal expansion.

Step 4: Conclusion

Since the prime factorization of \(q\) meets the criteria required for a terminative decimal expansion, we conclude that the rational number \(\frac{p}{q}\) indeed has a terminating decimal expansion.

Key Concepts

Rational Numbers

Rational numbers are a core concept in mathematics. They are numbers that can be expressed as a fraction where both the numerator and the denominator are integers. The denominator is not zero. For example, numbers like \( \frac{3}{4} \), \( \frac{-5}{2} \), and \( \frac{7}{1} \) are all rational numbers. Rational numbers include:

• Fractions like \( \frac{1}{2} \)
• Whole numbers like 3 (written as \( \frac{3}{1} \))
• Negative fractions like \( \frac{-6}{7} \)

A key characteristic of rational numbers is that their decimal expansion can either terminate or become repeating. This means a rational number's decimal form ends or the digits start repeating after a certain point.Understanding rational numbers is important for grasping many mathematical concepts including decimal expansions.

Prime Factorization

Prime factorization is the process of breaking down a number into its basic building blocks, which are prime numbers. A prime number is a whole number greater than 1 that can only be divided by 1 and itself without leaving a remainder, such as 2, 3, 5, 7, etc. To find the prime factorization of a number, you repeatedly divide it by the smallest prime numbers until the quotient itself is a prime number. For instance, the prime factorization of 20 is \( 2^2 \times 5^1 \). ### Why is Prime Factorization Important?

• It helps in simplifying fractions.
• It determines the behavior of numbers in decimal form, including whether a fraction has a terminating or repeating decimal.
• In this exercise, knowing that the denominator's prime factors are only 2s and 5s tells us the decimal expansion will terminate.

Decimal Expansion

Decimal expansion refers to the way a number is expressed in its decimal form. Decimal expansions can either be terminating or non-terminating. A terminating decimal is a decimal representation that ends after a finite number of digits. For example, \( 0.75 \) is a terminating decimal because it ends after two decimal places. This happens when the denominator of the rational number, in its simplest form, is composed of only the prime factors 2 and 5. ### Terminating vs. Non-Terminating Decimals

• Terminating decimals stop at a certain point.
• Non-terminating decimals keep going. These can be repeating, like \( 0.333... \) which repeats the digit 3 indefinitely.

Understanding decimal expansion helps give insight into the nature of numbers and how they relate to fractions and their denominators.

Denominator in Fractions

In fractions, the denominator is the number below the line. It shows the number of equal parts the whole is divided into. The role of a denominator is crucial in determining the form of a fraction when converted to a decimal. When the denominator of a fraction is composed entirely of the prime numbers 2 and 5 (in any combination), the fraction can be written as a terminating decimal. This means something like \( \frac{1}{8} \) (since 8 is \(2^3\)) will convert to a terminating decimal because 8 can be broken down to only 2s.### Key Points about Denominators:

• A denominator cannot be zero; division by zero is undefined.
• The prime factorization of the denominator (such as just 2s and 5s) largely influences the decimal form of a fraction.

Comprehending the denominator's role helps in predicting whether a fraction will have a terminating or repeating decimal form.