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

A rational number with a denominator of only the primes 2 and 5 has a terminating decimal expansion.

Step 1: Understanding Termination Criteria

A decimal expansion terminates if it can be expressed as a fraction where the denominator is a power of 10. This equivalently means that a number has a terminating decimal if its denominator, in simplest form, consists only of the prime factors 2 and 5, because 10 = 2 x 5.

Step 2: Expressing the Rational Number

Express the given rational number as \( \frac{p}{q} \) where \(q\) is the denominator whose prime factors are only 2s and 5s. The fraction is given that the denominator consists entirely of 2 and 5 as prime factors.

Step 3: Converting to Power of 10

If the denominator \(q\) can be expressed as \(2^a \times 5^b\), find the maximum power max(a, b). Multiply both numerator \(p\) and denominator \(q\) by \(2^{b-a}\) if \(a < b\) or by \(5^{a-b}\) if \(b < a\) to make both exponents equal. This converts it to a power of 10.

Step 4: Conclusion

Since we can convert \(q\) into a form \(10^n\) by multiplying appropriately, the rational number \( \frac{p}{q} \) where \(q\) consists of 2s and 5s in prime factorization, has a denominator as a power of 10, proving that it has a terminating decimal expansion.

Key Concepts

Rational Numbers

Rational numbers are numbers that can be expressed as a fraction of two integers, where the numerator is an integer \( p \) and the denominator \( q \) is a non-zero integer. The general form is \( \frac{p}{q} \). Understanding rational numbers is crucial as they represent all numbers that can be written without a remainder. These include not only integers but also numbers with decimal or fractional parts. Rational numbers can be positive, negative, or zero, and the key feature is that they can be expressed precisely as a fraction of two integers.

• Examples include: \( \frac{1}{2} \), \( -4 \), and \( 3.75 \).

A rational number will have a terminating or repeating decimal pattern, which helps us in determining how it can be expressed numerically.

Prime Factorization

The prime factorization of a number involves expressing it as a product of prime numbers. Each number has a unique set of prime numbers that multiply together to equal the original number. For example, the number 60 has a prime factorization of \(2^2 \times 3 \times 5\). Understanding prime factorization is important in many areas of mathematics, especially in simplifying fractions and finding least common multiples.

• Prime numbers are numbers greater than 1 that have no divisors other than 1 and themselves.
• Examples of prime numbers include 2, 3, 5, 7, and 11.

In the context of terminating decimals, prime factorization helps determine if the decimal expansion of a fraction is finite. If the denominator's prime factors are only 2s and 5s, then the fraction's decimal form will terminate, as these are the factors of 10.

Decimal Expansion

Decimal expansion refers to how a number is represented in the decimal system. It reflects either a terminating sequence or a repeating sequence of digits following the decimal point. Terminating decimals end or "terminate" after a certain number of digits. These are closely related to rational numbers where the denominator has specific properties.

• For example, \( \frac{1}{8} = 0.125 \) is a terminating decimal.
• A repeating decimal like \( \frac{1}{3} = 0.333... \) continues indefinitely with a repeating sequence.

Understanding decimal expansion is critical in recognizing and working with rational numbers in decimal form. A rational number will have a terminating decimal expansion if, in its simplest form, the denominator consists only of the prime factors 2 and 5. This would mean after a set number of decimal places, it would end.

Power of 10

Powers of 10 are exponential numbers where the base is 10 and the exponent indicates how many times to multiply 10. For example, \(10^3 = 1000\) means 10 is multiplied by itself three times. They are fundamental in understanding the metric system and how our decimal number system works.

• Any non-negative integer power of 10 results in a number with 1 followed by that many zeros.
• Conversely, negative powers of 10 such as \(10^{-1} = 0.1\), represent fractions of 10.

In the context of the exercise, the power of 10 is used to convert a fraction into a terminating decimal. When a denominator of a fraction is manipulated to become a power of 10, it ensures that the decimal representation of the fraction stops or terminates after a certain number of places. This property is used to transform a fraction into a terminating decimal by ensuring the prime factors of its denominator are only 2 and 5, allowing it to be expressed as an appropriate power of 10.