Rational Numbers as Terminating and Repeating Decimals

Introduction

Key Concepts

Definition of Rational Numbers

A rational number is any number that can be expressed as the ratio of two integers, where the denominator is not zero. Formally, a rational number can be written as:

where \( p \) and \( q \) are integers, and \( q \neq 0 \). This definition encompasses integers, fractions, and certain decimals, making rational numbers a versatile and essential component of the number system.

Terminating Decimals

Terminating decimals are decimal numbers that have a finite number of digits after the decimal point. A rational number will terminate if, after simplifying the fraction, the denominator contains only the prime factors 2 and/or 5. This is because the base-10 system is based on these prime factors.

For example:

• \(\frac{1}{2} = 0.5\) – Terminates
• \(\frac{3}{8} = 0.375\) – Terminates
• \(\frac{7}{20} = 0.35\) – Terminates

In each case, the denominator after simplification has prime factors 2 and/or 5, resulting in a decimal that ends after a certain number of digits.

Repeating Decimals

Repeating decimals are decimals that have one or more repeating digits or sequences of digits after the decimal point. A rational number will result in a repeating decimal if, after simplifying the fraction, the denominator contains prime factors other than 2 and 5.

For example:

• \(\frac{1}{3} = 0.\overline{3}\) – Repeats infinitely
• \(\frac{7}{6} = 1.1\overline{6}\) – Repeats infinitely
• \(\frac{5}{11} = 0.\overline{45}\) – Repeats infinitely

These decimals never terminate and instead continue indefinitely with a repeating pattern.

Converting Fractions to Decimals

To convert a fraction to a decimal, divide the numerator by the denominator. The result will either terminate or repeat based on the factors of the denominator, as previously discussed.

Example: Convert \(\frac{4}{25}\) to a decimal.

Divide 4 by 25:

Since 25 = \(5^2\), a denominator with only the prime factor 5, the decimal terminates.

Identifying Terminating and Repeating Decimals

To determine whether a decimal is terminating or repeating:

1. Simplify the fraction to its lowest terms.
2. Factorize the denominator.
3. If the denominator has prime factors other than 2 and 5, the decimal will repeat.
4. If the denominator has only 2 and/or 5 as prime factors, the decimal will terminate.

Example: Determine if \(\frac{14}{28}\) is a terminating or repeating decimal.

Simplify the fraction:

The denominator is 2, which is a prime factor of 10. Therefore, the decimal terminates:

Theoretical Explanation

The behavior of a rational number when expressed as a decimal is intrinsically linked to the denominator's prime factors after simplification. Since the decimal system is base-10, which factors as \(2 \times 5\), having a denominator composed solely of these primes allows the decimal to terminate by finiteness. If other primes are present, the decimal cannot be expressed finitely and hence repeats infinitely.

Mathematically, if a fraction \(\frac{p}{q}\) can be expressed as: $$ \frac{p}{q} = \frac{p \times k}{q \times k} = \frac{pk}{qk} $$ where \(k\) is chosen such that \(qk\) is a power of 10, then the decimal terminates. Otherwise, the decimal repeats.

Examples and Applications

Understanding whether a rational number results in a terminating or repeating decimal has practical applications in various fields such as engineering, computer science, and finance.

Example 1: Calculate \(\frac{9}{40}\) as a decimal.

Since 40 = \(2^3 \times 5\), the decimal terminates:

Example 2: Calculate \(\frac{7}{12}\) as a decimal.

Simplify the fraction:

12 = \(2^2 \times 3\), which includes a prime factor other than 2 and 5, hence the decimal repeats:

Application: In finance, understanding repeating decimals is crucial for calculating interest rates and amortization schedules, ensuring accuracy in long-term financial planning.

Equations and Formulas

Several formulas help in determining the nature of decimals derived from rational numbers:

• Terminating Decimal: If \( q = 2^m \times 5^n \), then \(\frac{p}{q}\) is a terminating decimal.
• Repeating Decimal: If \( q \) has any prime factors other than 2 or 5, then \(\frac{p}{q}\) is a repeating decimal.

Example Formula Application:

Determine if \(\frac{22}{7}\) is terminating or repeating:

Factorize the denominator:

Therefore, \(\frac{22}{7}\) is a repeating decimal: $$ \frac{22}{7} \approx 3.\overline{142857} $$

Visual Representation

Graphs and number lines can visually represent the concepts of terminating and repeating decimals. Terminating decimals can be pinpointed precisely on the number line, whereas repeating decimals indicate an ongoing, cyclical pattern.

Example: Plotting \(\frac{1}{4}\) and \(\frac{1}{3}\) on a number line.

\(\frac{1}{4} = 0.25\) – A fixed point on the number line.

\(\frac{1}{3} = 0.\overline{3}\) – An approximation needs to be used as it never precisely terminates.

Common Misconceptions

One common misconception is that all fractions result in repeating decimals. In reality, only fractions with denominators containing prime factors other than 2 and 5 produce repeating decimals. Another misconception is confusing rational numbers with irrational numbers, which by definition cannot be written as terminating or repeating decimals.

Clarification: \(\frac{1}{2} = 0.5\) terminates, whereas \(\frac{1}{\sqrt{2}}\) is irrational and does not terminate or repeat.

Implications in Advanced Mathematics

Understanding terminating and repeating decimals is foundational for more complex areas such as calculus, number theory, and computer algorithms. It aids in grasping concepts like limits, series, and the properties of real numbers.

In number theory, recognizing the decimal expansion of rational numbers helps in exploring periodicity and patterns within numerical systems. In computer science, it is crucial for data representation and algorithm efficiency.

Comparison Table

Summary and Key Takeaways