Which Fraction Has A Terminating Decimal As Its Decimal Expansion

Which Fractions Have a Terminating Decimal as Their Decimal Expansion?

Understanding which fractions produce terminating decimals is crucial for anyone working with numbers, from students mastering arithmetic to professionals using numerical computation. A terminating decimal is a decimal representation that ends, unlike a repeating decimal which continues infinitely with a repeating pattern. This article delves deep into the fascinating world of fractions and their decimal expansions, exploring the conditions that determine whether a fraction results in a terminating or repeating decimal. We'll uncover the underlying mathematical principles, providing you with a clear and comprehensive understanding of this important concept.

The Key to Termination: The Denominator's Prime Factorization

The secret to determining whether a fraction will have a terminating decimal expansion lies entirely within its denominator. More specifically, it's all about the denominator's prime factorization. Let's break it down:

A fraction can be represented as a/b, where 'a' is the numerator and 'b' is the denominator. To understand if a/b has a terminating decimal expansion, we need to examine the prime factorization of 'b'.

Prime Factorization Refresher

Prime factorization is the process of expressing a number as a product of its prime factors. Prime numbers are numbers greater than 1 that are only divisible by 1 and themselves (e.g., 2, 3, 5, 7, 11, etc.). For example:

• 12 = 2 x 2 x 3 = 2² x 3
• 20 = 2 x 2 x 5 = 2² x 5
• 35 = 5 x 7

The Terminating Decimal Rule: Only 2s and 5s

The crucial rule is this: A fraction a/b has a terminating decimal expansion if and only if the denominator b, when expressed in its prime factorization, contains only the prime factors 2 and/or 5 (or a combination of both).

Understanding the Rule: Why 2 and 5?

Our decimal system is base-10, meaning it's based on powers of 10. And 10 itself is 2 x 5. When converting a fraction to a decimal, we are essentially dividing the numerator by the denominator. If the denominator's prime factorization consists only of 2s and/or 5s, the division process will eventually terminate because we can manipulate the fraction to create an equivalent fraction with a denominator that's a power of 10. This leads to a finite number of digits after the decimal point.

Examples of Terminating Decimals

Let's illustrate this with some examples:

1. 1/4:

The prime factorization of 4 is 2 x 2 = 2². Therefore, 1/4 will have a terminating decimal expansion. Indeed, 1/4 = 0.25.

2. 7/20:

The prime factorization of 20 is 2 x 2 x 5 = 2² x 5. Since the denominator contains only 2s and 5s, 7/20 will also have a terminating decimal expansion. In fact, 7/20 = 0.35.

3. 3/8:

The prime factorization of 8 is 2 x 2 x 2 = 2³. Only the prime factor 2 is present, so 3/8 will have a terminating decimal: 3/8 = 0.375.

4. 17/50:

The prime factorization of 50 is 2 x 5 x 5 = 2 x 5². Both 2 and 5 are present, leading to a terminating decimal: 17/50 = 0.34.

Examples of Non-Terminating (Repeating) Decimals

Now let's examine fractions that don't have terminating decimal expansions:

1. 1/3:

The prime factorization of 3 is simply 3. Since 3 is neither 2 nor 5, the decimal expansion of 1/3 is non-terminating and repeating: 1/3 = 0.333...

2. 5/12:

The prime factorization of 12 is 2 x 2 x 3 = 2² x 3. The presence of the prime factor 3 means that 5/12 will have a repeating decimal expansion: 5/12 = 0.41666...

3. 7/18:

18 = 2 x 3 x 3 = 2 x 3². The presence of 3 results in a repeating decimal: 7/18 = 0.3888...

The Process: Converting Fractions to Decimals

Let's revisit the conversion process itself to further illuminate why only 2s and 5s in the denominator yield terminating decimals. We convert a fraction to a decimal by performing long division. If the denominator contains only 2s and 5s, we can always find an equivalent fraction with a denominator that's a power of 10. This allows the division to terminate.

For instance, consider 7/20:

• We can rewrite 20 as 2² x 5.
• To get a denominator of 10², we can multiply both numerator and denominator by 5: (7 x 5) / (20 x 5) = 35/100.
• 35/100 is equivalent to 0.35, a terminating decimal.

However, if the denominator contains prime factors other than 2 and 5, we cannot create an equivalent fraction with a power of 10 as the denominator, and the division process will continue indefinitely, resulting in a repeating decimal.

Practical Applications and Further Exploration

Understanding terminating decimals has broad practical implications in various fields. It's essential in:

• Engineering and Physics: Accurate calculations often require precise decimal representations.
• Finance: Calculations involving money invariably require terminating decimals for practical reasons.
• Computer Science: Numerical computations in computer programs often need to deal with the precision and limitations of terminating versus repeating decimals.

Further exploration could include:

• Investigating the patterns in repeating decimals: The lengths of repeating sequences can be analyzed mathematically.
• Exploring different number bases: The concept of terminating decimals changes slightly when we move beyond base-10.
• The relationship between rational and irrational numbers: Terminating decimals represent a subset of rational numbers.

Conclusion

The determination of whether a fraction results in a terminating or repeating decimal is entirely dependent on the prime factorization of its denominator. Only fractions with denominators containing only the prime factors 2 and/or 5 (or a combination of both) will have terminating decimal expansions. This fundamental understanding provides a powerful tool for numerical analysis and problem-solving across various disciplines. By mastering this concept, you gain a deeper appreciation for the elegance and underlying structure within the seemingly simple world of fractions and decimals.