Decimal Expansion of Rational Numbers: Rules and Examples

What is a decimal expansion?

A decimal expansion is the representation of a number using a decimal point. It converts a fraction into a decimal by dividing the numerator by the denominator. This process yields the rational decimal representation, revealing exactly how the number behaves past the decimal point.

Decimal expansions of rational numbers

When we divide the numerator of a rational fraction by its denominator, the resulting quotient is its decimal expansion. Analyzing the decimal forms of rational numbers reveals that every rational number decimal expansion takes one of two possible forms: a decimal that ends or a decimal that continues endlessly with a repeating pattern of digits.

Terminating versus recurring expansions

A decimal expansion is either terminating or recurring based on how the division process unfolds. Evaluating terminating, recurring decimals allows us to classify fractions accurately without ambiguity.

Proper terminating decimals end after a finite number of digits. This happens when the division reaches a remainder of zero. For instance, $\dfrac{1}{4} = 0.25$.

In contrast, recurring decimals continue forever, but a specific digit or sequence of digits repeats endlessly. This occurs when the division yields a repeating remainder, meaning it never reaches zero. For instance, $\dfrac{1}{3} = 0.333\ldots$.

Use denominator factors

You can determine the type of decimal expansion without performing long division by analyzing the prime factors of the fraction's denominator. The fraction must first be in its simplest form.

If the prime factorization of the denominator contains only the numbers $2$, $5$, or both, the fraction will have a terminating decimal expansion. If the denominator contains any prime factor other than $2$ or $5$, the decimal expansion will be recurring.

Contrast rational and irrational decimals

All real numbers can be written as decimals, but their behavior separates the rational from the irrational. While rational numbers always terminate or repeat, irrational numbers have decimal expansions that never terminate and never settle into a repeating pattern.

Worked examples

Working through examples builds confidence in predicting decimal behavior without relying on long division.

Example 1: Identifying a terminating decimal

Question: Without performing long division, determine if the decimal expansion of $\dfrac{13}{25}$ is terminating or recurring.

Method:

1. Check if the fraction is in simplest form. The numbers $13$ and $25$ share no common factors.
2. Find the prime factorization of the denominator.
3. The denominator is $25$, and its prime factorization is $5 \times 5$.
4. Apply the denominator rule. Since the only prime factor is $5$, the expansion terminates.

Answer: The decimal expansion is terminating.

Check: Using division, $\dfrac{13}{25} = 0.52$, which confirms it terminates.

Example 2: Identifying a recurring decimal

Question: Is the decimal expansion of $\dfrac{5}{12}$ terminating or recurring?

Method:

1. Verify the fraction is in simplest form. The numbers $5$ and $12$ share no common factors.
2. Factor the denominator $12$ into its prime components.
3. The prime factorization of $12$ is $2 \times 2 \times 3$.
4. Check for factors other than $2$ and $5$. The presence of the factor $3$ means the decimal will not terminate.

Answer: The decimal expansion is recurring.

Check: Using division, $\dfrac{5}{12} = 0.41666\ldots$, which confirms the digit $6$ recurs.

Example 3: Applying the rule after simplification

Question: Determine the type of decimal expansion for $\dfrac{21}{60}$.

Method:

1. Check if the fraction can be simplified. Both $21$ and $60$ are divisible by $3$.
2. Simplify the fraction: $\dfrac{21}{60} = \dfrac{7}{20}$.
3. Factor the new denominator $20$ into primes: $20 = 2 \times 2 \times 5$.
4. Evaluate the factors. Since they are only $2$ and $5$, the expansion will terminate.

Answer: The decimal expansion is terminating.

Check: Using division, $\dfrac{21}{60} = 0.35$, which clearly terminates.

Common mistakes

A frequent error is checking the denominator's prime factors before simplifying the fraction. If a fraction contains a factor like $3$ in the denominator that can be cancelled out by the numerator, you might incorrectly assume the decimal is recurring.

Always reduce the fraction to its simplest form before analyzing the denominator.

Simplify fractions completely before finding prime factors.

Frequently asked questions

Understanding the properties of decimals clarifies how different number sets overlap.

Is every decimal a rational number?

No. Decimals that terminate or have a repeating pattern are rational numbers. Decimals that go on forever without any repeating pattern are irrational numbers instead.

Why do factors of 2 and 5 create terminating decimals?

Our number system is base ten, and the prime factors of $10$ are $2$ and $5$. Any denominator composed entirely of $2$s and $5$s can be scaled to become a power of $10$, which results naturally in a terminating decimal.

Does 0.999... count as a rational number?

Yes. It is a non-terminating, recurring decimal. In fact, $0.999\ldots$ is exactly equal to the integer $1$, which can be written as the fraction $\dfrac{1}{1}$.