Decimal Expansion of Rational Numbers

Decimal Expansion of Rational Numbers: Terminating (2ⁿ5^m denominator), Non-terminating Recurring

This section explores the concept of decimal expansions of rational numbers. Rational numbers can be represented as fractions in the form of $\frac{p}{q}$, where $p$ and $q$ are integers and $q \neq 0$. The decimal expansion of a rational number can either terminate or repeat (recur).

A rational number will have a terminating decimal expansion if, when expressed in simplest form, the denominator has only prime factors of 2 and/or 5. This is because our base-10 number system is perfectly aligned with the prime factors 2 and 5. If the denominator contains any other prime factors (like 3, 7, 11, etc.), the decimal expansion will be non-terminating and recurring.

In simpler terms, if we can rewrite the denominator as $2^n \times 5^m$ where ‘n’ and ‘m’ are non-negative integers, the decimal terminates. If not, it repeats.

Formulae

The following are the key ideas:

• Terminating Decimal: A rational number $\frac{p}{q}$ terminates if and only if the prime factorization of $q$ is of the form $2^n 5^m$, where $n, m \in \mathbb{N} \cup \{0\}$.
• Non-terminating Recurring Decimal: A rational number $\frac{p}{q}$ is non-terminating and recurring if the prime factorization of $q$ contains prime factors other than 2 and 5.
• Converting to decimal To convert a terminating decimal, manipulate the fraction to have a denominator that is a power of 10, ($10^k$) where ‘k’ is the highest power of 2 or 5.

Examples

Example-1: $\frac{7}{20}$

The denominator is 20. Prime factorization of 20 is $2^2 \times 5^1$. Since the denominator only contains the prime factors 2 and 5, the decimal expansion terminates.

Converting: $\frac{7}{20} = \frac{7}{2^2 \times 5} = \frac{7 \times 5}{2^2 \times 5 \times 5} = \frac{35}{100} = 0.35$

Example-2: $\frac{11}{6}$

The denominator is 6. Prime factorization of 6 is $2 \times 3$. Since the denominator contains the prime factor 3 (other than 2 and 5), the decimal expansion is non-terminating and recurring.

Converting: $\frac{11}{6} = 1.8333… = 1.8\overline{3}$

Common mistakes by students

• Not simplifying the fraction first: Students often forget to simplify the fraction to its lowest terms before checking the denominator. For example, $\frac{4}{10}$ might look non-terminating until simplified to $\frac{2}{5}$.
• Incorrectly identifying prime factors: Sometimes students miscalculate the prime factorization.
• Forgetting to add zeros: When multiplying the numerator by a number to create the power of 10, students may forget to add zeros to the numerator when converting the result to a decimal.

Real Life Application

Understanding decimal expansions is crucial for everyday calculations involving money, measurements, and proportions. For example:

• Currency Conversion: When converting between currencies, decimal expansions are used to calculate and represent the exchange rates.
• Measurement conversions: Representing fractions of inches, or meters.
• Financial planning Calculating simple interest.

Fun Fact

The concept of repeating decimals has fascinated mathematicians for centuries! The length of the repeating block in a non-terminating decimal can sometimes be predicted by the denominator. For instance, when the denominator is a prime number other than 2 or 5, the maximum length of the repeating block is always one less than that prime number.

Recommended YouTube Videos for Deeper Understanding

Q.1 Which of the following rational numbers has a terminating decimal expansion?

Check Solution

Ans: D

A rational number has a terminating decimal expansion if its denominator can be expressed in the form $2^n 5^m$, where n and m are non-negative integers. For the given options: A) $30 = 2 \times 3 \times 5$ (Non-terminating) B) $28 = 2^2 \times 7$ (Non-terminating) C) $6 = 2 \times 3$ (Non-terminating) D) $80 = 2^4 \times 5$ (Terminating)

Q.2 The decimal expansion of $\frac{23}{2^2 \times 5}$ will terminate after how many decimal places?

Check Solution

Ans: B

We need to make the denominator a power of 10. $\frac{23}{2^2 \times 5} = \frac{23 \times 5}{2^2 \times 5^2} = \frac{115}{100} = 1.15$. The decimal terminates after two places.

Q.3 Which of the following fractions will result in a non-terminating recurring decimal expansion?

Check Solution

Ans: D

A fraction has a non-terminating, recurring decimal expansion if its denominator, when simplified, has prime factors other than 2 and 5. A) $125 = 5^3$ (Terminating) B) $16 = 2^4$ (Terminating) C) $2^3 \times 5^2$ (Terminating) D) $30 = 2 \times 3 \times 5$. The factor 3 makes it non-terminating recurring.

Q.4 What is the decimal expansion of $\frac{7}{20}$?

Check Solution

Ans: A

$\frac{7}{20} = \frac{7}{2^2 \times 5} = \frac{7 \times 5}{2^2 \times 5^2} = \frac{35}{100} = 0.35$.

Q.5 If $\frac{a}{b}$ is a rational number and the prime factorization of $b$ is $2^x \times 5^y$, where $x$ and $y$ are non-negative integers, then what type of decimal expansion does $\frac{a}{b}$ have?

Check Solution

Ans: B

If the denominator only contains prime factors 2 and/or 5, then the decimal expansion terminates.

Next Topic: Zeros of a Polynomial: Geometrical Meaning