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

Practice Questions

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

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