Decimal Expansion of Rational Numbers for CBSE Class 10 Mathematics

In real analysis, every real number can be classified as either a rational number or an irrational number. A fundamental question in arithmetic is: When does a rational number $\frac{p}{q}$ have a terminating decimal expansion, and when does it produce an infinitely repeating (recurring) decimal?

In CBSE Class 10 Mathematics, we learn an elegant criterion based on the prime factorisation of the denominator that allows us to determine the exact decimal nature of any rational fraction without performing tedious long division.

What You Will Learn

• Classification of decimal expansions (Terminating, Non-Terminating Recurring, Non-Terminating Non-Recurring)
• The Fundamental Theorem on Terminating Decimal Expansions
• The essential condition on the denominator: $q = 2^m \times 5^n$
• Determining the number of decimal places after which the expansion terminates
• Converting rational fractions into decimals without actual division
• Solved CBSE board exam problems and common pitfalls

1. Classification of Real Decimal Expansions

Every real number corresponds to a unique point on the number line and can be written as a decimal.

                  Real Numbers
                  /          \
        Rational (p/q)        Irrational
          /         \             |
     Terminating   Non-Terminating  Non-Terminating
                   & Repeating     & Non-Repeating

1. Terminating Decimals: The digits stop after a finite number of decimal places (e.g., $0.75$, $0.125$, $3.4$). These are always rational numbers.
2. Non-Terminating Repeating (Recurring) Decimals: The digits never end, but a digit or block of digits repeats infinitely (e.g., $0.333\dots = 0.\bar{3}$, $0.142857142857\dots = 0.\overline{142857}$). These are also rational numbers.
3. Non-Terminating Non-Repeating Decimals: The digits continue indefinitely without any repeating periodic pattern (e.g., $\sqrt{2} = 1.4142135\dots$, $\pi = 3.1415926\dots$). These are irrational numbers.

2. Theorems on Rational Numbers and Decimal Expansions

Theorem 1: Terminating Decimal Criterion

Let $x = \frac{p}{q}$ be a rational number such that the prime factorisation of $q$ is of the form: $q = 2^m \times 5^n$ where $m$ and $n$ are non-negative integers (i.e., $m, n \in \{0, 1, 2, 3, \dots\}$). Then, $x$ has a terminating decimal expansion.

Theorem 2: Non-Terminating Repeating Criterion

Let $x = \frac{p}{q}$ be a rational number where $p$ and $q$ are co-prime (in simplest form). If the prime factorisation of $q$ is not of the form $2^m \times 5^n$ (meaning it contains at least one prime factor other than $2$ or $5$, such as $3, 7, 11,$ etc.), then $x$ has a non-terminating repeating (recurring) decimal expansion.

Important: <u>Before checking the denominator $q$, you MUST reduce the fraction $\frac{p}{q}$ to its simplest form by cancelling out all common factors so that $\text{HCF}(p, q) = 1$. Failing to simplify first is the most common student error!</u>

3. How Many Places Does the Decimal Terminate After?

If $x = \frac{p}{q}$ is in simplest form and $q = 2^m \times 5^n$, then: $\text{Number of decimal places after which expansion terminates} = \max(m, n)$

Why Does This Rule Work?

Because our number system is base-$10$, and $10 = 2 \times 5$. To turn the denominator into a pure power of $10$, we multiply the numerator and denominator by appropriate powers of $2$ or $5$ until the exponents of $2$ and $5$ are equal to $\max(m, n)$.

4. Solved Board Exam Questions

Solved Example 1: Testing and Finding Decimal Expansion

Problem: Without actual division, determine whether $\frac{13}{3125}$ has a terminating or non-terminating repeating decimal. If terminating, find its decimal expansion.

Solution:

• Step 1 (Check co-primality): $13$ is a prime number and does not divide $3125$. Thus, $\text{HCF}(13, 3125) = 1$.
• Step 2 (Prime factorise denominator): $3125 = 5 \times 5 \times 5 \times 5 \times 5 = 5^5 = 2^0 \times 5^5$
• Step 3 (Apply theorem): The denominator is of the form $2^m \times 5^n$ with $m = 0$ and $n = 5$. Therefore, $\frac{13}{3125}$ has a terminating decimal expansion.
• Step 4 (Determine number of decimal places): $\max(m, n) = \max(0, 5) = 5 \text{ decimal places}$
• Step 5 (Compute decimal expansion without division): Make the powers of $2$ and $5$ equal by multiplying numerator and denominator by $2^5$: $\frac{13}{3125} = \frac{13}{5^5} = \frac{13 \times 2^5}{5^5 \times 2^5} = \frac{13 \times 32}{(5 \times 2)^5} = \frac{416}{10^5} = \frac{416}{100000} = 0.00416$

Solved Example 2: The Simplification Trap (CBSE Classic)

Problem: Determine whether $\frac{6}{15}$ has a terminating or non-terminating decimal expansion.

Solution:

• Wrong Approach: $15 = 3 \times 5$. Since $3$ is present, someone might think it is non-terminating. This is incorrect!
• Correct Approach: First reduce the fraction to simplest form: $\frac{6}{15} = \frac{2 \times 3}{5 \times 3} = \frac{2}{5}$
• In simplest form, the denominator is $q = 5 = 2^0 \times 5^1$.
• Since $q$ is strictly of the form $2^m \times 5^n$, the decimal expansion is terminating.
• Value: $\frac{2}{5} = \frac{2 \times 2}{5 \times 2} = \frac{4}{10} = 0.4$.

Exam Tip: Always cancel common factors between numerator and denominator before factorising the denominator!

Solved Example 3: Non-Terminating Repeating Case

Problem: Check whether $\frac{77}{210}$ has a terminating or non-terminating decimal expansion.

Solution:

1. Reduce to simplest form: $\frac{77}{210} = \frac{7 \times 11}{7 \times 30} = \frac{11}{30}$
2. Prime factorise the denominator $30$: $30 = 2 \times 3 \times 5$
3. The denominator contains the prime factor $3$, which is neither $2$ nor $5$.
4. Hence, $\frac{77}{210}$ has a <u>non-terminating repeating (recurring) decimal expansion</u>.

5. Summary Cheat Sheet

Remember: Non-negative integers include zero ($0, 1, 2, 3, \dots$). Therefore, a denominator with only powers of 2 ($2^m \times 5^0$) or only powers of 5 ($2^0 \times 5^n$) is still terminating!

Common Mistake: Forgetting to convert the denominator to powers of $10$ when asked to find the decimal expansion without division. Never use long division when the question states "without actual division"!