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Q6:
Look at several examples of rational numbers in the form $\frac{p}{q}$ ($q \neq 0$), where $p$ and $q$ are integers with no common factors other than 1 and having terminating decimal representations (expansions). Can you guess what property $q$ must satisfy?
Solution :
Step 1: Theoretical Foundation & Initial Setup
We are tasked with analyzing rational numbers of the form $\frac{p}{q}$, where:
- $p$ and $q$ are integers.
- $q \neq 0$.
- $\gcd(p, q) = 1$ [This indicates that $p$ and $q$ are coprime; they share no common factors other than 1, meaning the fraction is in its simplest form].
- The decimal representation of $\frac{p}{q}$ is terminating (it has a finite number of digits after the decimal point).
Our objective is to deduce the specific mathematical property that the denominator $q$ must satisfy to guarantee a terminating decimal expansion.
Step 2: Empirical Observation & Data Collection
To identify the pattern, we will generate several examples of rational numbers that yield terminating decimals, ensure they are in their simplest form, and examine the prime factorization of their denominators.
| Terminating Decimal | Fraction (Base $10^k$) | Simplest Form ($\frac{p}{q}$) | Denominator ($q$) | Prime Factorization of $q$ |
|---|---|---|---|---|
| $0.5$ | $\frac{5}{10}$ | $\frac{1}{2}$ | $2$ | $2^1$ |
| $0.25$ | $\frac{25}{100}$ | $\frac{1}{4}$ | $4$ | $2^2$ |
| $0.8$ | $\frac{8}{10}$ | $\frac{4}{5}$ | $5$ | $5^1$ |
| $0.375$ | $\frac{375}{1000}$ | $\frac{3}{8}$ | $8$ | $2^3$ |
| $0.44$ | $\frac{44}{100}$ | $\frac{11}{25}$ | $25$ | $5^2$ |
| $0.15$ | $\frac{15}{100}$ | $\frac{3}{20}$ | $20$ | $2^2 \times 5^1$ |
| $0.016$ | $\frac{16}{1000}$ | $\frac{2}{125}$ | $125$ | $5^3$ |
Step 3: Analyzing the Denominators ($q$)
By observing the "Prime Factorization of $q$" column, a distinct pattern emerges. The prime factors of every denominator $q$ in these terminating examples consist exclusively of the numbers $2$ and $5$.
To verify this exclusivity, let us briefly look at fractions where the denominator contains prime factors other than $2$ or $5$:
- $\frac{1}{3} = 0.3333...$ [Non-terminating, repeating; prime factor is $3$]
- $\frac{1}{7} = 0.142857...$ [Non-terminating, repeating; prime factor is $7$]
- $\frac{5}{6} = \frac{5}{2 \times 3} = 0.8333...$ [Non-terminating, repeating; prime factors are $2$ and $3$]
Step 4: Theoretical Justification
Any terminating decimal can be expressed as a fraction where the denominator is a power of $10$ (i.e., $10^k$ for some integer $k \geq 1$).
[Per the Fundamental Theorem of Arithmetic], the prime factorization of $10$ is strictly $2 \times 5$. Therefore, the prime factorization of any power of $10$ is:
$10^k = (2 \times 5)^k = 2^k \times 5^k$
When a fraction $\frac{x}{10^k}$ is reduced to its simplest form $\frac{p}{q}$ by canceling common factors, the only prime factors that can possibly remain in the denominator $q$ are $2$ and $5$. No new prime factors can be introduced during simplification.
Step 5: Visualizing the Logical Flow
The following diagram illustrates the structural transformation from a terminating decimal to its simplest fractional form, proving why $q$ is restricted to powers of $2$ and $5$.
Step 6: Formulating the General Property
Based on the empirical data and theoretical proof, for any rational number $\frac{p}{q}$ (where $p$ and $q$ are coprime) to have a terminating decimal expansion, the prime factorization of the denominator $q$ must not contain any prime numbers other than $2$ and $5$.
Mathematically, this is expressed as:
$q = 2^m \times 5^n$
where $m$ and $n$ are non-negative integers ($m, n \geq 0$).
Final Solution: For a rational number $\frac{p}{q}$ (where $p$ and $q$ are coprime integers) to have a terminating decimal representation, the prime factorization of the denominator $q$ must consist exclusively of powers of $2$ and/or $5$. That is, $q$ must satisfy the property $q = 2^m \times 5^n$, where $m$ and $n$ are non-negative integers.
More Questions from Class 9 Mathematics Number Systems EXERCISE 1.3
- Q1(i): Write the following in decimal form and say what kind of decimal expansion each has : (i) $\frac{36}{100}$
- Q1(ii): Write the following in decimal form and say what kind of decimal expansion each has : (ii) $\frac{1}{11}$
- Q1(iii): Write the following in decimal form and say what kind of decimal expansion each has : (iii) $4\frac{1}{8}$
- Q1(iv): Write the following in decimal form and say what kind of decimal expansion each has : (iv) $\frac{3}{13}$
- Q1(v): Write the following in decimal form and say what kind of decimal expansion each has : (v) $\frac{2}{11}$
- Q1(vi): Write the following in decimal form and say what kind of decimal expansion each has : (vi) $\frac{329}{400}$
- Q2: You know that $\frac{1}{7} = 0.\overline{142857}$. Can you predict what the decimal expansions of $\frac{2}{7}, \frac{3}{7}, \frac{4}{7}, \frac{5}{7}, \frac{6}{7}$ are, without actually doing the long division? If so, how? [Hint : Study the remainders while finding the value of $\frac{1}{7}$ carefully.]
- Q3(i): Express the following in the form $\frac{p}{q}$, where $p$ and $q$ are integers and $q \neq 0$. (i) $0.\overline{6}$
- Q3(ii): Express the following in the form $\frac{p}{q}$, where $p$ and $q$ are integers and $q \neq 0$. (ii) $0.4\overline{7}$
- Q3(iii): Express the following in the form $\frac{p}{q}$, where $p$ and $q$ are integers and $q \neq 0$. (iii) $0.\overline{001}$
- Q4: Express $0.99999 ....$ in the form $\frac{p}{q}$. Are you surprised by your answer? With your teacher and classmates discuss why the answer makes sense.
- Q5: What can the maximum number of digits be in the repeating block of digits in the decimal expansion of $\frac{1}{17}$? Perform the division to check your answer.
- Q7: Write three numbers whose decimal expansions are non-terminating non-recurring.
- Q8: Find three different irrational numbers between the rational numbers $\frac{5}{7}$ and $\frac{9}{11}$.
- Q9(i): Classify the following numbers as rational or irrational : (i) $\sqrt{23}$
- Q9(ii): Classify the following numbers as rational or irrational : (ii) $\sqrt{225}$
- Q9(iii): Classify the following numbers as rational or irrational : (iii) $0.3796$
- Q9(iv): Classify the following numbers as rational or irrational : (iv) $7.478478...$
- Q9(v): Classify the following numbers as rational or irrational : (v) $1.101001000100001...$
CBSE Solutions for Class 9 Mathematics Number Systems
Chapters in CBSE - Class 9 Mathematics
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