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Q9(iii):
Classify the following numbers as rational or irrational :
(iii) $0.3796$
Solution :
Initial Setup & Theoretical Foundation
To classify a given real number as rational or irrational, we must analyze its decimal expansion or its ability to be expressed as a fraction. The classification is governed by the following fundamental definitions in real analysis:
- Rational Number ($\mathbb{Q}$): A number that can be expressed in the form $\frac{p}{q}$, where $p$ and $q$ are integers ($\mathbb{Z}$) and $q \neq 0$. In decimal form, rational numbers have expansions that are either terminating or non-terminating but repeating.
- Irrational Number ($\mathbb{I}$): A number that cannot be expressed as a simple fraction. Its decimal expansion is strictly non-terminating and non-repeating.
Step 1: Analyzing the Decimal Expansion
The given number is $0.3796$.
By observing the digits after the decimal point, we can see that the expansion ends exactly after the fourth decimal place (the ten-thousandths place). There are no trailing ellipses ($\dots$) or bar notations ($\overline{3796}$) indicating infinite continuation.
[Per the properties of real numbers, any decimal number that comes to an end after a finite number of digits is classified as a terminating decimal].
Step 2: Conversion to Fractional Form ($\frac{p}{q}$)
To rigorously prove that a terminating decimal is a rational number, we must demonstrate that it can be written in the standard fractional form $\frac{p}{q}$.
Since there are four digits after the decimal point, we multiply and divide the number by $10^4$ ($10000$):
$0.3796 = \frac{0.3796 \times 10000}{10000} = \frac{3796}{10000}$
Here, $p = 3796$ and $q = 10000$. Both $3796$ and $10000$ are integers, and the denominator $10000 \neq 0$. This satisfies the strict definition of a rational number.
Step 3: Simplification and Prime Factorization Analysis (Verification)
While $\frac{3796}{10000}$ is sufficient to prove rationality, reducing the fraction to its lowest terms provides complete mathematical rigor. We find the Greatest Common Divisor (GCD) of $3796$ and $10000$.
- Both numbers are divisible by $4$.
- Numerator: $3796 \div 4 = 949$
- Denominator: $10000 \div 4 = 2500$
Thus, the simplest fractional form is $\frac{949}{2500}$.
[By the Rational Number Theorem, a fraction $\frac{p}{q}$ in its simplest form yields a terminating decimal if and only if the prime factorization of $q$ is of the form $2^n \times 5^m$ for non-negative integers $n, m$].
Checking the denominator: $2500 = 25 \times 100 = 5^2 \times (2^2 \times 5^2) = 2^2 \times 5^4$. Because the prime factors of the denominator consist exclusively of $2$ and $5$, the mathematical theorem perfectly corroborates that $0.3796$ is a terminating, rational number.
Final Solution: The number $0.3796$ has a terminating decimal expansion and can be expressed in the form $\frac{p}{q}$ as $\frac{3796}{10000}$ (or $\frac{949}{2500}$). Therefore, it is a Rational Number.
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.
- 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?
- 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(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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