Q9(iii):

Classify the following numbers as rational or irrational : (iii) $0.3796$

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.