Understanding Natural, Rational, and Prime Numbers

The question asks us to evaluate two statements about different types of numbers: natural numbers, rational numbers, and prime numbers. Let's examine each statement carefully.

The two statements are:

• Statement I: All natural numbers are rational numbers.
• Statement II: 1 is the smallest prime number.

Analyzing Statement I: Natural Numbers and Rational Numbers

Statement I claims that all natural numbers belong to the set of rational numbers. To check if this is true, we need to understand the definitions of these two types of numbers.

Natural Numbers: These are the counting numbers. The set of natural numbers, often denoted by $\mathbb{N}$, includes $\{1, 2, 3, 4, ...\}$. Some definitions also include 0, but commonly in number theory, natural numbers start from 1.

Rational Numbers: These are numbers that can be expressed as a fraction $\frac{p}{q}$, where $p$ and $q$ are integers, and $q$ is not equal to 0. The set of rational numbers is denoted by $\mathbb{Q}$. Examples of rational numbers include $\frac{1}{2}$, $-\frac{3}{4}$, $5$ (since $5 = \frac{5}{1}$), and $0.75$ (since $0.75 = \frac{3}{4}$).

Now, let's consider any natural number, say $n$. Can we write this natural number $n$ as a fraction $\frac{p}{q}$ where $p$ and $q$ are integers and $q \neq 0$? Yes, we can always write any natural number $n$ as $\frac{n}{1}$. Here, $p=n$ and $q=1$. Since $n$ is a natural number, it is also an integer. And $q=1$ is also an integer and is not 0.

Therefore, any natural number $n$ can be expressed in the form $\frac{p}{q}$ where $p$ and $q$ are integers and $q \neq 0$. This means every natural number is a rational number.

Based on this analysis, Statement I is correct.

Analyzing Statement II: The Smallest Prime Number

Statement II claims that 1 is the smallest prime number. To evaluate this, we need to recall the definition of a prime number.

Prime Number: A prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself. By definition, a prime number must be a natural number greater than 1.

Let's look at the first few natural numbers and their divisors:

• The number 1: Its only positive divisor is 1.
• The number 2: Its positive divisors are 1 and 2. It is greater than 1, and its only positive divisors are 1 and itself. So, 2 is a prime number.
• The number 3: Its positive divisors are 1 and 3. It is greater than 1, and its only positive divisors are 1 and itself. So, 3 is a prime number.
• The number 4: Its positive divisors are 1, 2, and 4. It has a divisor (2) other than 1 and itself. So, 4 is not a prime number.

According to the definition, a prime number must be greater than 1. The number 1 is not greater than 1. Also, 1 only has one positive divisor (1), whereas prime numbers must have exactly two distinct positive divisors (1 and the number itself). Therefore, 1 is not a prime number.

The smallest natural number that meets the definition of a prime number is 2. So, the smallest prime number is 2.

Based on this analysis, Statement II is incorrect.

Conclusion: Evaluating the Statements

We have determined the correctness of both statements:

• Statement I: All natural numbers are rational numbers. (Correct)
• Statement II: 1 is the smallest prime number. (Incorrect)

We need to choose the option that reflects these findings.

Option 1: Both Statement I and Statement II are correct (Incorrect)

Option 2: Both Statement I and Statement II are incorrect (Incorrect)

Option 3: Statement I is correct but Statement II is incorrect (Correct)

Option 4: Statement I is incorrect but Statement II is correct (Incorrect)

The most appropriate answer is that Statement I is correct, but Statement II is incorrect.

Revision Table: Understanding Number Sets

Additional Information on Number Properties

Understanding different sets of numbers is fundamental in mathematics. Here's a bit more detail:

• The set of natural numbers is a subset of the set of integers. Every natural number is an integer.
• The set of integers is a subset of the set of rational numbers. Every integer can be written as a fraction with a denominator of 1, like $m = \frac{m}{1}$.
• Rational numbers can also be expressed as terminating or repeating decimals. For example, $\frac{1}{2} = 0.5$ (terminating) and $\frac{1}{3} = 0.333...$ (repeating).
• Numbers that are not rational are called irrational numbers (e.g., $\sqrt{2}$, $\pi$). The set of rational and irrational numbers together form the set of real numbers ($\mathbb{R}$).
• Prime numbers are a special subset of natural numbers (excluding 1). The definition of a prime number is strictly "a natural number greater than 1". This is why 1 is not considered a prime number.