Which of the following is not a rational number ?

Q: Which of the following is not a rational number ?

Option 3, $\sqrt{2}$, is not a rational number. A rational number is defined as any number that can be expressed as the quotient or fraction $\frac{p}{q}$ of two integers, where $p$ and $q$ are integers and $q \neq 0$. 1. **Explanation for Option 3 ($\sqrt{2}$)**: - $\sqrt{2}$ is an irrational number. It cannot be expressed as a fraction of two integers. The decimal representation of $\sqrt{2}$ is non-terminating and non-repeating, approximately 1.41421356..., which further confirms its irrationality. The proof of $\sqrt{2}$ being irrational is a classic example in mathematics, often demonstrated by contradiction, showing that no fraction $\frac{p}{q}$ can exactly equal $\sqrt{2}$. 2. **Explanation for why the other options are incorrect**: - **Option 1 ($\frac{15}{4}$)**: This is a rational number because it is expressed as a fraction of two integers, 15 and 4. The division results in a terminating decimal, 3.75, which is characteristic of rational numbers. - **Option 2 ($\frac{10}{3}$)**: This is also a rational number. It is expressed as a fraction of two integers, 10 and 3. The division results in a repeating decimal, 3.333..., which is another characteristic of rational numbers. - **Option 4 ($\frac{21}{4}$)**: This is a rational number as well, expressed as a fraction of two integers, 21 and 4. The division results in a terminating decimal, 5.25, confirming its rationality. In conclusion, Option 3 ($\sqrt{2}$) is the correct answer as it is the only option that represents an irrational number, while all other options are rational numbers.

Question

Accepted Answer

Option 3, $\sqrt{2}$, is not a rational number. A rational number is defined as any number that can be expressed as the quotient or fraction $\frac{p}{q}$ of two integers, where $p$ and $q$ are integers and $q \neq 0$.

1. **Explanation for Option 3 ($\sqrt{2}$)**:
   - $\sqrt{2}$ is an irrational number. It cannot be expressed as a fraction of two integers. The decimal representation of $\sqrt{2}$ is non-terminating and non-repeating, approximately 1.41421356..., which further confirms its irrationality. The proof of $\sqrt{2}$ being irrational is a classic example in mathematics, often demonstrated by contradiction, showing that no fraction $\frac{p}{q}$ can exactly equal $\sqrt{2}$.

2. **Explanation for why the other options are incorrect**:
   - **Option 1 ($\frac{15}{4}$)**: This is a rational number because it is expressed as a fraction of two integers, 15 and 4. The division results in a terminating decimal, 3.75, which is characteristic of rational numbers.
   - **Option 2 ($\frac{10}{3}$)**: This is also a rational number. It is expressed as a fraction of two integers, 10 and 3. The division results in a repeating decimal, 3.333..., which is another characteristic of rational numbers.
   - **Option 4 ($\frac{21}{4}$)**: This is a rational number as well, expressed as a fraction of two integers, 21 and 4. The division results in a terminating decimal, 5.25, confirming its rationality.

In conclusion, Option 3 ($\sqrt{2}$) is the correct answer as it is the only option that represents an irrational number, while all other options are rational numbers.