$\sqrt{2}$ is a/an

### Concept & Logic The square root of any prime number, or any positive integer that is not a perfect square, cannot be expressed as a finite or repeating decimal. These are the hallmark traits of **irrational numbers**. ### Step-by-Step Solution **Deduction:** 1. Check the integer under the root symbol: $2$. 2. The number $2$ is not a perfect square (like $1, 4, 9, 16$). 3. Because it is not a perfect square, $\sqrt{2}$ cannot be written as a simple fraction $\frac{p}{q}$ consisting of integers. 4. Its actual decimal value is approximately $1.41421356...$ which extends infinitely without repeating. 5. Therefore, it is mathematically classified as an irrational number. ### Exam Strategy & Shortcut Learn this absolute rule: Any radical $\sqrt{x}$ where $x$ is a positive integer but not a perfect square is automatically irrational. Since $2$ isn't a perfect square, pick "irrational" immediately. ### Common Pitfall Confusing irrational numbers with fractional approximations. Some test-takers try to estimate $\sqrt{2}$ as $1.4$ or $\frac{7}{5}$ to simplify math, and then incorrectly label the root itself as a rational number based on that approximation. ### Final Answer Therefore, the correct answer is **irrational number**.