A number, whose decimal expression is non-terminating, non-recurring, is called :

Question

Accepted Answer

A number whose decimal expression is non-terminating and non-recurring is called an irrational number. Let's delve into why Option 2 is the correct choice and why the other options are incorrect.

1. **Explanation for Option 2 (Irrational):**
   - An irrational number is defined as a number that cannot be expressed as a fraction of two integers, meaning it cannot be written in the form $ \frac{p}{q} $, where $ p $ and $ q $ are integers and $ q \neq 0 $.
   - The decimal representation of an irrational number is non-terminating (it goes on forever) and non-recurring (it does not repeat a pattern). Examples of irrational numbers include $\pi$ and $\sqrt{2}$.
   - Since the question specifies a number with a non-terminating, non-recurring decimal expression, it fits the definition of an irrational number perfectly, making Option 2 the correct answer.

2. **Why Option 1 (Rational) is Incorrect:**
   - A rational number is one that can be expressed as a fraction $ \frac{p}{q} $, where $ p $ and $ q $ are integers and $ q \neq 0 $.
   - The decimal representation of a rational number is either terminating (e.g., 0.5) or recurring (e.g., 0.333...).
   - Since the question describes a non-terminating, non-recurring decimal, it does not fit the characteristics of a rational number, making Option 1 incorrect.

3. **Why Option 3 (Both rational and irrational) is Incorrect:**
   - A number cannot be both rational and irrational simultaneously. These are mutually exclusive categories.
   - A number is either expressible as a fraction (rational) or not (irrational), but not both.
   - Therefore, Option 3 is logically incorrect.

4. **Why Option 4 (Neither rational nor irrational) is Incorrect:**
   - All real numbers are either rational or irrational by definition.
   - There is no category of real numbers that is neither rational nor irrational.
   - Hence, Option 4 is incorrect because it contradicts the fundamental classification of real numbers.

In conclusion, the correct answer is Option 2, as it accurately describes a number with a non-terminating, non-recurring decimal expression as irrational.