Question

The quotient of a non-zero rational and irrational number will be:

• A. Natural number
• B. Irrational number
• C. Rational number
• D. Whole number

Hint:

The quotient of a non-zero rational number and an irrational number is always irrational.

Answer 1

The Correct Option is B

Step 1: Understanding rational and irrational numbers.
- A rational number is any number that can be expressed as the quotient of two integers, i.e., in the form \( \frac{p}{q} \), where \( p \) and \( q \) are integers and \( q \neq 0 \). - An irrational number is a number that cannot be expressed as a fraction of two integers and has a non-terminating, non-repeating decimal expansion.
Step 2: Analyzing the quotient.
Let the rational number be \( r = \frac{a}{b} \) (where \( a \) and \( b \) are integers, and \( b \neq 0 \)), and let the irrational number be \( i \). The quotient of a non-zero rational number and an irrational number is given by: \[ \frac{r}{i} = \frac{\frac{a}{b}}{i} = \frac{a}{b \cdot i} \] Since the product of a rational number and an irrational number is irrational, the quotient \( \frac{r}{i} \) will also be irrational.
Step 3: Conclusion.
Therefore, the quotient of a non-zero rational number and an irrational number is always an irrational number.