The product of a non zero rational and an irrational number is

Options

A. always irrational
B. always rational
C. rational or irrational
D. one

Answer

To determine the product of a non-zero rational number and an irrational number, we can follow these steps:

### Step 1: Define a non-zero rational number
Let’s choose a non-zero rational number. For example, we can take \( r = 3 \).

### Step 2: Define an irrational number
Next, we need to choose an irrational number. A common example of an irrational number is \( \sqrt{2} \).

### Step 3: Calculate the product
Now, we calculate the product of the rational number and the irrational number:
\[ \text{Product} = r \times \text{irrational number} = 3 \times \sqrt{2} \]

### Step 4: Analyze the product
To analyze the product \( 3\sqrt{2} \), we need to determine if it is rational or irrational.

- A rational number can be expressed as a fraction \( \frac{a}{b} \) where \( a \) and \( b \) are integers and \( b \neq 0 \).
- An irrational number cannot be expressed in this form.

Since \( \sqrt{2} \) is irrational, multiplying it by a non-zero rational number (like 3) will always yield an irrational number.

### Conclusion
Thus, the product of a non-zero rational number and an irrational number is always irrational.

### Final Answer
The product of a non-zero rational number and an irrational number is irrational.

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