Question 8

The product of a non-zero rational number with an irrational number is always a/an

Solution:

To solve the question, we need to determine the nature of the product of a non-zero rational number and an irrational number.

1. Understanding Rational and Irrational Numbers:
– A rational number is a number that can be expressed as the quotient of two integers (e.g., 1/2, 3, -4).
– An irrational number is a number that cannot be expressed as a simple fraction (e.g., √2, π).

2. Choosing a Non-Zero Rational Number:
– Let’s select a simple non-zero rational number. For example, we can choose $$2$$.

3. Choosing an Irrational Number:
– Now, we need to choose an irrational number. A common example is $$\sqrt{3}$$.

4. Calculating the Product:
– We will multiply the chosen rational number by the irrational number:
$$2\times \sqrt{3}=2\sqrt{3}$$

5. Determining the Nature of the Product:
– Now, we need to determine if $$2\sqrt{3}$$ is rational or irrational.
– Since $$\sqrt{3}$$ is irrational, multiplying it by a non-zero rational number (2 in this case) will still yield an irrational number.

6. Conclusion:
– Therefore, the product of a non-zero rational number and an irrational number is always an irrational number.

Final Answer:
The product of a non-zero rational number with an irrational number is always an irrational number.