State, in each case, whether the given statements is ture or false.
(i) The sum of two rational numbers is rational.
(ii) The sum of two irrational numbers is irrational.
(iii) The product of two rational numbers is rational.
(iv) The product of two irrational numbers is irrational.
(v) The sum of a rational number and irrational number is irrational.
(vi) The product of a nonzero rational number and an irrational number is a rational number.
(vii)Every real number is rational or irrational.
(viii) Every number is either rational or irrational.
(ix) `pi ` is irrational and `(22)/(7)` is rational.

Answer

To determine whether each statement is true or false, we will analyze them one by one.

### Step-by-Step Solution:

**(i)** The sum of two rational numbers is rational.
- **Explanation:** Let the two rational numbers be \( \frac{p_1}{q_1} \) and \( \frac{p_2}{q_2} \). The sum is given by:
\[ \frac{p_1}{q_1} + \frac{p_2}{q_2} = \frac{p_1 \cdot q_2 + p_2 \cdot q_1}{q_1 \cdot q_2} \]
Since both the numerator and denominator are integers, the result is a rational number.
- **Conclusion:** True.

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**(ii)** The sum of two irrational numbers is irrational.
- **Explanation:** Consider two irrational numbers, such as \( \sqrt{2} \) and \( \sqrt{3} \). Their sum \( \sqrt{2} + \sqrt{3} \) is irrational. However, if we take \( \sqrt{2} \) and \( 2 - \sqrt{2} \), their sum is \( 2 \), which is rational.
- **Conclusion:** False.

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**(iii)** The product of two rational numbers is rational.
- **Explanation:** Let the two rational numbers be \( \frac{p_1}{q_1} \) and \( \frac{p_2}{q_2} \). The product is given by:
\[ \frac{p_1}{q_1} \times \frac{p_2}{q_2} = \frac{p_1 \cdot p_2}{q_1 \cdot q_2} \]
Since both the numerator and denominator are integers, the result is a rational number.
- **Conclusion:** True.

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**(iv)** The product of two irrational numbers is irrational.
- **Explanation:** Consider \( \sqrt{2} \) and \( \sqrt{2} \). Their product is \( 2 \), which is rational. Therefore, the product of two irrational numbers can sometimes be rational.
- **Conclusion:** False.

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**(v)** The sum of a rational number and an irrational number is irrational.
- **Explanation:** Let the rational number be \( \frac{p}{q} \) and the irrational number be \( r \). The sum is:
\[ \frac{p}{q} + r \]
This cannot be expressed as a fraction, hence it remains irrational.
- **Conclusion:** True.

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**(vi)** The product of a nonzero rational number and an irrational number is a rational number.
- **Explanation:** Let the nonzero rational number be \( \frac{p}{q} \) and the irrational number be \( r \). The product is:
\[ \frac{p}{q} \times r \]
This product is irrational, as multiplying a nonzero rational number by an irrational number yields an irrational number.
- **Conclusion:** False.

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**(vii)** Every real number is rational or irrational.
- **Explanation:** By definition, real numbers are classified as either rational (can be expressed as a fraction) or irrational (cannot be expressed as a fraction).
- **Conclusion:** True.

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**(viii)** Every number is either rational or irrational.
- **Explanation:** This statement is essentially the same as the previous one, as it covers all numbers, including real numbers.
- **Conclusion:** True.

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**(ix)** \( \pi \) is irrational and \( \frac{22}{7} \) is rational.
- **Explanation:** It is well-known that \( \pi \) is an irrational number (it cannot be expressed as a fraction), and \( \frac{22}{7} \) is a rational approximation of \( \pi \).
- **Conclusion:** True.

### Summary of Statements:
1. True
2. False
3. True
4. False
5. True
6. False
7. True
8. True
9. True