Sums & Products of Rational & Irrational Numbers Quick Check

Certainly! Let's go over the sums and products of rational and irrational numbers. Here’s a quick reference guide:

### Definitions
- **Rational Number**: A number that can be expressed as the quotient of two integers (e.g., \( \frac{1}{2}, -3, 4.75 \)).
- **Irrational Number**: A number that cannot be expressed as a simple fraction (e.g., \( \sqrt{2}, \pi, e \)).

### Sums
1. **Rational + Rational**:
- Always rational.
- Example: \( 2 + 3 = 5 \)

2. **Irrational + Irrational**:
- May be rational or irrational.
- Example: \( \sqrt{2} + (-\sqrt{2}) = 0 \) (rational); \( \sqrt{2} + \sqrt{3} \) (irrational).

3. **Rational + Irrational**:
- Always irrational.
- Example: \( 2 + \sqrt{3} \) is irrational.

### Products
1. **Rational × Rational**:
- Always rational.
- Example: \( 2 \times 3 = 6 \)

2. **Irrational × Irrational**:
- May be rational or irrational.
- Example: \( \sqrt{2} \times \sqrt{2} = 2 \) (rational); \( \sqrt{2} \times \sqrt{3} \) (irrational).

3. **Rational × Irrational**:
- Always irrational (as long as the rational number is not zero).
- Example: \( 2 \times \sqrt{3} \) is irrational.

### Summary
- **Rational + Rational = Rational**
- **Irrational + Irrational = May be Rational or Irrational**
- **Rational + Irrational = Irrational**
- **Rational × Rational = Rational**
- **Irrational × Irrational = May be Rational or Irrational**
- **Rational × Irrational = Irrational (if rational is non-zero)**

If you need further examples or deeper explanation, feel free to ask!