Certainly! Let's go over the sums and products of rational and irrational numbers. Hereโs 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 ร Rational**:
- Always rational.
- Example: \( 2 \times 3 = 6 \)
2. **Irrational ร Irrational**:
- May be rational or irrational.
- Example: \( \sqrt{2} \times \sqrt{2} = 2 \) (rational); \( \sqrt{2} \times \sqrt{3} \) (irrational).
3. **Rational ร 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 ร Rational = Rational**
- **Irrational ร Irrational = May be Rational or Irrational**
- **Rational ร Irrational = Irrational (if rational is non-zero)**
If you need further examples or deeper explanation, feel free to ask!