Sure! Here’s a quick overview of the sum and product of rational and irrational numbers:
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Sum:
- Rational + Rational = Rational: The sum of two rational numbers is always rational. For example, \( \frac{1}{2} + \frac{1}{3} = \frac{5}{6} \).
- Irrational + Irrational: The sum of two irrational numbers can be either rational or irrational. For example, \( \sqrt{2} + (-\sqrt{2}) = 0\) (rational), while \( \sqrt{2} + \sqrt{3} \) is irrational.
- Rational + Irrational = Irrational: The sum of a rational and an irrational number is always irrational. For example, \( \frac{1}{2} + \sqrt{2}\) is irrational.
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Product:
- Rational × Rational = Rational: The product of two rational numbers is always rational. For example, \( \frac{1}{2} \times \frac{1}{3} = \frac{1}{6} \).
- Irrational × Irrational: The product of two irrational numbers can be either rational or irrational. For example, \( \sqrt{2} \times \sqrt{2} = 2\) (rational), but \( \sqrt{2} \times \sqrt{3} \) is irrational.
- Rational × Irrational = Irrational: The product of a non-zero rational number and an irrational number is always irrational. For example, \( \frac{1}{2} \times \sqrt{2} \) is irrational.
Summary Table:
| Operation | Result | |--------------------------|------------------------| | Rational + Rational | Rational | | Irrational + Irrational | Rational or Irrational | | Rational + Irrational | Irrational | | Rational × Rational | Rational | | Irrational × Irrational | Rational or Irrational | | Rational × Irrational | Irrational |
If you have any specific questions or need more examples, feel free to ask!