PREVIOUSLY ASKED IN:
WBPSC Miscellaneous Preliminary 2023
Answer
Sum of two irrational numbers is not always an irrational.
Explanation
The sum of two irrational numbers can be either rational or irrational, meaning it is not *always* irrational. For example, \\( (2 + \\sqrt{5}) \\) and \\( (2 - \\sqrt{5}) \\) are both irrational numbers. However, their sum is \\( 4 \\), which is a rational number. On the other hand, the sum of \\( \\sqrt{2} \\) and \\( \\sqrt{3} \\) remains irrational. Therefore, option C is the most accurate statement.
Key Points
- > Irrational numbers have non-terminating and non-repeating decimal expansions.
- > Examples: \\( \\pi \\), \\( \\sqrt{2} \\), Euler's number \\( e \\).
- > The sum or difference of two irrationals can be rational.
- > The product of two irrationals can be rational (e.g., \\( \\sqrt{3} \\times \\sqrt{3} = 3 \\)).
- > Rational + Irrational = Always Irrational.
- > Zero (0) is a rational integer.
- > The set of rational and irrational numbers make up the Real Numbers.
Additional Information
Rules of Real Numbers
| Operation | Result | Example |
|---|---|---|
| Rational + Rational | Always Rational | \\( 2 + 3 = 5 \\) |
| Rational + Irrational | Always Irrational | \\( 2 + \\sqrt{3} \\) |
| Irrational + Irrational | Rational or Irrational | \\( \\sqrt{5} + ( -\\sqrt{5} ) = 0 \\) |
| Irrational \\( \\times \\) Irrational | Rational or Irrational | \\( \\sqrt{2} \\times \\sqrt{8} = \\sqrt{16} = 4 \\) |
Memory Tips
- Conjugate Pairs: Whenever you add or multiply irrational conjugates like \\( (a + \\sqrt{b}) \\) and \\( (a - \\sqrt{b}) \\), the irrational part cancels out, leaving a rational number.