Question:
The sum of a rational number and an irrational number is which type of number?
The sum of a rational number and an irrational number is which type of number?
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Rationals are closed under addition/subtraction; irrationals are not. Adding any nonzero rational to an irrational keeps it irrational.
Updated On: Oct 27, 2025
- An integer
- Irrational number
- Natural number
- None of these
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The Correct Option is B
Solution and Explanation
Step 1: Let \(r\) be rational and \(i\) be irrational.
Assume for contradiction that \(r+i\) is rational.
Step 2: Use closure of rationals under subtraction.
If \(r+i\) were rational, then \(i=(r+i)-r\) would be the difference of two rationals, hence rational—contradiction.
Step 3: Conclude.
Therefore, \(r+i\) must be irrational.
Assume for contradiction that \(r+i\) is rational.
Step 2: Use closure of rationals under subtraction.
If \(r+i\) were rational, then \(i=(r+i)-r\) would be the difference of two rationals, hence rational—contradiction.
Step 3: Conclude.
Therefore, \(r+i\) must be irrational.
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