Let x and y be rational and irrational numbers , respectively. Is x+y necessarily an irrational number ?

Options

A. True
B. False
C. Can not be determined
D. None of these

Explanation

To determine whether the sum of a rational number \( x \) and an irrational number \( y \) is necessarily an irrational number, we can follow these steps:

### Step 1: Define the Numbers
Let \( x \) be a rational number and \( y \) be an irrational number. For example, let \( x = 2 \) (which is rational) and \( y = \sqrt{3} \) (which is irrational).

### Step 2: Calculate the Sum
Now, we calculate the sum \( x + y \):
\[ x + y = 2 + \sqrt{3} \]

### Step 3: Assume the Result is Rational
Assume that \( x + y \) is a rational number. Let's denote this sum as \( a \):
\[ a = 2 + \sqrt{3} \]

### Step 4: Isolate the Irrational Part
To isolate \( \sqrt{3} \), we rearrange the equation:
\[ \sqrt{3} = a - 2 \]

### Step 5: Analyze the Right Side
Since \( a \) is assumed to be rational (as we are assuming \( x + y \) is rational), and \( 2 \) is also rational, the right side \( a - 2 \) must also be rational (the difference of two rational numbers is rational).

### Step 6: Contradiction
However, we have \( \sqrt{3} \) on the left side, which is known to be an irrational number. This leads to a contradiction because a rational number cannot equal an irrational number.

### Conclusion
Therefore, our assumption that \( x + y \) is rational must be false. Hence, we conclude that the sum of a rational number and an irrational number is necessarily an irrational number.

### Final Answer
Thus, \( x + y \) is necessarily an irrational number.

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