Question

How does the Closure Property prove that the sum of a rational and irrational number is irrational?(1 point)
Responses
The sum or difference of two rational numbers is a rational number.
The sum or difference of two rational numbers is a rational number.
The sum or difference of a rational and irrational number is a rational number.
The sum or difference of a rational and irrational number is a rational number.
Irrational numbers can be added or subtracted.
Irrational numbers can be added or subtracted.
The sum of an irrational and rational number can’t be predicted.

Answer 1

To prove that the sum of a rational and an irrational number is irrational using the Closure Property, we can analyze the properties of rational and irrational numbers.

1. The sum or difference of two rational numbers is a rational number. (True, but not directly relevant to this question.)
2. The sum or difference of a rational and irrational number is a rational number. (False; this is the statement we want to prove is incorrect.)
3. Irrational numbers can be added or subtracted. (True, but does not help in proving the specific statement.)
4. The sum of an irrational and rational number can’t be predicted. (Vague, but does imply uncertainty regarding the outcome.)

The most relevant to the question is to note that:

- If \( r \) is a rational number and \( i \) is an irrational number, the assumption that \( r + i \) is rational leads to a contradiction. Specifically, if \( r + i \) were rational, then \( i \) could be expressed as the difference of two rational numbers (\( i = (r + i) - r \)), which contradicts the definition of an irrational number.

Therefore, the statement that “the sum or difference of a rational and irrational number is a rational number” is incorrect; it is the opposite that is true: the sum of a rational number and an irrational number is always irrational.

In this case, none of the provided responses correctly capture the necessary conclusion, which can be summarized as follows: "The sum of a rational number and an irrational number is irrational."