How does the Closure Property prove that the sum of a rational and irrational number is irrational?(1 point)

Question

Responses

The sum of an irrational and rational number can’t be predicted.
The sum of an irrational and rational number can’t be predicted.

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.

Answer 1

The correct response to how the Closure Property proves that the sum of a rational and an irrational number is irrational is:

The sum or difference of two rational numbers is a rational number.

Here's a brief explanation:

1. By definition, a rational number can be expressed as a fraction of integers (a/b, where a and b are integers and b ≠ 0).
2. If you add a rational number (let's call it r) to an irrational number (let's call it i), the result (r + i) cannot be expressed as a fraction of two integers, as that would imply that the irrational part somehow 'cancels out' and produces a rational number, which is not possible.
3. Thus, the sum of a rational and an irrational number remains irrational.

The closure property applies to rational numbers specifically (i.e., the sum of two rational numbers is also rational), but in this case, it shows that when a rational number is added to an irrational number, it cannot produce a rational number.