Question

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

Answer 1

The correct response that explains how the Closure Property helps prove that the sum of a rational and irrational number is irrational is:

**The sum or difference of a rational and irrational number is a rational number.**

This statement is actually not true; the sum (or difference) of a rational number and an irrational number is always irrational. The Closure Property itself states that when you operate (add, subtract, multiply, divide) two numbers within the same set (like rational numbers), you still get a number within that same set. Since the set of rational numbers is closed under addition, the sum of two rational numbers is rational. However, when a rational number is added to an irrational number, it cannot be expressed as a fraction of integers, thus resulting in an irrational number.

The most accurate summary of the principle is:

The sum (or difference) of a rational number and an irrational number is always irrational.