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

Question

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

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

Irrational numbers can be added or subtracted.

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

Answer 1

The correct response to prove that the sum of a rational and an irrational number is irrational is:

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

However, this statement is actually incorrect. The accurate statement is that the sum (or difference) of a rational and an irrational number is an irrational number.

The closure property concerning rational and irrational numbers indicates that while the sum or difference of two rational numbers is always rational, when you include an irrational number, the result will always be irrational. Thus, when a rational number is added to or subtracted from an irrational number, the result cannot be expressed as a fraction, which means it is irrational.

So, in summary, the correct conclusion is:

The sum of a rational and an irrational number is irrational.