The correct response is: The sum or difference of two rational numbers is a rational number.
To elaborate on the reasoning:
-
Definition of Rational and Irrational Numbers:
- A rational number can be expressed as the fraction of two integers, \( \frac{a}{b} \), where \( b \neq 0 \).
- An irrational number cannot be expressed as a simple fraction, meaning its decimal representation is non-repeating and non-terminating.
-
Closure Property:
- The closure property states that when you combine two numbers within a specific set (like rational numbers), the result will also be within that set. For rational numbers, the sum or difference of two rational numbers is a rational number.
-
Adding Rational and Irrational Numbers:
- Let \( r \) be a rational number and \( i \) be an irrational number. If the sum \( r + i \) were rational, then it could be expressed in the form \( \frac{a}{b} \) (rational).
- Rearranging gives us \( i = (r + i) - r \). Since \( (r + i) \) is assumed rational and \( r \) is rational, their difference must also be rational.
- However, this contradicts the fact that \( i \) is irrational.
Thus, the sum of a rational number and an irrational number must be irrational, confirming that the closure property leads to an understanding that the mix results in an irrational number.