Question

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

Answer 1

To understand how the closure property applies to proving that the sum of a rational number and an irrational number is irrational, we first need to clarify what rational and irrational numbers are:

- A **rational number** can be expressed as the fraction \(\frac{a}{b}\), where \(a\) and \(b\) are integers and \(b \neq 0\).
- An **irrational number** cannot be expressed as a fraction of integers; it has a non-repeating, non-terminating decimal expansion.

Now, let's analyze the provided statements:

1. **Irrational numbers can be added or subtracted.**
- This statement is true, but it doesn't directly prove anything about the sum of a rational and irrational number.

2. **The sum or difference of a rational and an irrational number is a rational number.**
- This statement is actually **false**. If this were true, it would contradict the definition of irrational numbers. Therefore, it cannot be used in the proof.

3. **The sum or difference of two rational numbers is a rational number.**
- This statement is true due to the closure property of rational numbers, but it does not pertain directly to the sum of a rational and an irrational number.

4. **The sum of an irrational and rational number can’t be predicted.**
- This statement is vague. While it touches on the uncertainty regarding the outcome of adding a rational and an irrational number, it's not a formal proof.

To prove that the sum of a rational number and an irrational number is irrational, we typically use the following reasoning:

- Let \( r \) be a rational number and \( x \) be an irrational number.
- If the sum \( r + x \) were to be rational, we could isolate \( x \) as follows: \( x = (r + x) - r\).
- Since \( r \) is rational, and if the sum \( r + x \) is also rational, then \((r + x) - r\) would also be rational (as the difference of rational numbers is rational).
- This would mean that \( x \) would have to be rational, which contradicts our assumption that \( x \) is irrational.

Thus, the conclusion is that the sum \( r + x \) must be irrational.

In summary, the only statements that would logically lead you toward proving the irrationality of the sum would be the properties of rational and irrational numbers, with 2 being crucially important to note as false in proving the original statement.