To determine whether \(5 - \sqrt{3}\) is irrational, we can use a similar approach to the one in the proof you provided.

### Step 1: Assume \(5 - \sqrt{3}\) is Rational
Assume, for the sake of contradiction, that \(5 - \sqrt{3}\) is a rational number. This means we can write:
\[
5 - \sqrt{3} = a
\]
where \(a\) is a rational number.

### Step 2: Isolate the Square Root
Rearrange the equation to isolate \(\sqrt{3}\):
\[
\sqrt{3} = 5 - a
\]

### Step 3: Analyze the Right-Hand Side
Since \(a\) is rational, \(5 - a\) is also rational (because the difference of two rational numbers is rational).

### Step 4: Contradiction
However, \(\sqrt{3}\) is known to be an irrational number. This leads to a contradiction because we have just stated that \(\sqrt{3}\) equals a rational number, which is impossible.

### Step 5: Conclusion
The assumption that \(5 - \sqrt{3}\) is rational must be false. Therefore, \(5 - \sqrt{3}\) is indeed an irrational number.

This proof uses the method of contradiction, similar to the earlier proof with \(\sqrt{2} + \sqrt{3}\).

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Would you like more details or have any questions?

Here are 8 related questions for further exploration:
1. How can we extend this proof to show that \(5 + \sqrt{3}\) is irrational?
2. What happens if we assume \(\sqrt{3} + \sqrt{2}\) is rational?
3. Can a sum of a rational number and an irrational number ever be rational?
4. Why is the assumption that \(5 - \sqrt{3}\) is rational incorrect?
5. What are some common methods for proving a number is irrational?
6. Can the product of a rational and an irrational number be rational?
7. How does this proof relate to the closure properties of rational and irrational numbers?
8. Are there any other irrational numbers that are similar to \(\sqrt{3}\)?

**Tip:** Always remember that the square root of a non-perfect square is irrational, which is a key fact often used in these types of proofs.

Explore a detailed proof that 5 - √3 is an irrational number using the method of contradiction. Understand the concept of irrational numbers and learn how to apply proof techniques in mathematics. Suitable for advanced high school students and anyone interested in number theory.

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