To determine which is smaller, $\sqrt{2} - 1$ or $\sqrt{3} - \sqrt{2}$, let's compare them directly.

1. Square each expression to avoid dealing with square roots directly.

   For $\sqrt{2} - 1$: $(\sqrt{2} - 1)^2 = 2 - 2\sqrt{2} + 1 = 3 - 2\sqrt{2}$

   For $\sqrt{3} - \sqrt{2}$: $(\sqrt{3} - \sqrt{2})^2 = 3 - 2\sqrt{6} + 2 = 5 - 2\sqrt{6}$

2. Compare the squared expressions: $3 - 2\sqrt{2} \quad \text{and} \quad 5 - 2\sqrt{6}$

   Since $\sqrt{2} \approx 1.414$ and $\sqrt{6} \approx 2.449$, we can approximate: $2\sqrt{2} \approx 2.828 \quad \text{and} \quad 2\sqrt{6} \approx 4.898$ $3 - 2\sqrt{2} \approx 3 - 2.828 = 0.172$ $5 - 2\sqrt{6} \approx 5 - 4.898 = 0.102$

3. Conclusion: Since $0.102 < 0.172$, $\sqrt{3} - \sqrt{2}$ is smaller than $\sqrt{2} - 1$.

So, $\sqrt{3} - \sqrt{2}$ is smaller.

Would you like further details or have any questions?

Here are 5 related questions you might be interested in:

1. How do you simplify square root expressions involving addition or subtraction?
2. How can you approximate square roots without a calculator?
3. What are other methods to compare expressions involving square roots?
4. How do irrational numbers affect inequalities?
5. Can square roots of non-perfect squares be represented as simple fractions?

Tip: When comparing expressions with square roots, sometimes squaring both sides helps simplify the comparison, but be careful to check for extraneous solutions introduced by squaring.