Prove that: 2√3 is an irrational number.

Sure, let's prove that 2√3 is an irrational number.

Step 1: Assume the opposite, that is, 2√3 is a rational number.

Step 2: If 2√3 is rational, it can be expressed as a ratio of two integers. Let's say 2√3 = a/b, where a and b are integers and b ≠ 0.

Step 3: We can rearrange this equation to isolate √3. So, √3 = a/(2b).

Step 4: Now, we know that √3 is an irrational number. This is a well-known fact and can be proven separately.

Step 5: But we have just expressed √3 as a ratio of two integers (a and 2b), which contradicts the fact that √3 is irrational.

Step 6: Therefore, our initial assumption that 2√3 is rational must be incorrect.

So, 2√3 is an irrational number.