To prove that the product of 2 and an irrational number is irrational, let's take an irrational number, like the square root of 2 (โ2). If we multiply 2 by โ2, we get 2โ2.
Now, we need to show that 2โ2 is irrational. If we assume that 2โ2 is rational, this would mean that it can be expressed as a fraction of two integers (a/b). However, if we divide both sides of the equation by 2, we would have โ2 = (a/b) / 2 = a/(2b), which implies that โ2 is rational. But we know that โ2 is irrational. Therefore, our initial assumption must be false, so 2โ2 is irrational.
Thus, the product of 2 and an irrational number (like โ2) results in another irrational number, which is 2โ2.