2 = 1
All kinds of falsehoods can be "proven" true if subtle errors in reasoning are allowed to go unnoticed. I get a kick out of debunking these faulty arguments. See if you can find what's wrong with the well-known "proof" that 2 = 1, outlined below: --- Given . a = b Multiply both sides by a . a² = ab Add a² on both sides . a² + a² = a² + ab Simplify the left side . 2a² = a² + ab Subtract 2ab from both sides . 2a² - 2ab = a² + ab - 2ab Simplify right side . 2a² - 2ab = a² - ab Factor 2 out of each term on left side . 2(a² - ab) = a² - ab Divide both sides by a² - ab . 2(a² - ab)/(a² - ab) = (a² - ab)/(a² - ab) Which "proves:" 2 = 1 --- Can you find the error? Come on ... don't look ahead until you at least give it a try! All right, here's the mistake: Look at the eighth step. If a = b (given), then a² - ab = 0. Division by zero is impossible, and therefore not allowed. Breaking the "never divide by zero...