Proof that one equals two

Given: a = b

Invalid? Step Justification
Check a = b Given
Check ab = b^2 Multiply both sides by b
Check ab - a^2 = b^2 - a^2 Subtract a^2 from both sides
Check a(b-a) = (b + a)(b - a) Factor both sides
Check a = b + a Divide by (b - a)
Check a = a + a Given a = b
Check a = 2 * a Simple addition
Check 1 = 2 Divide by a

Can you spot the fallacy in this proof?

There is only one illegal step in this proof. Click on the button next to the step you think is invalid. The correctness of your choice will be displayed after a brief waiting period. The correct answer has the shortest waiting period.

Created by Andrew Freed, on April 7, 2002.
Modified by Andrew Freed, on April 15, 2002.

Development of this resource was sponsored by the Penn State Fund for Excellence in Learning and Teaching (FELT), project "Java-based Teaching of Mathematics in Information Sciences and Technology", supervised by Frank Ritter and David Mudgett.