The following example uses division by zero to "prove" that 2 = 1, but can be modified to prove that any number equals any other number.1. Let a and b be equal non-zero quantitiesA = B2. Multiply through by aA² = AB3. SubtractA² - B²4. Factor both sides(A - B)(A + B) = B(A - B)5. Divide out (A - B)A + B = B6. Observing thatB + B = B7. Combine like terms on the left2B = B8. Divide by the non-zero b2 = 1The fallacy is in line 5: the progression from line 4 to line 5 involves division by a − b, which is zero since a equals b. Since division by zero is undefined, the argument is invalid. Deriving that the only possible solution for lines 5, 6, and 7, namely that a = b = 0, this flaw is evident again in line 7, where one must divide by b (0) in order to produce the fallacy (not to mention that the only possible solution denies the original premise that a and b are nonzero). A similar invalid proof would be to say that since 2 × 0 = 1 × 0 (which is true), one can divide by zero to obtain 2 = 1. An obvious modification "proves" that any two real numbers are equal.Many variants of this fallacy exist. For instance, it is possible to attempt to "repair" the proof by supposing that a and b have a definite nonzero value to begin with, for instance, at the outset one can suppose that a and b are both equal to one:A = B = 1However, as already noted the step in line 5, when the equation is divided by a − b, is still division by zero. As division by zero is undefined, the argument is invalid.