1. Let *a* and *b* be equal non-zero quantities * a = b*

2. Multiply by *a**a** = ab*²

3. Subtract *b²*

*a*²

*– b*²

*= ab – b*²

4. Factor both sides

(*a – b*)(*a + b*) = *b*(*a – b*)

5. Divide out (*a – b*)*a + b = b*

6. Observing that *a = b**b + b = b*

7. Combine like terms on the left* *2*b = b*

8. Divide by b

2 = 1

9. Add 1 to each side and flip over equation

* *1 + 1 = 3

** Q.E.D. **(Thus we have proved)

How is this proof possible?

The source of the fallacy is the fifth step, where (*a – b*) is divided out.

As *a = b*,* a – b* = 0, and dividing by zero is impossible in mathematics.