Algebraic manipulation can lead to certain fallacies can give raise to surpising results. Here is a sample of 2 of them taken from the book "one equals zero" by Nitsa Movshovitz-Hadar and John Webb (1998). These paradoxes can be used in class for students to think about.
1) suppose we have the equation x-x^2=1.
Since 0 is not a root of this equation, we can divide both sides by x:
subtracting both sides by x,
But -1 isn't the answer. What went wrong?
2) Given the equation
log(x-1)^2=2 log 3
Is this the only solution? If not? What went wrong in the working above?
Overall, the book proved very interesting for me. There is a variety of paradoxes ranging from algebra to statistics. It also includes calculus and geometry.