TITLE: Examples of fallacies in arithmetic and/or algebra
QUESTION [11 upvotes]: I'm currently preparing for a talk to be delivered to a general audience, consisting primarily of undergraduate students from diverse majors.  My proposed topic would be Examples of fallacies in arithmetic and/or algebra.
So my question would be:

What are some examples of arithmetic/algebraic fallacies that you know of?

One example per answer please.
Let me give my own example, which is one of my personal favorites:

Let $$a = b.$$
  Multiplying both sides by $a$, we get $$a^2 = ab.$$
  Subtracting $b^2$ from both sides, we obtain $$a^2 - b^2 = ab - b^2.$$
  Factoring both sides, we have $$(a + b)(a - b) = b(a - b).$$
  Dividing both sides by $(a - b)$, $$a + b = b.$$
  Substituting $a = b$ and simplifying, $$b + b = b,$$ and $$2b = b.$$
  Dividing both sides by $b$, $$2 = 1.$$

Of course, this fallacious argument breaks down because we divided by $a - b = 0$, since $a = b$ by assumption, and division by zero is not allowed.

REPLY [5 votes]: I am fond of fallacies where a property of members of set of things, and the properties of the limit of that set, are assumed to be equal. But the limit need not be a member of the set, and therefore need have nothing in common with members of the set.
For example, imagine a collection of line segments that goes straight up one unit and straight right one unit. The total length is two.
Now imagine it goes up a half, right a half, up a half, right a half. Again, the length is two. And now we have something that looks like a staircase.
Now up a third, right a third, up a third, right a third, up a third, right a third.  Another staircase. Length is still two.
Obviously as we continue this sequence the line segments more and more closely approximate a line of length root-two going diagonally.  The conclusion we fallaciously reach is that two and its square root are equal.