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.