Examples of fallacies in arithmetic and/or algebra

I like the following one. It's kinda silly, but still interesting.

We know $1\$=100c$. But then:

$$\begin{align}1\$&=100c\\ &=10c\times 10c\\ &=0.1\$\times 0.1\$\\ &=0.01\$\\ &=1c\end{align}$$

So a dollar is worth just a penny!

$$1 = \sqrt{1} = \sqrt{(-1)(-1)} = \sqrt{-1}\sqrt{-1} = i\cdot i = -1$$

You might want to look at Edward J. Barbeau's books "Mathematical Fallacies, Flaws, and Flimflam" (MAA, 2000) and "More Fallacies, Flaws and Flimflam" (MAA, 2013).