False beliefs in mathematics (conceptual errors made despite, or because of, mathematical education)

Many well-educated people believe that a p-value is the probability that a study conclusion is wrong. For example, they believe that if you get a 0.05 p-value, there's a 95% chance that your conclusion is correct. In fact there may be less than a 50% chance that the conclusion is correct, depending on the context. Read more here.

I recently caught myself thinking that the formula for the determinant of a 2-by-2 matrix also works for a block matrix, i.e. $\det (A B; C D) = \det(A)\det(D) - \det(B)\det(C)$.

Every torsion-free Abelian group is free.

(This only holds for finitely generated Abelian groups.)