Over on mathoverflow, there is a popular CW question titled: Examples of common false beliefs in mathematics. I thought it would be nice to have a parallel question on this site to serve as a reference for false beliefs within less obscure mathematics. That said, it would be good not get bogged down with misconceptions that are generally assumed to be elementary such as: $(x + y)^{2} = x^{2} + y^{2}$.
2026-02-24 17:55:43.1771955743
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False beliefs in mathematics (conceptual errors made despite, or because of, mathematical education)
3.1k Views Asked by Bumbble Comm https://math.techqa.club/user/bumbble-comm/detail At
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Every torsion-free Abelian group is free.
(This only holds for finitely generated Abelian groups.)
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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. $\det (A B; C D) = \det(A)\det(D) - \det(B)\det(C)$.
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The question I've heard on many levels (including the grad level): what is the square root of $a^2$? And everyone says: it's $a$!
In fact it is $|a|$.
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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.
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Both my students and some of my colleagues (!) believe that the graph of a function cannot cross a horizontal asymptote. Obviously this implies that they misunderstand the definition of an asymptote. More worryingly (in my eyes), it also seems to imply that they don't understand why we even care about asymptotes.
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To generalize a few of the answers, for pretty much any function, someone somewhere will make the mistake of treating it as if it is linear in all of its variables. Thus we get: $e^a + e^b = e^{a+b}$, $\sqrt{a + b} = \sqrt{a} + \sqrt{b}$, $a/(b+c) = a/b + a/c$, ...
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A question on this site asks for examples of normed vector spaces, an answer says something to the effect:
The field of rationals , seen as a vector space over itself, together with the norm inherited from $\mathbb{R}$, forms an incomplete normed space.
This is wrong because the definition of a normed vector space requires a vector space over $\mathbb{R}$ or $\mathbb{C}$ from the outset. Besides, the vector space of $\mathbb{Q}$ over $\mathbb{Q}$ is one-dimensional, but finite-dimensional normed spaces are complete.
These are 2 instances which i have seen to happen with my friends. If $A$ and $B$ are 2 matrices, then they believe that $(A+B)^{2}=A^{2}+ 2 \cdot A \cdot B +B^{2}$.
Another mistake is if one i asked to solve this equation, $ \displaystyle\frac{\sqrt{x}}{2}=-1$, people generally square both the sides and do get $x$ as $4$.