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

Question

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: $(x + y)^{2} = x^{2} + y^{2}$.

Answer 1

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.

Answer 2

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. $\det (A B; C D) = \det(A)\det(D) - \det(B)\det(C)$.

Answer 3

Every torsion-free Abelian group is free.

(This only holds for finitely generated Abelian groups.)

Answer 4

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 $|a|$.

Answer 5

I have seen this one time too many $$\frac{a}{b}+\frac{c}{d}=\frac{a+c}{b+d}$$

Answer 6

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: $e^a + e^b = e^{a+b}$, $\sqrt{a + b} = \sqrt{a} + \sqrt{b}$, $a/(b+c) = a/b + a/c$, ...

Answer 7

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 $4$.

Answer 8

Recently, a friend of mine pointed out the following to me:

The open unit disc $D\subset\mathbb{C}$ is not biholomorphic to all of $\mathbb{C}$.

Indeed they are diffeomorphic, but we can easily see that they are not biholomorphic since if there was a biholomorphism $\phi:\mathbb{C}\rightarrow D$, then consider the function $f:D\rightarrow\mathbb{C}$ given by $f(z)=z$. It is obviously holomorphic and non-constant. But then $f\circ\phi$ would be holomorphic, non-constant and bounded, which is a contradiction to Liouville's theorem.

(In fact, the same argument holds to prove that there is no surjective holomorphic function from the whole of $\mathbb{C}$ to any bounded domain.)

Answer 9

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.

Answer 10

The null factor law is as follows: $$ab = 0 \Rightarrow (a = 0) \vee (b = 0).$$ This law applies for real numbers, as well as polynomials which is where the law is most commonly envoked. I have seen far too many instances of the following incorrect generalisation: $$ab = c \Rightarrow (a = c)\vee(b = c).$$

Answer 11

I have seen this many times:

$$a^2 + a^3 = a^5$$

Answer 12

• If $V$ is a finite dimensional normed vector space and $A$ is convex compact, then there is a unique $y\in A$ minimizing the distance from $0$ to $A$ (uniqueness doesn't need to hold, for example, for $\mathbb{R}^n$ with the max-norm).
• If $\exp(A) \exp(B)=\exp(A+B)$, then $A$ and $B$ commute (doesn't hold for some matrices)

Answer 13

An error that I often see with my students, and that I made myself when I was a student :

Let $f$ be a function on $\mathbb R\mapsto \mathbb R$ and $f'$ its derivative.

Then if $\lim_{x\rightarrow\infty} f'(x)=0$ then $f$ is bounded ($\lim_{x\rightarrow\infty} f(x)<\infty$).

Answer 14

Many students struggle to understand why the dream of Freshmen is true in some cases. More precisely, they just cannot accept in a commutative ring of characteristic $p$, the fomula $$ (x+y)^p=x^p+y^p $$ is true. Probably many people believe such an equlity is false for their whole life, because it is false in $\mathbb{R}$.

Answer 15

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.